text:
package org.estatio.module.lease.dom.amendments;

import java.math.BigDecimal;
import java.util.Arrays;
import java.util.List;

import org.joda.time.LocalDate;

import org.estatio.module.lease.dom.InvoicingFrequency;
import org.estatio.module.lease.dom.LeaseItemType;

import lombok.Getter;

public enum LeaseAmendmentTemplate {

    COVID2_BEL(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2020,11,2), // min date because of lease selection
            new BigDecimal("50.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2020,11,2),
            new LocalDate(2020,11,30),
            Arrays.asList(
                    new Tuple<>(null, "BE2054") // DEFAULT
            ),
            new LocalDate(2020,11,2),
            null,
            null,
            null,
            null,
            "-C2",
            new LocalDate(2020,10,1),
            new LocalDate(2021,3,31),
            false,
            "/BEL"),
    COVID_BEL(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,3,18), // min date because of lease selection
            new BigDecimal("50.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2020,3,18),
            new LocalDate(2020,5,10),
            Arrays.asList(
                    new Tuple<>(null, "BE2053") // DEFAULT
            ),
            new LocalDate(2020,7,1),
            null,
            null,
            null,
            null,
            "-A",
            new LocalDate(2020,1,1),
            new LocalDate(2020,12,31),
            false,
            "/BEL"),
    COVID1_BEL(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,3,18), // min date because of lease selection
            new BigDecimal("50.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2020,3,18),
            new LocalDate(2020,5,10),
            Arrays.asList(
                    new Tuple<>(null, "BE2053") // DEFAULT
            ),
            new LocalDate(2020,7,1),
            null,
            null,
            null,
            null,
            "-A1",
            new LocalDate(2020,1,1),
            new LocalDate(2020,12,31),
            false,
            "/BEL"),
    COVID2_FRA_66_PERC(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2020,11,1),
            new BigDecimal("66.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED
            ),
            new LocalDate(2020,11,1),
            new LocalDate(2020,11,30),
            Arrays.asList(
                    new Tuple<>(null, "FR2055") // DEFAULT
            ),
            new LocalDate(2020,11,1),
            null,
            null,
            null,
            null,
            "-66C2",
            new LocalDate(2020,10,1),
            new LocalDate(2020,12,31),
            false,
            "/FRA"),
    COVID2_FRA_100_PERC(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2020,11,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED
            ),
            new LocalDate(2020,11,1),
            new LocalDate(2020,11,30),
            Arrays.asList(
                    new Tuple<>(null, "FR2055") // DEFAULT
            ),
            new LocalDate(2020,11,1),
            null,
            null,
            null,
            null,
            "-100C2",
            new LocalDate(2020,10,1),
            new LocalDate(2020,12,31),
            false,
            "/FRA"),
    COVID2_FRA_FREESTYLE(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2020,11,1),
            new BigDecimal("0.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED
            ),
            new LocalDate(2020,11,1),
            new LocalDate(2020,11,30),
            Arrays.asList(
                    new Tuple<>(null, "FR2056") // DEFAULT
            ),
            new LocalDate(2020,11,1),
            null,
            null,
            null,
            null,
            "-FSC2",
            new LocalDate(2020,10,1),
            new LocalDate(2020,12,31),
            false,
            "/FRA"),
    COVID_FRA_50_PERC(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,3,16), // min date because of lease selection
            new BigDecimal("50.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED
            ),
            new LocalDate(2020,3,16),
            new LocalDate(2020,5,10),
            Arrays.asList(
                    new Tuple<>(null, "FR2054") // DEFAULT
            ),
            new LocalDate(2020,7,1),
            Arrays.asList(
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE, InvoicingFrequency.MONTHLY_IN_ADVANCE),
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ARREARS, InvoicingFrequency.MONTHLY_IN_ARREARS)
            ),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED,
                    LeaseItemType.SERVICE_CHARGE,
                    LeaseItemType.MARKETING
            ),
            new LocalDate(2020,7,1),
            new LocalDate(2020,12,31),
            "-A50",
            new LocalDate(2020,1,1),
            new LocalDate(2020,12,31),
            false,
            "/FRA"),
    COVID_FRA_100_PERC(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,4,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED
            ),
            new LocalDate(2020,4,1),
            new LocalDate(2020,6,30),
            Arrays.asList(
                    new Tuple<>(null, "FR2054") // DEFAULT
            ),
            new LocalDate(2020,7,1),
            Arrays.asList(
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE, InvoicingFrequency.MONTHLY_IN_ADVANCE),
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ARREARS, InvoicingFrequency.MONTHLY_IN_ARREARS)
            ),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED,
                    LeaseItemType.SERVICE_CHARGE,
                    LeaseItemType.MARKETING
            ),
            new LocalDate(2020,7,1),
            new LocalDate(2020,12,31),
            "-A100",
            new LocalDate(2020,1,1),
            new LocalDate(2020,12,31),
            false,
            "/FRA"),
    COVID_ITA_100_PERC_1M(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,7,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2020,7,1),
            new LocalDate(2020,7,31),
            Arrays.asList(
                    new Tuple<>("6001", "6014"),
                    new Tuple<>("6002", "6015"),
                    new Tuple<>("6031", "6014"), // current discount charge on any discount item
                    new Tuple<>("6032", "6015"), // current discount charge on any discount item
                    new Tuple<>(null, "6015")
            ),
            new LocalDate(2020,7,1),
            null,
            null,
            null,
            null,
            "-A1M",
            new LocalDate(2020,4,1), // because of _PLUSM1 / _PLUSM2 calcs
            new LocalDate(2020,12,31),
            false,
            "/ITA"),
    COVID_ITA_100_PERC_2M(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,7,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2020,7,1),
            new LocalDate(2020,8,31),
            Arrays.asList(
                    new Tuple<>("6001", "6014"),
                    new Tuple<>("6002", "6015"),
                    new Tuple<>("6031", "6014"), // current discount charge on any discount item
                    new Tuple<>("6032", "6015"), // current discount charge on any discount item
                    new Tuple<>(null, "6015")
            ),
            new LocalDate(2020,7,1),
            null,
            null,
            null,
            null,
            "-A2M",
            new LocalDate(2020,4,1), // because of _PLUSM1 / _PLUSM2 calcs
            new LocalDate(2020,12,31),
            false,
            "/ITA"),
    COVID2_ITA_100_PERC_1M(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2021,3,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2021,3,1),
            new LocalDate(2021,3,31),
            Arrays.asList(
                    new Tuple<>("6001", "6034"),
                    new Tuple<>("6002", "6035"),
                    new Tuple<>("6031", "6034"), // current discount charge on any discount item
                    new Tuple<>("6032", "6035"), // current discount charge on any discount item
                    new Tuple<>(null, "6035")
            ),
            new LocalDate(2021,1,1),
            null,
            null,
            null,
            null,
            "-C21M",
            new LocalDate(2020,10,1), // because of _PLUSM1 / _PLUSM2 calcs
            new LocalDate(2021,6,30),
            false,
            "/ITA"),
    COVID2_ITA_100_PERC_2M(
            LeaseAmendmentType.COVID_WAVE_2,
            new LocalDate(2021,3,1),
            new BigDecimal("100.00"),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT
            ),
            new LocalDate(2021,3,1),
            new LocalDate(2021,4,30),
            Arrays.asList(
                    new Tuple<>("6001", "6034"),
                    new Tuple<>("6002", "6035"),
                    new Tuple<>("6031", "6034"), // current discount charge on any discount item
                    new Tuple<>("6032", "6035"), // current discount charge on any discount item
                    new Tuple<>(null, "6035")
            ),
            new LocalDate(2021,1,1),
            null,
            null,
            null,
            null,
            "-C22M",
            new LocalDate(2020,10,1), // because of _PLUSM1 / _PLUSM2 calcs
            new LocalDate(2021,6,30),
            false,
            "/ITA"),
    COVID_ITA_FREQ_CHANGE_ONLY(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,7,1),
            null,
            null,
            null,
            null,
            null,
            null,
            Arrays.asList(
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE, InvoicingFrequency.MONTHLY_IN_ADVANCE),
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ARREARS, InvoicingFrequency.MONTHLY_IN_ADVANCE),
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE_PLUS1M, InvoicingFrequency.MONTHLY_IN_ADVANCE),
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE_PLUS2M, InvoicingFrequency.MONTHLY_IN_ADVANCE)
            ),
            Arrays.asList(
                    LeaseItemType.RENT,
                    LeaseItemType.RENT_DISCOUNT,
                    LeaseItemType.RENT_DISCOUNT_FIXED,
                    LeaseItemType.SERVICE_CHARGE,
                    LeaseItemType.MARKETING
            ),
            new LocalDate(2020,7,1),
            new LocalDate(2020,12,31),
            "-AF",
            new LocalDate(2020,4,1), // because of _PLUSM1 / _PLUSM2 calcs
            new LocalDate(2020,12,31),
            true,
            "/ITA"),
    DEMO_TYPE(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,1,1),
            new BigDecimal("50"),
            Arrays.asList(LeaseItemType.RENT, LeaseItemType.RENT_DISCOUNT, LeaseItemType.RENT_DISCOUNT_FIXED),
            new LocalDate(2020,3,16),
            new LocalDate(2020,5,10),
            Arrays.asList(
                    new Tuple<>(null, "GBR_DISCOUNT")
            ),
            new LocalDate(2020,7,1),
            Arrays.asList(
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE, InvoicingFrequency.MONTHLY_IN_ARREARS)
            ),
            Arrays.asList(LeaseItemType.RENT, LeaseItemType.SERVICE_CHARGE),
            new LocalDate(2020,7,1),
            new LocalDate(2020, 12, 31),
            "-DEM",
            new LocalDate(2020,1,1),
            new LocalDate(2020,12,31),
            false,
            "/GBR"),
    DEMO_TYPE2(
            LeaseAmendmentType.COVID_WAVE_1,
            new LocalDate(2020,7,1),
            new BigDecimal("100"),
            Arrays.asList(LeaseItemType.RENT, LeaseItemType.RENT_DISCOUNT, LeaseItemType.RENT_DISCOUNT_FIXED),
            new LocalDate(2020,7,1),
            new LocalDate(2020,8,31),
            Arrays.asList(
                    new Tuple<>(null, "GBR_DISCOUNT")
            ),
            new LocalDate(2020,7,1),
            Arrays.asList(
                    new Tuple<>(InvoicingFrequency.QUARTERLY_IN_ADVANCE, InvoicingFrequency.MONTHLY_IN_ADVANCE)
            ),
            Arrays.asList(LeaseItemType.RENT, LeaseItemType.SERVICE_CHARGE),
            new LocalDate(2020,7,1),
            new LocalDate(2020, 12, 31),
            "-DEM2",
            new LocalDate(2020,7,1),
            new LocalDate(2020,12,31),
            false,
            "/GBR")
    ;

    @Getter
    private final LeaseAmendmentType leaseAmendmentType;

    @Getter
    private final LocalDate amendmentStartDate;

    @Getter
    private final BigDecimal discountPercentage;

    @Getter
    private final List<LeaseItemType> discountAppliesTo;

    @Getter
    private final LocalDate discountStartDate;

    @Getter
    private final LocalDate discountEndDate;

    @Getter
    private final List<Tuple<String, String>> chargeReferenceForDiscountItem;

    @Getter
    private final LocalDate minimalAmortisationReferenceDate;

    @Getter
    private final List<Tuple<InvoicingFrequency, InvoicingFrequency>> frequencyChanges;

    @Getter
    private final List<LeaseItemType> frequencyChangeAppliesTo;

    @Getter
    private final LocalDate frequencyChangeStartDate;

    @Getter
    private final LocalDate frequencyChangeEndDate;

    @Getter
    private String ref_suffix;

    @Getter
    private final LocalDate previewInvoicingStartDate;

    @Getter
    private final LocalDate previewInvoicingEndDate;

    @Getter
    private Boolean allowsBulkApply;

    @Getter
    private String atPath;

    LeaseAmendmentTemplate(
            final LeaseAmendmentType leaseAmendmentType,
            final LocalDate amendmentStartDate,
            final BigDecimal discountPercentage,
            final List<LeaseItemType> discountAppliesTo,
            final LocalDate discountStartDate,
            final LocalDate discountEndDate,
            final List<Tuple<String, String>> chargeReferenceForDiscountItem,
            final LocalDate minimalAmortisationReferenceDate,
            final List<Tuple<InvoicingFrequency, InvoicingFrequency>> frequencyChanges,
            final List<LeaseItemType> frequencyChangeAppliesTo,
            final LocalDate frequencyChangeStartDate,
            final LocalDate frequencyChangeEndDate,
            final String ref_suffix,
            final LocalDate previewInvoicingStartDate,
            final LocalDate previewInvoicingEndDate,
            final boolean allowsBulkApply,
            final String atpath) {
        this.leaseAmendmentType = leaseAmendmentType;
        this.amendmentStartDate = amendmentStartDate;
        this.discountPercentage = discountPercentage;
        this.discountAppliesTo = discountAppliesTo;
        this.discountStartDate = discountStartDate;
        this.discountEndDate = discountEndDate;
        this.chargeReferenceForDiscountItem = chargeReferenceForDiscountItem;
        this.minimalAmortisationReferenceDate = minimalAmortisationReferenceDate;
        this.frequencyChanges = frequencyChanges;
        this.frequencyChangeAppliesTo = frequencyChangeAppliesTo;
        this.frequencyChangeStartDate = frequencyChangeStartDate;
        this.frequencyChangeEndDate = frequencyChangeEndDate;
        this.ref_suffix = ref_suffix;
        this.previewInvoicingStartDate = previewInvoicingStartDate;
        this.previewInvoicingEndDate = previewInvoicingEndDate;
        this.allowsBulkApply = allowsBulkApply;
        this.atPath = atpath;
    }

    public static class Tuple<X, Y> {
        public final X oldValue;
        public final Y newValue;
        public Tuple(X oldValue, Y newValue) {
            this.oldValue = oldValue;
            this.newValue = newValue;
        }
    }
}


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We can also assist you in evaluating the likely impact of a whole range of investment and divestment decisions. Among our Romanian projects, this has included a number of site evaluations for major investments (shopping malls), a network of showrooms for an international vehicle manufacturer, a complete evaluation and overhaul of a large banking network and studies for a major insurer.
Interestingly, and considering the size of the investments required, these studies are often carried out in parallel with other research – the difference being that our work is entirely data-driven: it is objective and all sources are traceable. This provides enormous confidence for future studies as the results have been demonstrated in a wide variety of situations.
Under the Resources tab, there are a number of Application Notes related to specific industries.


text:
\section{Introduction}

Let $p$ be a prime and  $n > 0$ an integer; these will be fixed throughout and we will always suppress $p$ and
mostly suppress $n$ from the notation. Let $\E =E_n$ denote the Lubin--Tate spectrum associated to the Honda formal group law of height $n$ over $\F_{p^n}$, and let $\Kn=K(n)$ be the corresponding Morava $K$-theory at height $n$ at the prime $p$. As is the usual convention, given any spectrum $X$, we write
\[
\E_\ast X = \pi_\ast L_{\Kn}(\E \wedge X)
\]
where $L_{\Kn}$ denotes $\Kn$-localization.

We are interested in the $\Kn$-local category and, in particular, one very interesting
spectrum therein which arises from comparing two dualities. The first of these duality functors is Spanier--Whitehead duality, sending $X$ to $D_nX=F(X,L_{\Kn}S^0)$. If $X$ is a dualizable
spectrum -- for example if $X$ is a finite spectrum --  then $\E_\ast D_nX \cong \E^{-\ast}X$ and can be
computed by a universal coefficient spectral sequence. The second is Gross--Hopkins duality, sending
$X$ to $I_nX =F(M_nX, I_{\mathbb{Q}/\mathbb{Z}})$, the Brown--Comenetz dual of its monochromatic layer. Specifically, $M_nX$ is the fiber of $L_nX \to L_{n-1}X$ and $I_{\mathbb{Q}/\mathbb{Z}}$ is the spectrum representing the cohomology theory $I_{\mathbb{Q}/\mathbb{Z}}^*(X)=\Hom_{\mathbb{Z}}(\pi_*X, \mathbb{Q}/\mathbb{Z})$.
It is a consequence of the work of Gross and Hopkins that the dual $I_n$ of
the sphere $L_{\Kn}S^0$ is invertible in the $\Kn$-local category and, hence, we have for any spectrum $X$ a natural equivalence
\[
I_nX \simeq D_nX \wedge I_n.
\]

At this point, information about the homotopy type of $I_n$ becomes vital, and one gets a handle on it using that the spectrum $\E$ has an action by the Morava
stabilizer group $\G = \G_n$. Consequently, the graded $\E_\ast$-module $\E_\ast X$ has
a continuous action by $\G$, giving it the structure of a Morava module.

The key to the invertibility of $I_n$ is the calculation of the Morava module $\E_\ast I_n$. The group $\G$ is a semidirect product $\mathbb{S} \rtimes Gal(\mathbb{F}_{p^n}/\F_p)$, where $\mathbb{S}=\mathbb{S}_n$ is the automorphism group of the formal group law of $\Kn$. The group $\mathbb{S}$ can be identified with a subgroup of the general linear group ${\mathrm{Gl}}_n(\W)$, where $\W$ denotes the Witt vectors on the finite field $\F_{p^n}$. The group $\mathbb{S}$ has enough symmetry that the
determinant ${\mathrm{Gl}}_n(\W) \to \W^\times$ restricts to a homomorphism
\[
\det\colon\mathbb{S} \longrightarrow \Z_p^\times,
\]
which can be extended to $\G$ as the composite
\[ \det\colon\G = \mathbb{S} \rtimes Gal(\F_{p^n}/\F_p) \xrightarrow{\det \times \id} \Z_p^\times \times Gal(\F_{p^n}/\F_p) \xrightarrow{proj_1} \Z_p^\times. \]

This gives a $\G$-action on $\Z_p$, and we write the corresponding representation as $\mathbb{Z}_{p}\langle {\det} \rangle$. If $M$ is a Morava module, we can
define a new Morava module by $M\langle {\det} \rangle = M \otimes_{\Z_p} \mathbb{Z}_{p}\langle {\det} \rangle$ with the diagonal $\G$-action. Then
we have by \cite{grosshopkins1} and \cite{StrickGrossHop} an isomorphism of Morava modules
\[
\E_\ast I_n = \E_\ast(S^{n^2-n})\langle \det\rangle.
\]
If the prime is large ($2p > \mathrm{max}\{n^2+1,2n+2\}$) this determines the homotopy type of $I_n$. If the prime
is not large, then we would like a fixed model $S\langle {\det} \rangle$ of an invertible spectrum in the $\Kn$-local category equipped
with an isomorphism
\[
\E_\ast S\langle {\det} \rangle \cong \E_\ast\langle \det\rangle.
\]
Then we have a $\Kn$-local equivalence
\[
I_n \simeq S^{n^2-n} \wedge S\langle {\det} \rangle \wedge P_n,
\]
where $P_n$ is an invertible $\Kn$-local spectrum with $\E_\ast P_n \cong \E_\ast S^0$ as Morava modules. Attention then
turns to identifying $P_n$. In the known cases this comes down to calculating the homotopy groups of $I_nX$ for $X$ a particularly nice type $n$ complex. See \cite{GHM_pic} for analysis of $P_n$ at $n=2=p-1$; the case $n=1=p-1$ was done by \cite{HMS_pic} and also appears in \cite{HS-KRD,GHM_pic}.

The point of this note is to give a construction of a model of $S\langle {\det} \rangle$ valid at all primes $p$ and all $n > 0$.
We actually give two constructions of $S\langle {\det} \rangle$, one using homotopy fixed points, following an idea of Mike Hopkins, and another,
more naive and direct one, following ideas from \cite{GHM_pic}, fixing the typos therein and extending the construction to the prime $2$.

The first model will evidently have the property that $L_{\Kn}(\E^{h\K} \wedge S\langle {\det} \rangle) =
\E^{h\K}$ for all closed subgroups $\K$ in the kernel of the determinant. The key to this construction is to introduce a spectrum $S(1)$ with a continuous $\mathbb{G}$--action, non-equivariantly equivalent to the $p$-complete sphere spectrum $S^0=S^0_p$, and such that smashing with it naturally twists $\mathbb{G}$--actions by the determinant representation. Then we define
\[ S\langle {\det} \rangle = (\E \wedge S(1))^{h\G},\]
the action on the right-hand side being diagonal. The following is our main result.

\begin{repthm}{thm:pretty-damn-cool}
There is a canonical $\G$-equivariant equivalence $f\colon \E \wedge S\langle {\det} \rangle \to \E \wedge S(1)$, where the action of $\G$ on the source is via the action on $\E$, while on the target it is diagonal. This induces an isomorphism of Morava modules $\E_*S\langle {\det} \rangle \cong \E_*{\left<\det \right>}$.
\end{repthm}

If $\K$ is a closed subgroup of $\mathbb{G}$ in the kernel of the determinant, taking $\K$-homotopy fixed points in this equivalence gives the desired result (\cref{cor:invariance})
\[ \E^{h\K} \wedge S\langle {\det} \rangle \simeq (\E \wedge S(1) )^{h\K} \simeq \E^{h\K}.\]

This project gives a chance to revisit and give an encomium on the amazing paper of Devinatz and Hopkins on
fixed point spectra in the $\Kn$-local category \cite{DH}. Distilled down we have the following question: let $X$ be
a spectrum with a continuous action of the Morava stabilizer group $\G$. We can then form the
$\G$-spectrum $Z= \E \wedge X$ with diagonal $\G$-action and discuss the homotopy type of
$Z^{h\G}=(\E \wedge X)^{h\G}$. Note that $\E_\ast Z=\pi_\ast L_{\Kn}(\E \wedge Z)$ has two
$\G$-actions: the Morava module action on $\E$ and the action on $Z$. A consequence of our results is that
if $X$ is dualizable in the $\Kn$-local category, then
\begin{equation}\label{eq:maineq}
\E_\ast\left(Z^{h\G}\right) = \E_\ast (\E \wedge X)^{h\G} \cong \E_\ast X
\end{equation}
and the Morava module action on $\E_\ast (\E \wedge X)^{h\G}$ corresponds to the diagonal action on
\[
\E_*X = \pi_\ast L_{\Kn}(\E \wedge X).
\]
An analogue of this result for arbitrary spectra $X$ with \emph{trivial} $\G$-action was proven by Davis and Torii~\cite{davistorii}. The equivalence \eqref{eq:maineq} is not hard to prove once we have come to terms with the notion of a continuous
$\G$-action. Since we are making a homology calculation we need cosimplicial techniques, and this is exactly what
Devinatz and Hopkins supply.

To end this introduction, we remark that in \cite{craig_imj}, Westerland considers a different variant of the determinant map on $\mathbb{G}$ and defines a corresponding determinant sphere by using a fiber sequence as in \Cref{prop:fiber2}. Presumably that determinant sphere could also be constructed by the methods in our \Cref{sec:TateDet}, but we have not checked the details.

\section{Continuous $\mathbb{G}$ actions and their homotopy fixed points}\label{sec:continuousactions}

As is perhaps apparent from the introduction, we will assume our readership has access to the standard framework
of $\Kn$-local homotopy theory. The usual source for an in-depth study of the technicalities
is Hovey and Strickland \cite{hovstrmemoir} and basic introductions can be found in almost any paper on chromatic homotopy
theory.  We were especially thorough in \cite[\S 2]{BGH}.

Less familiar is the analysis of point-set properties of the action of Morava stabilizer group $\G$ on the spectrum $\E$. We
will need to use an explicit construction of the homotopy fixed points. For our purposes the original definition by
Devinatz and Hopkins~\cite{DH} will do. The reader interested in extensions and variations of the original notion may want to consult the work of Davis and collaborators, notably Behrens and Quick, e.g., \cite{behrensquick, quick2, davisquick}.

We will also not access the full power and structure of equivariant stable
homotopy theory.
Our $G$-spectra will simply be $G$-objects in some suitable category of spectra; when $G$ is profinite, we will also use a simple notion of continuity (see \cref{def:cont-g-spec}).

We start with some algebra. Recall that $\E_\ast = \W\llbracket u_1,\ldots,u_{n-1}\rrbracket[u^{\pm 1}]$ where the power series ring is in
degree zero and the degree of $u$ is $-2$ and let $\mm \subset \E_0$ be the maximal ideal.

\begin{rem}\label{rem:compl-tower}
Before we proceed further, we need to establish some more notation. Using the periodicity results of Hopkins and Smith \cite{nilpotence2}, Hovey and
Strickland produce a sequence of ideals
$J(i) \subseteq \mm \subseteq \E_0$ and finite type $n$ spectra $\MJ{i}$ with the following properties:
\begin{enumerate}

\item $J(i+1) \subseteq J(i)$ and $\bigcap_i J(i) = 0$;

\item $\E_0/J(i)$ is finite;

\item $\E_0(\MJ{i}) \cong \E_0/J(i)$ and there are spectrum maps $q\colon \MJ{i+1} \to \MJ{i}$ realizing the quotient
$\E_0/J(i+1) \to \E_0/J(i)$;

\item There are maps $\eta=\eta_i\colon S^0 \to \MJ{i}$ inducing the quotient map $\E_0 \to \E_0/J(i)$ and $q\eta_{i+1} = \eta_i\colon
S^0 \to \MJ{i}$;

\item If $X$ is a finite type $n$ spectrum, then the map $X \to \holim_i(X \wedge \MJ{i})$ induced by the maps
$\eta$ is an equivalence.

\item  If $X$ is any $L_n$-local spectrum then by \cite{hovstrmemoir} we have $L_{\Kn}X \simeq \holim_i X \wedge \MJ{i}$. In
particular we have $ \E \simeq \holim_i \E \wedge \MJ{i}$.

\end{enumerate}
Most of this is proved in \cite[\S~4]{hovstrmemoir}, and (6) is proved in \cite[Prop.~7.10]{hovstrmemoir}.
Hovey and Strickland also prove that items (1)-(5) characterize the tower $\{\MJ{i}\}$ up to equivalence
in the pro-category of towers under $S^0$. See Proposition 4.22 of \cite{hovstrmemoir}.
Note that the sequence $\{ J(i)\}$ of ideals defines the same topology on $\E_0$ as the $\mm$-adic topology and
that $\mathbb{G}$ acts on $\E_0/J(i)$ through a finite quotient.
\end{rem}

For profinite sets $T=\lim_j T_j$ and $A=\lim_i A_i$, recall that the set of continuous maps from $T$ to $A$ is defined as
\begin{equation*}
\Map^c(T,A) = {\lim}_i \colim_j \Map(T_j,A_i).
\end{equation*}
Let $M$ be a Morava module and always assume $M$ is $\mm$-complete. An important example of the previous construction is the Morava module of continuous maps
\begin{equation*}
\Map^c(\mathbb{G},M) = {\lim}_i \Map^c(\mathbb{G},M/\mm^i) = {\lim}_i \colim_j \Map(\mathbb{G}/U_j,M/\mm^i)
\end{equation*}
where $U_{j+1} \subseteq U_j \subseteq \mathbb{G}$ is a nested sequence of open normal subgroups so that $\cap\  U_j = \{e\}$; then
$\mathbb{G} = \lim_j \mathbb{G}/U_j$.

We now begin to make these constructions topological by giving a definition of a spectrum of continuous maps in the $\Kn$-local category.
\begin{defn}\label{rem:func-defined}
Suppose  $T = \lim_j T_j$ is a profinite set, and
$A \simeq \holim_i A\wedge \MJ{i}$ is a $\Kn$-local spectrum. Define
\[
F_c(T_+,A) = \holim_i \hocolim_j F( {T_j}_+, A\wedge \MJ{i}).
\]
\end{defn}

In applications $T$ will be $\G$ or $\G/\K \times \G^s$ with $s \geq 0$ and $\K \subseteq \G$ a closed subgroup, or $\G=\mathbb{Z}_p^{\times}$.

We now calculate $\pi_\ast F_c(T_+,A)$, at least for some $A$. For later applications,
we will need a slightly more general result about $\pi_\ast F(Z,F_c(T_+,A))$ with $Z$ arbitrary. If $Z$ is any spectrum we may write $Z \simeq \hocolim Z^\alpha$ for some filtered collection of finite spectra. If
$A \simeq \holim_i A\wedge \MJ{i}$ is a $\Kn$-local spectrum, then we have a topology on
$\pi_t F(Z,A) = A^{-t}(Z)$ defined by the open system of neighborhoods of zero given by the kernels of the map
\[
\pi_t F(Z,A) \longrightarrow \pi_tF(Z^\alpha,A \wedge \MJ{i}).
\]
This is the {\it natural topology} of \cite[Section 11]{hovstrmemoir}. The groups $\pi_\ast F(Z,A)$ are complete
in this topology if
\[
\pi_\ast  F(Z,A) \cong \lim_{\alpha,i} \pi_*F(Z^\alpha,A \wedge \MJ{i}).
\]
In applying the following result our main example will be $A=\E\wedge X$ with $X$ dualizable in the $\Kn$-local category.

\begin{lem}\label{lem:ContinuousMaps}
Suppose $Z$ is any spectrum, $T = \lim_j T_j$ is a profinite set, and
$A \simeq \holim_i A\wedge \MJ{i}$ is a $\Kn$-local spectrum. Further suppose $\pi_t (A \wedge \MJ{i})$ is finite
for all $i$ and $t$. We then have an isomorphism
\[
\pi_* F(Z, F_c(T_+, A)) \cong \Map^c(T, A^{-*}Z)
\]
where $A^{-*}Z$ is equipped with the natural topology.
\end{lem}

\begin{proof} Let $Z \simeq \hocolim_{\alpha} Z^\alpha$ be some cellular filtration on $Z$ by finite spectra. Our
finiteness hypothesis on $A$ implies
\[
A^{-*}Z = \pi_*F(Z,A) \cong \lim_{\alpha, i} \pi_* F(Z^\alpha, A\wedge \MJ{i}).
\]
Now we have that
\[
F(Z, F_c(T_+, A)) \simeq \holim_{\alpha} \holim_i F(Z^\alpha, \hocolim_j F({T_j}_+, A\wedge \MJ{i}))
\]
is equivalent to
\[
\holim_{\alpha} \holim_i   \hocolim_j  F(Z^\alpha,F({T_j}_+, A\wedge \MJ{i})),
\]
since $Z^\alpha$ is dualizable. The homotopy groups of
\[
F(Z^\alpha,F({T_j}_+, A\wedge \MJ{i})) \simeq F({T_j}_+, F(Z^\alpha, A\wedge \MJ{i}))
\]
are $\Map(T_j, \pi_* F(Z^\alpha, A\wedge \MJ{i}))$ and the claim follows using the Milnor sequence and our finiteness hypotheses for the vanishing of the ${\lim}^1$ term.
\end{proof}

\begin{rem}\label{rem:func} For a $\Kn$-local spectrum $X \simeq \holim X \wedge \MJ{i}$, we can give
\[F((\G/U_j)_+,X \wedge \MJ{i})\] a left $\G  = \lim_i \G/U_i$ action
by operating on the right on the source. (Note that the subgroups $U_j$ are normal.) This assembles into an action on $F_c(\G_+,X)$. If the homotopy groups $\pi_t (X \wedge \MJ{i})$ are finite,
\Cref{lem:ContinuousMaps} gives an isomorphism of continuous $\G$-modules
\begin{equation}\label{eq:alg-func}
\pi_\ast F_c(\G_+,X) \cong \Map^c (\G,\pi_\ast X)
\end{equation}
where again $\G$ acts on the source.

Writing
$\G^s = \lim (\G/U_i)^s$ we define $F_c(\G_+^s,X)$ for $s \geq 1$ as in \Cref{rem:func-defined}. We have that
\[
F_c(\G_+^s,F_c(\G_+^t,X)) \simeq F_c(\G_+^{s+t},X).
\]
The equation $F_c(\G_+^{s+1},X) \simeq F_c(\G_+,F_c(\G_+^s,X))$ defines an action of $\G$ on $F_c(\G_+^{s+1},X)$
using the right action on the first factor of $\G^{s+1}$.

Evaluation defines a map $\G_+ \wedge F((\G/U_j)_+,X \wedge \MJ{i}) \to X \wedge \MJ{i}$. Here $\G$ is simply regarded
as a set, with no topology. These fit together to give a map
\[
\G_+ \wedge F_c(\G_+,X) \longrightarrow X.
\]
\end{rem}

Recall that if $G$ is a discrete group, a $G$-action on a spectrum $X$ consists of the data of a map of spaces $BG \to B\mathrm{haut}(X)$, where $\mathrm{haut}(X)$ is the space of self-homotopy equivalences of $X$. Unpacking this data, with the bar resolution of $G$ giving a simplicial structure on $BG$, this amounts to
\begin{enumerate}
\item a map $\eta=\eta_X\colon X \to F(G_+,X)$, whose adjoint, explicitly given as the composition
\[
G_+ \wedge X \xrightarrow{G_+\wedge \eta} G_+ \wedge F(G_+,X) \xrightarrow{ev} X,
\]
is our action map $G_+ \wedge X \to X$, such that
\item the resulting diagram
\[ X \to F(G^{\bullet+1}_+, X)\]
is an augmented cosimplicial spectrum.
This diagram starts as
\[\xymatrix{ X \ar[r]^-{\eta} & F(G_+,X) \ar@<1ex>[r]^-{F(m_+,X)} \ar@<-1ex>[r]_-{F(G_+,\eta)} & F(G_+^2, X) \ar[l] \cdots }\]
and continues so that $\eta$ gives the last of the coface maps at each stage, while the rest of the maps come from the simplicial structure on $G^{\bullet+1}$. In particular, $m$ here denotes the multiplication $G\times G\to G$.
\end{enumerate}

Our definition of a continuous $\G$--spectrum will mimic this one; it is for this paper only and is not meant to replace any of the more sophisticated definitions
provided by others; for example, in work of Davis and Quick (e.g., \cite{davis_thesispaper, davisquick}).
\begin{defn}\label{def:cont-g-spec}
Let $X$ be a $\Kn$-local spectrum. The data of a \emph{continuous} $\G$--action on $X$ consists of:
\begin{enumerate}
\item\label{item:act} A map $\eta = \eta_X\colon X \to F_c(\G_+,X)$, and
\item\label{item:augm} An extension of $\eta$ into an augmented cosimplicial diagram
\begin{equation}\label{eq:cosim-g}
X \longrightarrow F_c(\G^{\bullet + 1}_+,X),
\end{equation}
in which the coface maps come from the simplicial structure on $\G^{\bullet+1}$ together with $\eta$.
\end{enumerate}
If $X$ is a continuous $\G$--spectrum and $\K \subseteq \G$ is a closed subgroup, we define $F_c(\mathbb{G}_+,X)^\K =
F_c(\G/\K_+,X)$ and
\begin{align}\label{eq:fixed-pnts}
X^{h\K} &= \holim_\Delta F_c(\G_+^{\bullet + 1},X)^\K\\
& \simeq \holim_\Delta F_c(\G/\K_+ \times \G_+^{\bullet},X).\nonumber
\end{align}

When presented with a cosimplicial $\G$-spectrum of the type \eqref{eq:cosim-g} we will say we have
an augmented cosimplicial $\G$-spectrum so that the augmentation refines the action map, as in Part \eqref{item:act}. In practice we may simply produce a cosimplicial diagram as in Part \eqref{item:augm}, checking that the coface maps are as required.

A map of continuous $\mathbb{G}$--spectra consists of a map of the respective augmented cosimplicial diagrams.
\end{defn}

\begin{rem}\label{rem:logic}
Composing with the natural map $F_c(\mathbb{G}_+^{\bullet+1},X) \to F(\mathbb{G}_+^{\bullet+1},X) $, we get that a continuous $\mathbb{G}$--spectrum is in particular a $\mathbb{G}$--spectrum in the discrete sense described above. Often the logic is flipped: one starts with a discrete action, and attempts to lift it to a continuous one, i.e., one tries to find the dashed lifts
\[\xymatrix{
& F_c(\G_+,X) \ar[d] && & F_c(\G_+^{\bullet+1},X) \ar[d]\\
X \ar[r] \ar@{-->}[ru] & F(\G_+,X), && X \ar[r] \ar@{-->}[ru]  & F(\G_+^{\bullet+1},X),
} \]
where the horizontal maps are already given. When such lifts exist, we say that the given discrete action is continuous.

This is what Devinatz--Hopkins \cite{DH} accomplish, where the discrete $\mathbb{G}$--action on $\E$ was already given by the Goerss--Hopkins--Miller theorem. We discuss this example further in \cref{rem:unpack-DH} below.
\end{rem}

\begin{example}\label{ex:trivial}
A much easier example of \cref{rem:logic} is the following: For any $\Kn$-local spectrum $X$, the trivial action of $\mathbb{G}$ on $X$ is continuous. Here we start with $\eta\colon X\to F(\mathbb{G}_+, X) $ adjoint to the ``projection" map $\mathbb{G}_+ \wedge X \to X$.
\end{example}

\begin{rem}\label{rem:HFPSS} Suppose further that $\K \subseteq \mathbb{G}$ is a closed subgroup and
that $X \simeq \holim_i X\wedge \MJ{i}$ is a $\Kn$-local spectrum such that $\pi_t (X \wedge \MJ{i})$ is finite
for all $i$ and $t$. Using \Cref{lem:ContinuousMaps}, one sees that these definitions are designed
so that the Bousfield--Kan spectral sequence associated to \eqref{eq:fixed-pnts} is the homotopy fixed point spectral sequence
\[
E_2^{s,t} \cong H_c^s(\K,\pi_tX) \Longrightarrow \pi_{t-s} X^{h\K}
\]
with $E_2$-term given by the continuous group cohomology.
\end{rem}

\begin{rem}\label{rem:otherG}
There is an obvious generalization of this definition to other settings, for example the group may be any profinite group. Likewise, the spectrum $X$ may live in another category where analogues of the generalized Moore spectra $\MJ{i}$ play a similar role. For example $X$ may be a $p$-complete spectrum, so $X \simeq \holim_i X \wedge S/p^i$. While we will in effect construct a continuous $p$-complete $\mathbb{Z}_p^\times$ spectrum in this sense in \Cref{sec:TateDet}, we refrain from setting up a general theory, as this would be beyond the scope of this paper.
\end{rem}

The following is an easy but useful property, which we record as a lemma for convenient future reference.
\begin{lem}\label{lem:inclusion-of-fixed}
Let $X$ be a continuous $\mathbb{G}$--spectrum. If $X^{h\mathbb{G}}$ is given the trivial $\mathbb{G}$--action, the ``inclusion of fixed points" map $X^{h\mathbb{G}} \to X$ is $\mathbb{G}$-equivariant.
\end{lem}
\begin{proof}
The map in question is $\holim_\Delta $ of the cosimplicial map
\[
F_c(\mathbb{G}^{\bullet}_+,X) \simeq F_c(\mathbb{G}^{\bullet+1}_+,X)^{\mathbb{G}} \longrightarrow F_c(\mathbb{G}_+^{\bullet+1},X),
\]
given by the inclusion of fixed points, which by construction has the required properties.
\end{proof}

One way to summarize the results of Devinatz and Hopkins \cite{DH} is as follows. The phrase ``essentially unique"
means the space of choices is contractible.

\begin{thm}\label{thm:DH-fixed-pts} The $\G$-spectrum $\E$ has an essentially unique structure as a continuous
$\G$-spectrum with the property that if $\K \subseteq \G$ is closed, then the map of Morava modules
$\E_\ast \E^{h\K} \to \E_\ast \E$ is naturally isomorphic to the inclusion
\[
\Map^c(\G/\K,\E_\ast) \longrightarrow \Map^c(\G,\E_\ast).
\]
\end{thm}
The Morava modules $\E_\ast \E^{h\K}$ and $\E_\ast \E$ are discussed in more details immediately after \Cref{rem:unpack-DH}.

\begin{rem}\label{rem:unpack-DH} The statement of \Cref{thm:DH-fixed-pts}  at once disguises quite a bit of difficult
work and obscures the logic of the Devinatz--Hopkins argument; thus, it is surely worth going into a bit of detail.

Suppose for a moment that we knew that \Cref{thm:DH-fixed-pts} was true. As above, choose a nested sequence of
open normal subgroups $U_{j+1} \subseteq U_j \subseteq \G$ with $\cap\ U_j = \{e\}$. Then we would have a sequence
of spectra
\begin{equation}\label{eq:nested-fp}
\cdots \longrightarrow \E^{hU_{j}} \longrightarrow \E^{hU_{j+1}}  \longrightarrow\cdots \longrightarrow\E
\end{equation}
with the following properties
\begin{enumerate}

\item $\E^{hU_j}$ is a $\G/U_j$ spectrum and all the maps of \eqref{eq:nested-fp} are $\G$-equivariant;

\item the map $\E_\ast \E^{hU_{j}} \longrightarrow \E_\ast \E$ of Morava modules is isomorphic to the inclusion
\[
\Map^c(\G/U_j,\E_\ast) \longrightarrow \Map^c(\G,\E_\ast);
\]

\item the induced map $\hocolim_j \E^{hU_j} \to \E$ is a $\Kn$-local equivalence.
\end{enumerate}

Let us give some detail on Part (3). By \Cref{rem:compl-tower}, Part (6) we have that if
 $X$ is $L_n$-local then $L_{\Kn}X = \holim X \wedge \MJ{i}$.
The spectra $\E^{hU_j}$ are $\Kn$-local and, hence $L_n$-local. Since $L_n$ is smashing the homotopy colimit is $L_n$-local,
so Part (3) is equivalent to the statement that
\[
\hocolim_j \E^{hU_j} \wedge \MJ{i} \longrightarrow \E \wedge \MJ{i}
\]
is an equivalence for all $i$. This follows from (2) and the fact that $\cap\ U_j = \{e\}$.

Next observe that since $\G/U_j$ is finite, $\E_\ast \E^{hU_{j}}$ is finitely generated as an $\E_\ast$-
module, hence $\E^{hU_{j}}$ is dualizable in the $\Kn$-local category, by \cite[Thm 8.6]{hovstrmemoir}.
Putting all this together -- and still
assuming we know \Cref{thm:DH-fixed-pts} --  we would have
the following diagram of cosimplicial spectra, with the vertical maps being $\Kn$-local equivalences
\begin{equation}\label{eq:define-sq}
\xymatrix{
\hocolim_j \E^{hU_j} \rto \dto_\simeq & \hocolim_j F((\G/U_j)_+^{\bullet+1},\E^{hU_j}) \dto^\simeq \\
\E \rto &F_c(\G_+^{\bullet+1},\E).
}
\end{equation}

Devinatz and Hopkins prove \Cref{thm:DH-fixed-pts} by reversing the logical order of this discussion: Recall that the Goerss--Hopkins--Miller theorem provides $\E$ with an essentially unique structure as an $E_\infty$-ring spectrum,
the space $\map_{E_\infty}(\E,\E)$ has contractible components, and $\pi_0\map_{E_\infty}(\E,\E)\cong \G$. This
gives $\E$ an essentially unique structure as $\G$-spectrum, with the action through $E_\infty$-ring maps.

Using the Goerss--Hopkins--Miller Theorem, Devinatz and Hopkins define a sequence of spectra which they call $\E^{hU_j}$ and
maps as in \eqref{eq:nested-fp} satisfying Parts (1)--(3) above. They then define the continuous $\G$-structure on $\E$
using the diagram of \eqref{eq:define-sq}. Then they must justify the notation $\E^{hU_j}$; that is, they must show
the spectra defined this way agree, up to equivalence, with the fixed points as defined in \eqref{eq:fixed-pnts}.
Finally,  they must calculate $\E_\ast \E^{h\K}$. For this they use the remarkable \Cref{prop:DHSS} below.
\end{rem}

We further unpack the statement of \Cref{thm:DH-fixed-pts} and generalize it (\Cref{prop:CommuteHFP} and \Cref{cor:ActMoravaModule}).
For any $X$,
\[
\E_\ast (\E \wedge X) = \pi_\ast L_{\Kn}(\E \wedge \E \wedge X)
\]
is a Morava module, using the action of $\G$ on the left factor
$\E$. Now, suppose $X$ itself has a $\G$-action so that the diagonal action on $\E \wedge X$ is continuous. If $h \in \G$ and $x \in \E_\ast X$, then we write $h\ast_d x$ for this action. The adjoint of the diagonal action of $\G$ on $\E \wedge X$ gives rise to a map
\begin{equation}\label{eq:fund-iso}
\eta\colon \E_\ast (\E \wedge X) \longrightarrow \Map^c(\G,\E_\ast X).
\end{equation}
Explicitly, if $x\colon S^t \to \E \wedge \E \wedge X$ and $g \in \G$, then $\eta_x(g)$ is the composite
\[
\xymatrix{
S^t \rto^-x & \E \wedge \E \wedge X \rto^-{1 \wedge g \wedge g} & \E \wedge \E \wedge X \rto^-{\mu \wedge 1} & \E \wedge X,
}
\]
where $\mu$ is multiplication and we have suppressed the $\Kn$-localizations.

When $X$ is $S^0$ with the trivial action and, then $\eta$ gives the identification $\E_*\E \cong \Map^c(\mathbb{G}, \E_*)$ which appeared in \Cref{thm:DH-fixed-pts}. The following result covers every other case that arises in this note.
\begin{lem}\label{lem:etaiso} Suppose $X = Y \wedge Z$ where $\Kn_\ast Y$ is zero in odd degrees and $Z$ is a $\Kn$-locally dualizable spectrum. Then the map $\eta$ in \eqref{eq:fund-iso} is an isomorphism.
\end{lem}

\begin{proof}
As in the proof of \cite[Prop.~2.4]{ghmr}, it suffices to show that the natural map
\[
\E_*(\E \wedge X) \longrightarrow \lim_i\E_*(\E \wedge \MJ{i} \wedge X)
\]
occurring in the Milnor sequence associated to $\holim_i\E \wedge \E \wedge X \wedge \MJ{i}$ is an isomorphism, i.e., that $\lim_i^1\E_{*}(\E \wedge \MJ{i} \wedge X) = 0$. The assumption on $Y$ implies that $\E_*(Y)$ is a flat $\E_*$-module, so there is an isomorphism
\[
\E_{*}(\E \wedge \MJ{i} \wedge X) \cong \E_*\E\otimes_{\E_*} \E_*(Y) \otimes_{\E_*} \E_*(\MJ{i} \wedge Z).
\]
Since $Z$ is dualizable, $\MJ{i} \wedge Z$ is $\Kn$-locally compact, hence $\E_*(\MJ{i} \wedge Z)$ is finite. This shows that the tower $(\E_*(\MJ{i} \wedge Z))_i$ is Mittag-Leffler, which implies that the required $\lim^1$ vanishes.
\end{proof}

\begin{rem}\label{rem:all-the-actions} We now have (at least) two actions to keep straight.
\begin{enumerate}
\item For the Morava module structure on  $\E_\ast (\E \wedge X)$
the isomorphism $\eta$ becomes $\G$-equivariant if we give the
module of functions the conjugation action
\[
(h\phi)(g) = h\ast_d \phi(h^{-1}g).
\]

\item The diagonal action on $\E \wedge X$ gives an action of $\G$ on $\E_\ast (\E \wedge X)$; this involves
the right factor of $\E$. With respect to this action $\eta$ becomes $\G$-equivariant if we give the
module of functions the action
\[
(h\star \phi)(g) = \phi(gh).
\]
\end{enumerate}
Note that the two actions commute.
\end{rem}

At this point, we need the following remarkable result due to Devinatz and Hopkins.
\begin{prop}\label{prop:DHSS} Let $W^\bullet$ be a cosimplicial spectrum. Suppose there exists
an integer $N$ and a finite type $0$ spectrum $Y$ so that for all spectra $Z$  the Bousfield--Kan spectral sequence
\[
\pi^s \pi_t F(Z, Y\wedge W^\bullet ) \Longrightarrow \pi_{t-s}F(Z, \holim_{\Delta} (Y \wedge W^\bullet) )
\]
has a horizontal vanishing line of intercept $s=N$ at the $E_\infty$-page. Then for any spectra $A $ and $F$ and maps $f\colon A\to A$,
there is an equivalence
\[
f^{-1}L_F(A \wedge \holim_{\Delta} W^\bullet) \simeq \holim_{\Delta} (f^{-1} L_F (A \wedge W^\bullet)).
\]
\end{prop}

\begin{proof} This is all contained in \cite[\S 5]{DH}, even if it is not explicitly stated this way. More specifically,
we combine the material before their Lemma 5.11, Lemma 5.12, and the argument using $Y$ given in the proof of their
Theorem 5.3.
\end{proof}

\begin{prop}\label{prop:CommuteHFP}
Let $X$ be a $\mathbb{G}$-spectrum, which is ($\Kn$-locally) dualizable, and such that the diagonal action of $\mathbb{G}$ on $\E\wedge X$ is continuous.
Then for a closed subgroup $\K$ of $\mathbb{G}$,
there is an equivalence
\[ \E \wedge (\E \wedge X)^{h\K} \simeq (\E\wedge \E \wedge X)^{h\K}, \]
where on the right-hand side, $\K$ is acting trivially on the first factor.
\end{prop}
\begin{proof}
We will prove this by applying \Cref{prop:DHSS} (with $A=\E$, $F = \Kn$, and $f=\mathrm{id}$) to the
cosimplicial spectrum which computes the homotopy fixed points $(\E \wedge X)^{h\K}$.
Specifically, $(\E \wedge X)^{h\K} \simeq \holim_{\Delta} W^\bullet$, with
\[
W^s = F_c(\mathbb{G}_+^{s+1}, \E\wedge X )^{\K} = F_c(\mathbb{G}/\K_+ \wedge \mathbb{G}^s_+, \E\wedge X).
\]
We need to check that the conditions of \Cref{prop:DHSS} are satisfied; then the result follows. The argument
we give exactly mirrors that of \cite[Theorem 5.3]{DH}.

We choose $Y$ to be a finite type 0 spectrum so that $\E_0Y$ is free as a $C$-module for every cyclic subgroup $C \subseteq \mathbb{G}$ of
order $p$ and so that $\E_1 Y = 0$. Moreover, $\E_* Y$ is free as an $\E_*$-module. Such a spectrum $Y$ is constructed by Jeff Smith; see \cite[\S 6.4, 8.3, 8.4]{RavNil}.

Since both $X$ and $Y$ are dualizable, \Cref{lem:ContinuousMaps} gives us that, for any spectrum $Z$, there is an isomorphism
\[
\pi_t F(Z,Y\wedge W^s) \cong   \Map^c(\mathbb{G}^{s+1}, \pi_tF(Z, \E\wedge X \wedge Y))^\K.
\]
Using again that $X$ and $Y$ are dualizable as well as that $\E_* Y$ is in even degrees and free over $\E_*$,
\[
\pi_t F(Z,\E\wedge X \wedge Y) \cong \E^{-t}(Z\wedge DX) \otimes_{\E_0} \E_0(Y).
\]
Now $\E_0(Y)$ is free as a $C$-module for every cyclic subgroup $C \subseteq \mathbb{G}$ of order $p$,
so the same is true for $\pi_t F(Z,\E\wedge X \wedge Y)$, and that fact implies that
\[
\pi^s \pi_t F(Z, Y\wedge W^\bullet) \cong H^s(\K, \E^{-t}(Z\wedge D X) \otimes_{\E_0} \E_0(Y))
\]
is zero for $s>n^2$ \cite[Lemma 8.3.5]{RavNil}.\footnote{The quoted results only claims the vanishing for $s > N$ where
$N$ depends only on $n$ and $p$. To get $N=n^2$ would require reworking the proof and using that $\G$ has
virtual Poincar\'e duality of dimension $n^2$.} In particular, this gives a horizontal vanishing line at the $E_2$-page, and \Cref{prop:DHSS}
applies to give the claim.
\end{proof}

\begin{cor}\label{cor:ActMoravaModule}
If $X$ and $\K$ are as in \Cref{prop:CommuteHFP}, then there is an isomorphism of Morava modules
\[\E_\ast((\E \wedge X)^{h\K})\cong \Map^c(\G/\K,\E_\ast X)\]
where the Morava module structure on the right-hand side is the conjugation action described in \Cref{rem:all-the-actions}.
\end{cor}
\begin{proof}
\Cref{prop:CommuteHFP} implies that
$\E_\ast((\E \wedge X)^{h\K}) \cong \pi_*(\E \wedge \E\wedge X)^{h\K}$. We will use the homotopy fixed point spectral sequence computing $\pi_*(\E \wedge \E\wedge X)^{h\K}$. As was discussed in \Cref{rem:all-the-actions}, there is a $\K$-equivariant isomorphism \[\E_*( \E\wedge X) \cong \Map^c(\mathbb{G}, \E_*X)\] with the $\K$-action on $\E_*(\E\wedge X) = \pi_*(\E \wedge \E \wedge X)$ the diagonal action on the right two factors and the $\K$-action on $\Map^c(\mathbb{G}, \E_*X)$ is right multiplication on the source. It follows that the $E_2$-term of the homotopy fixed point spectral sequence is
\[
H^*(\K, \pi_*(\E \wedge \E\wedge X)) \cong H^*(\K, \Map^c(\mathbb{G}, \E_*X)).
\]
Since $H^*(\K, \Map^c(\mathbb{G}, \E_*X)) \cong \Map^c(\mathbb{G}, \E_*X)^{\K}$, the homotopy fixed point spectral sequence collapses and the edge homomorphism gives an isomorphism of Morava modules
\begin{align*}
\E_\ast((\E \wedge X)^{h\K})&  \xrightarrow{\cong} (\E_\ast(\E \wedge X))^\K \cong \Map^c(\G,\E_\ast X)^\K \cong \Map^c(\G/\K,\E_\ast X).
\end{align*}
The actions were sorted out in \Cref{rem:all-the-actions}.
\end{proof}

\section{The Tate sphere and the determinant sphere}\label{sec:TateDet}

In order to define the determinant sphere, we need a spectrum-level construction which twists actions. This is accomplished by
a sphere spectrum we suggestively denote by $S(1)$, to be indicative of a Tate twist. Namely, $S(1)$ is the $p$-completed
sphere spectrum $S^0$ with a continuous action of $\mathbb{Z}_p^{\times}$ coming from its action as automorphisms on
$\pi_0S^0$, to be constructed below.

We can also consider $S(1)$ as a spectrum with a $\mathbb{G}$-action, where $\mathbb{G}$ acts through the determinant homomorphism
\[
\det \colon \mathbb{G} \longrightarrow \mathbb{Z}_p^{\times},
\]
defined as in \cite[Section 1.3]{ghmr}. The determinant is a surjection and we let $\mathrm{S}\G$ denote its kernel, so that there is an exact sequence
\[
1 \longrightarrow \mathrm{S}\G \longrightarrow \mathbb{G} \longrightarrow \mathbb{Z}_p^{\times} \longrightarrow 1.
\]
We will then define $S\langle {\det} \rangle$ as the homotopy fixed points of a particular $\G$-spectrum in the $\Kn$-local category.

We now begin the construction of $S(1)$; we will start by constructing a discrete action of a dense subgroup of $\mathbb{Z}_p^\times$. If $p > 2$, we have a decomposition
\[
(1 +p\Z_p) \times \bbmu \cong \Z_p^\times
\]
where $\bbmu=\F_p^\times$ is the cyclic group of order $p-1$ given by the Teichm\"uller lifts. Let $C \subseteq 1+p\Z_p$
be the infinite cyclic subgroup generated by $\tau = 1+p \in 1 +p\Z_p$.

If $p=2$, we have a slightly different decomposition
\[
(1 +4\Z_2) \times \bbmu \cong \Z_2^\times
\]
where now $\bbmu = \{ \pm 1 \}$. Let $C$ be generated by $\tau = 1+4 =5 \in 1+4\Z_2$.

With this setup, we write $G=C \times \bbmu$ for all primes. Note that $G$ is a dense subgroup of $\mathbb{Z}_p^\times$, and $\tau$ is a generator of the torsion-free subgroup $C\cong \Z$.
If $p> 2$ the inclusion $C \to 1+p\Z_p$ completes to an isomorphism $\Z_p \cong 1+p\Z_p$.
At $p=2$ we get a similar isomorphism $\Z_2 \cong 1 +4\Z_2$.

\begin{prop}\label{prop:the-disc-action} The inclusion $G \to \Z_p^\times = \pi_1B\mathrm{haut}(S^0)$ can be
canonically realized by a map
\[
BG \longrightarrow B\mathrm{haut}(S^0).
\]
\end{prop}

\begin{proof} Since $B\mathrm{haut}(S^0)$ is an infinite loop space we need only realize separately the maps $C \to \Z_p^\times$
and $\bbmu \to \Z_p^\times$ as maps $BC \to B\mathrm{haut}(S^0)$ and $B\bbmu \to B\mathrm{haut}(S^0)$. The map we want
will then be the composite
\[
BG \simeq BC \times B\bbmu \longrightarrow B\mathrm{haut}(S^0) \times B\mathrm{haut}(S^0) \longrightarrow B\mathrm{haut}(S^0)
\]
where the second map is the loop space multiplication.

At all primes $BC \simeq B\Z \simeq S^1$ and the choice of $\tau$ defines the required map $S^1 \to B\mathrm{haut}(S^0)$.

If $p=2$, then $B\bbmu \simeq B\Z/2 \simeq BO(1)$ and the map we need is defined by the composition
\[
BO(1) \longrightarrow BO \longrightarrow B\mathrm{haut}(S^0).
\]
Suppose $p>2$ and let $A$ be some $2$-skeleton of $B\bbmu$. The inclusion $\bbmu \subseteq \Z_p^\times$ defines a map
$A \to B\mathrm{haut}(S^0)$ by extending a generator of $\bbmu \subset \pi_1B\mathrm{haut}(S^0)$ to $A$. Since $\pi_i B\mathrm{haut}(S^0) \cong \pi_{i-1}S^0$ is $p$-complete for $i \geq 2$ and $\bbmu$ has order prime to $p$, the map out of $A$ extends uniquely to a map $B\bbmu \to B\mathrm{haut}(S^0)$.
\end{proof}

Let $k \geq 1$ and let $G_k \subseteq G$ be the kernel of the composition
\[
\xymatrix{
G \rto^-{\subseteq} & \Z_p^\times \rto & (\Z/p^k)^\times.
}
\]
If $p > 2$, then $G_1 =C$ and $G_k$ is infinite cyclic generated by  $\tau^{p^{k-1}}$. If $p = 2$,
then $G_2 = C$ and for $k > 1$ the group $G_k$ is infinite cyclic generated by $\tau^{p^{k-2}}$. We have that the intersection $\cap G_k $ is trivial, and $\lim_k G/G_k \cong \Z_p^\times$; thus, the subgroups $G_k$ define the usual topology on $\Z_p^\times$.

\begin{prop}\label{prop:moore-tower} Let $\widetilde\ST$ be the $p$-complete sphere spectrum with the discrete action of $G$ constructed above. If $p$ is odd let $k\geq 1$ and if $p =2$
let $k > 1$. Then there is
an equivalence
\[
S/p^k \simeq EG_+ \wedge_{G_k} \widetilde\ST
\]
and the residual action of $G/G_k \cong (\Z/p^k)^\times$ realizes the standard action of $(\Z/p^k)^\times$ on
$\Z/p^k \cong \pi_0S/p^k$.
\end{prop}

\begin{proof} The homotopy orbit spectrum $EG_+ \wedge_{G_k} \widetilde\ST$ is a connected spectrum and we have
a homotopy orbit spectral sequence for $H_\ast(-) = H_\ast(-,\Z)$:
\[
E^2_{p,q} \cong H_p(G_k,H_q\widetilde\ST) \Longrightarrow H_{p+q} (EG_+ \wedge_{G_k} \widetilde\ST).
\]
Let $p > 2$. The group $G_k$ is infinite cyclic generated by $\tau^{p^{k-1}}$ where $\tau = 1+p$.
Since $\tau^{p^{k-1}} \equiv 1+p^k$
modulo $p^{k+1}$ we have $E^2_{p,q} = 0$ unless $(p,q) = (0,0)$ and there is a surjection of $G$-modules
\[
\Z_p \cong H_0(\widetilde\ST) \longrightarrow H_0(G_k,H_0(\widetilde\ST)) \cong \Z/p^k.
\]
It follows that $EG_+ \wedge_{G_k} \widetilde\ST$ must be a Moore spectrum for $\Z/p^k$ with the standard action of $\Z/p^k$ on $\pi_0S/p^k$. The proof at the prime $2$ is completely analagous.
\end{proof}

Recall that continuous actions were discussed in \Cref{sec:continuousactions}. See in particular \Cref{def:cont-g-spec} and \Cref{rem:otherG}.
\begin{prop}\label{prop:cont-zz-Sone} The $G$-action on $\widetilde\ST$ extends to a continuous action of the
profinite group $\Z_p^\times$, in the sense that we have an augmented cosimplicial spectrum
\[
\widetilde\ST \longrightarrow F_c((\Z_p^\times)_+^{\bullet +1},\widetilde\ST),
\]
so that the augmentation refines the $\mathbb{Z}_p^\times$--action.
\end{prop}
\begin{proof} Write $S/p^k(1)$ for $EG_+ \wedge_{G_k} \widetilde\ST$ with its $G/G_k \cong (\Z/p^k)^\times$-action. Then the
augmented cosimplicial spectra
\[
S/p^k(1) \to F((G/G_k)_+^{\bullet +1},S/p^k(1))
\]
assemble to give a map
\begin{align*}
\widetilde\ST \simeq \holim_k S/p^k(1) \longrightarrow & \holim_k \hocolim_j F((G/G_j)_+^{\bullet +1},S/p^k(1)) \\
& = F_c((\Z_p^\times)_+^{\bullet +1},\widetilde\ST)
\end{align*}
as needed.
\end{proof}

\begin{defn}\label{defn:S(1)} We will write $S(1)$ for the $p$-complete sphere $S^0$ with the continuous
$\Z_p^\times$-action of \cref{prop:cont-zz-Sone}.
The same construction gives $S(1)$ as a continuous $p$-complete $\G$--spectrum, where $\G$ acts through the determinant surjection $\det\colon\G\to\mathbb{Z}_p^\times$.

We refer to this equivariant sphere as the \emph{Tate sphere}.
\end{defn}

Now we take the Morava $E$-theory spectrum $\E$ and give $\E \wedge S(1)$ the
diagonal $\G$-action. The next result indicates that this is an interesting construction.

\begin{prop}\label{prop:homotopy-ES} There is an isomorphism of Morava modules
\[
\E_\ast\langle\det\rangle \cong \pi_\ast (\E \wedge S(1)) = \E_\ast S(1).
\]
\end{prop}
\begin{proof}
The edge map of the Tor spectral sequence
\[
\E_\ast\langle\det\rangle = \E_\ast \otimes_{\pi_0S^0} \pi_0 S(1) \longrightarrow \pi_\ast(\E \wedge S(1))
\]
is an isomorphism, and respects the $\mathbb{G}$-action by the naturality of the spectral sequence.
\end{proof}

The following technical result is the key to our calculations.

\begin{prop}\label{cont-for-ES1}  The $\G$-spectrum $\E \wedge S(1)$ has the structure of a $\Kn$-local
continuous $\G$-spectrum.
\end{prop}

\begin{proof} As in \eqref{eq:cosim-g} we need to construct an augmented cosimplicial $\G$-spectrum
\[
\E \wedge S(1) \longrightarrow F_c(\G_+^{\bullet + 1},\E \wedge S(1))
\]
so that the augmentation refines the $\G$-action on $\E \wedge S(1)$.

As above, we continue writing $S/p^k(1)$ for $EG_+ \wedge_{G_k} \widetilde\ST$ with its $G/G_k \cong (\Z/p^k)^\times$ action.
Let us also write $S/p^k$ for the Moore spectrum when we do not need to refer to  the action.

Since $\MJ{i}$ and $S/p^k$ are finite spectra we have
\begin{align*}
F_c(\G_+^s,\E \wedge S(1)) &= \holim_i \hocolim_j F((\G/U_j)^s_+,\E \wedge S(1) \wedge \MJ{i})\\
& \xrightarrow{\simeq} \holim_k \holim_i \hocolim_j F((\G/U_j)^s_+,\E \wedge S/p^k(1) \wedge \MJ{i});
\end{align*}
indeed, both sides of the last equivalence are $p$-complete and the natural map between them is an equivalence after smashing with $S/p$. For all $j$ so that $U_j$ is in the kernel of
\[
\xymatrix{
\G \rto^-\det & \Z_p^\times \rto & (\Z/p^k)^\times,
}
\]
the diagonal action of $\G/U_j$ on $\E^{hU_j} \wedge S/p^k(1) \wedge \MJ{i}$ defines an augmented
cosimplicial $\G$-spectrum
\[
\E^{hU_j} \wedge S/p^k(1) \wedge \MJ{i} \longrightarrow F((\G/U_j)^{\bullet +1}_+,\E^{hU_j} \wedge S/p^k(1) \wedge \MJ{i}).
\]
Since $\hocolim_j \E^{hU_j} \simeq \E$, these assemble into the cosimplicial spectrum we need.
\end{proof}

We can now make our central definition.

\begin{defn}\label{def-Sdet} The determinant sphere is the spectrum
\[
S\langle {\det} \rangle = (\E \wedge S(1))^{h\G} = \holim_\Delta F_c(\G_+^{\bullet +1},\E\wedge S(1))^\G.
\]
\end{defn}

\begin{rem}\label{rem:some-ss} If $\K \subseteq \mathbb{G}$ is closed we define
\[
(\E \wedge S(1))^{h\K} \longrightarrow \holim_\Delta F_c(\mathbb{G}_+^{\bullet+1},\E \wedge S(1))^\K.
\]
Therefore, using \Cref{prop:homotopy-ES} and \Cref{rem:HFPSS}, we have a homotopy fixed point spectral sequence
\[
H^s_c(\K,\E_\ast \langle\det\rangle) \Longrightarrow \pi_{t-s}(\E\wedge S(1))^{h\K}.
\]
\end{rem}

We now must show that there is an isomorphism of Morava modules $\E_\ast S\langle {\det} \rangle \cong \E_\ast \langle\det\rangle$.
The key results are \Cref{prop:homotopy-ES} and \Cref{prop:CommuteHFP} used as its
\Cref{cor:ActMoravaModule}.

\begin{prop}\label{prop:what-were-here-for} There is an isomorphism of
Morava modules
\[
\E_\ast S\langle {\det} \rangle \cong \E_\ast\langle \det\rangle.
\]
\end{prop}

\begin{proof} Combine \Cref{cor:ActMoravaModule} and \Cref{prop:homotopy-ES}.
\end{proof}

We now extend this map to an equivalence of spectra. Let $\iota\colonS\langle {\det} \rangle = (\E\wedge S(1))^{h\G} \to \E \wedge S(1)$
be the inclusion of the fixed points from \cref{lem:inclusion-of-fixed}, and let $\mu\colon \E \wedge \E \to \E$ be the multiplication. Define
\[
f\colon \E \wedge S\langle {\det} \rangle \longrightarrow \E \wedge S(1)
\]
to be the composition
\begin{equation}\label{eq:keycomposite}
\xymatrix{
\E \wedge S\langle {\det} \rangle \ar[r]^-{1 \wedge \iota} & \E \wedge \E \wedge S(1) \ar[r]^-{\mu \wedge 1} & \E \wedge S(1).
}
\end{equation}
This map is $\G$-equivariant if we use the action on $\E$ on the source and the diagonal action on the target.

\begin{thm}\label{thm:pretty-damn-cool} The map $f\colon \E \wedge S\langle {\det} \rangle \to \E \wedge S(1)$
of \eqref{eq:keycomposite} is a $\G$-equivariant equivalence and induces the isomorphism of Morava modules
\[
\E_*S\langle {\det} \rangle \cong \E_*{\left<\det \right>}.
\]
of \Cref{prop:what-were-here-for}.
\end{thm}

\begin{proof} To check that $f$ is an equivalence we need only check that it induces the indicated map on Morava modules.
Applying $\pi_*(-)$ to \eqref{eq:keycomposite} gives
\begin{align}\label{eq:compo}
\xymatrix{
\E_*S\langle {\det} \rangle \ar[r] \ar[d]^-{\cong}& \E_*( \E \wedge S(1)) \ar[d]^-{\cong} \ar[r]^-{\mu \wedge 1} & \E_* S(1) \ar[d]^-{=}  \\
\Map^c(\mathbb{G}, \E_*S(1) )^{\mathbb{G}} \ar[r] & \Map^c(\mathbb{G}, \E_*S(1)) \ar[r]  & \E_*S(1).}
\end{align}
The first vertical isomorphism is from \Cref{cor:ActMoravaModule}, whereas the second is the isomorphism of \Cref{lem:etaiso}.
In the bottom row, the first map is the inclusion of fixed points and the second map is evaluation at the unit $e \in \mathbb{G}$.
The fixed points on the bottom left are exactly the constant functions, so the composite is an isomorphism as claimed.
\end{proof}

This yields the following practical invariance result.

\begin{cor}\label{cor:invariance}
If $\K $ is a closed subgroup of $\mathbb{G}$ which is in the kernel of the determinant, then $\E^{h\K} \wedge S\langle {\det} \rangle \simeq \E^{h\K}$.
\end{cor}

\begin{proof} We use \Cref{thm:pretty-damn-cool}.
When we restrict the $\mathbb{G}$-action on the Tate sphere $S(1)$ to $\K$, we get that $\K$ acts trivially, so $S(1)$ is $\K$-equivariantly equivalent to $S^0$.  We have
\[ \E^{h\K} \wedge S\langle {\det} \rangle \simeq (\E \wedge S\langle {\det} \rangle)^{h\K} \simeq (\E \wedge S(1))^{h\K} \simeq \E^{h\K}, \]
where the first equivalence follows since $S\langle {\det} \rangle$ is a $\Kn$-locally dualizable spectrum with trivial $\K$-action.
\end{proof}

\section{Deconstructing the determinant sphere}

Let $S\mathbb{G} \subseteq \G$ be the kernel of the determinant. Then we can form the fixed point spectrum $\E^{h \mathrm{S}\G}$.
This will have a residual action of $\G/\mathrm{S}\G \cong \Z_p^\times$. (See the paragraph before Theorem 4 in \cite{DH}.) Furthermore
\[
(\E \wedge S(1))^{h \mathrm{S}\G} \simeq \E^{h \mathrm{S}\G} \wedge S(1),
\]
where the right hand side has a diagonal $\Z_p^\times$-action.

At odd primes we get a simple description of $S\langle {\det} \rangle$ directly from Devinatz--Hopkins fixed point theory.

\begin{prop}\label{prop:fiber1} Let $p > 2$ and let $\phi \in \G$ be any element so that $\det(\phi)$ topologically
generates $\Z_p^\times$. Then there is a fiber sequence
\[
\xymatrix{
S\langle {\det} \rangle \rto & \E^{h \mathrm{S}\G} \ar[rr]^-{\det(\phi)\phi-1} && \E^{h \mathrm{S}\G}.
}
\]
\end{prop}

\begin{proof} Using that $S(1)$ is non-equivariantly the sphere $S^0$, we have  a commutative diagram
\[
\xymatrix{
\E^{h \mathrm{S}\G} \wedge S(1) \ar[rr]^-{\phi \wedge \det(\phi) - 1} \dto_\simeq && \E^{h \mathrm{S}\G} \wedge S(1)\dto^\simeq\\
\E^{h \mathrm{S}\G} \ar[rr]^-{\det(\phi)\phi-1} && \E^{h \mathrm{S}\G}.
}
\]
Let $F$ be fiber of the bottom map. The composition
\[
\xymatrix{
S\langle {\det} \rangle = (\E \wedge S(1))^{h\G} \rto & \E^{h \mathrm{S}\G} \wedge S(1) \ar[rr]^-{\phi \wedge \det(\phi) - 1} && \E^{h \mathrm{S}\G} \wedge S(1)
}
\]
is null-homotopic, so we get a map $f\colonS\langle {\det} \rangle \to F$. Using the fact that
\[
\E_\ast \E^{h \mathrm{S}\G} \cong \Map^c(\G/\mathrm{S}\G,\E_\ast) \cong  \Map^c(\Z_p^\times,\E_\ast)
\]
we compute that $f$ induces an isomorphism of Morava modules.
\end{proof}

We can refine the fiber sequence of \Cref{prop:fiber1}. We still have $p > 2$ and we have a splitting
\[
\bbmu \times (1+p\Z_p) \cong \Z_p^\times.
\]
The group $\bbmu \cong \F_p^\times$ is cyclic of order $p-1$ and  $(1+p\Z_p)$ is isomorphic to $\Z_p$ itself.

Let $\alpha \in \mathbb{W}^\times \subseteq \G$ be a $(p^n-1)$st
root of unity; then $\det(\alpha) \in \bbmu$ is a generator. The group $\bbmu \subseteq \Z_p^\times$ acts
on $\E^{h \mathrm{S}\G}$ and, since this group is abstractly isomorphic to $C_{p-1}$, the spectrum $\E^{h \mathrm{S}\G}$ splits as a
wedge of the
eingenspectra for this action. Let $\E_\chi^{h \mathrm{S}\G}$ be the summand defined by the equations
\[
\pi_\ast \E_\chi^{h \mathrm{S}\G} = \{\ x \in \pi_\ast \E^{h S \G}\ |\ \alpha_\ast x = \det(\alpha)^{-1} x\ \}.
\]
Note that the spectrum $\E_\chi^{h \mathrm{S}\G} $ corresponds to $(\E^{h\mathrm{S}\G}\wedge S(1))^{h\bbmu}$.
Indeed, forgetting the $\bbmu$-action and remembering that the underlying spectrum of $S(1)$ is the $p$-complete sphere, the map which sends $x\in \pi_*(\E^{h\mathrm{S}\G})$ to $x\wedge 1 \in \pi_*(\E^{h\mathrm{S}\G}\wedge S(1))$ is a non-equivariant isomorphism.
Now note that if $\alpha_*(x) =  \det(\alpha)^{-1} x$ in $\pi_*\E^{h\mathrm{S}\G}$ then $\alpha_*(x \wedge 1) = \alpha_*(x) \wedge \det(\alpha) = x \wedge 1$ in $\pi_*(\E^{h\mathrm{S}\G}\wedge S(1))$ so that
\[x\wedge 1 \in (\pi_*(\E^{h\mathrm{S}\G}\wedge S(1)))^{\bbmu} \cong \pi_*(\E^{h\mathrm{S}\G}\wedge S(1))^{h\bbmu}.\]

\begin{prop}\label{prop:fiber2} Let $p > 2$ and let $\psi \in \G$ be any element so that $\det(\psi)$ topologically
generates $1+p\Z_p \subseteq \Z_p^\times$. Then there is a fiber sequence
\[
\xymatrix{
S\langle {\det} \rangle \rto & \E_\chi^{h \mathrm{S}\G} \ar[rr]^-{\det(\psi)\psi-1} && \E_\chi^{h \mathrm{S}\G}.
}
\]
\end{prop}

The proof is very similar to  that of \Cref{prop:fiber1}. This fiber sequence appears in \cite[Rem.~2.5]{GHM_pic}
although there is a typo there: the factor of $\det(\psi^{p+1})$ should be replaced by $\det(\psi)^{-(p+1)}$ in Equation (2.6).

At the prime $2$ we have $\Z_2^\times \cong \bbmu \times (1+4\Z_p)$ for $\bbmu =\{\pm 1\}$ and the decomposition gets a little more
subtle. In particular, $\E^{h \mathrm{S}\G_2}$ does not decompose as a wedge of $\bbmu$-eigenspectra, where $\bbmu$ acts on $\E^{h \mathrm{S}\G_2}$ through $\Z_2^{\times} \cong \mathbb{G}_2/\mathrm{S}\G_2$. The following construction
 expands on ideas of Hans-Werner Henn.

Define a spectrum $\E^{h \mathrm{S}\G}_{-}$ as the cofiber of the inclusion of the fixed points
\[
(\E^{h \mathrm{S}\G})^{h\bbmu} \longrightarrow \E^{h \mathrm{S}\G} \longrightarrow \E^{h \mathrm{S}\G}_{-};
\]
this will end up modeling $(\E^{h\mathrm{S}\G} \wedge S(1))^{h\bbmu}$.

\begin{prop}\label{prop:fiber3} Let $p = 2$ and let $\psi \in \G$ be any element so that $\det(\psi)$ topologically
generates $1+4\Z_2 \subseteq \Z_2^\times$. Then there is an extension of the map
$\psi\colon\E^{h \mathrm{S}\G} \to \E^{h \mathrm{S}\G}$ to a commutative diagram
\[
\xymatrix{
\E^{h \mathrm{S}\G}\dto_\psi \rto & \E^{h \mathrm{S}\G}_{-}\dto^\psi\\
\E^{h \mathrm{S}\G} \rto & \E_{-}^{h \mathrm{S}\G}
}
\]
so that there is a fiber sequence
\[
\xymatrix{
S\langle {\det} \rangle \rto & \E_{-}^{h \mathrm{S}\G} \ar[rr]^-{\det(\psi)\psi-1} && \E_{-}^{h \mathrm{S}\G}.
}
\]
\end{prop}

\begin{proof} There is a cofiber sequence of $\bbmu$-spectra
\[
S^0 \longrightarrow \Sigma^\infty_+ {\bbmu} \longrightarrow S(1).
\]
Smashing it with $\E^{h \mathrm{S}\G} $ we obtain a cofiber sequence of $\bbmu$--spectra
\[
\E^{h \mathrm{S}\G} \longrightarrow \E^{h \mathrm{S}\G}\wedge {\bbmu}_+ \longrightarrow \E^{h \mathrm{S}\G} \wedge S(1).
\]
Now taking $\bbmu$ homotopy fixed points gives a cofiber sequence
\[
(\E^{h \mathrm{S}\G})^{h\bbmu} \longrightarrow \E^{h \mathrm{S}\G} \longrightarrow (\E^{h \mathrm{S}\G} \wedge S(1))^{h\bbmu}.
\]
Thus we have  $(\E^{h \mathrm{S}\G} \wedge S(1))^{h\bbmu} \simeq \E_{-}^{h \mathrm{S}\G}$, and the result
follows as in the proof of \Cref{prop:fiber1}.
\end{proof}

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redpajama_set_name: RedPajamaArXiv


text:
La gare du Camping Monticello est une gare ferroviaire française de la ligne de Ponte-Leccia à Calvi (voie unique à écartement métrique), située sur le territoire de la commune de Monticello, dans le département de la Haute-Corse et la Collectivité territoriale de Corse (CTC).

C'est une halte des Chemins de fer de la Corse (CFC) desservie par des trains « grande ligne ». Arrêt facultatif (AF), il faut signaler sa présence au conducteur pour qu'il y ait un arrêt du train.

Situation ferroviaire
Établie à  d'altitude, la gare du Camping Monticello est située au point kilométrique (PK) 95,9 de la ligne de Ponte-Leccia à Calvi (voie unique à écartement métrique), entre les gares du Regino (AF) et de l'Île-Rousse.

Histoire
La halte dénommée « Île-Rousse Plage », est mise en service en 1965 à l'occasion de la mise en place du service « trains-tramways de la Balagne » circulant de mai à octobre à l'intention des touristes Le nom est modifié plus tard.

Service des voyageurs

Accueil
Halte CFC, c'est un point d'arrêt non géré (PANG) à accès libre. Arrêt facultatif  (AF) : le train ne s'arrête que si la demande a été faite au conducteur.

Desserte
Camping Monticello est desservie par des trains CFC « grande ligne » de la relation Bastia (ou Ponte-Leccia) - Calvi. Les horaires sont fluctuants en fonction de la saison (juillet-Août) et du hors saison (le reste de l'année).

Intermodalité
Le stationnement des véhicules est possible à proximité.

Notes et références

Voir aussi

Articles connexes
 Chemins de fer de la Corse (CFC)
 Liste de gares de France
 Ligne de Ponte-Leccia à Calvi

Lien externe
 Site cf-corse.fr situation de la gare du Camping Monticello sur le réseau

Gare en Corse
Ligne de Ponte-Leccia à Calvi

redpajama_set_name: RedPajamaWikipedia


text:
10:33 AM Star Wars Jedi: Fallen Order May Be Getting A Photo Mode, New Game Plus
9:45 AM Pokemon GO Announces Special Crossover With One Piece
New Far Cry News Teased For The Video Game Awards (VIDEO)
by Chris Calles
The Game Awards are almost here and the upcoming show promises to be even bigger than ever, bringing somewhere around 10 new title reveals with it. Founder and Host Geoff Keighley has been fairly keen on providing teases and hints for fans in order to generate hype for the event. The latest teaser? More Far Cry news!
Ubisoft recently took to social media to spread the word of the Game Awards event. Having their latest title Far Cry 5 be released just this spring it makes sense that the company would show support for this year's Awards.
Tune in for a worldwide reveal at #TheGameAwards on December 6, at 5:30PM PST/8:30PM EST.
— Far Cry 5 (@FarCrygame) December 5, 2018
One of Far Cry 5's end sequences featured a bit of a cliffhanger, so naturally many are hoping this is a teaser going off one particular event that happens depending on choices (spoiler). A new trailer for an upcoming content reveal at the Game Awards indicates it may either be a direct sequel or at the very least some new story DLC set sometime after the nuclear apocalypse begins in Hope County, MT.
Keighley responded to the reveal of the upcoming Far Cry news by stating that there are some rumors of games set to be announced at the Game Awards that are true, though there are some that are false. Thankfully, this Far Cry tease is legitimate.
Fun to get the @FarCrygame news out there and surprise you. There are lots of rumors about games set to announce tomorrow night. Some are true, many are false, and then there's another category: The things you won't see coming.
— Geoff Keighley (@geoffkeighley) December 5, 2018
The Game Awards begin tomorrow, December 6th at 7:30 pm CT. What could the upcoming Far Cry reveal contain? Will it be a new game, some new content for Far Cry 5, or something else entirely? Let us know your thoughts on the upcoming Far Cry news and the Game Awards in the comments section below. As always, don't forget to follow Don't Feed the Gamers on Twitter for 24/7 coverage of all tlhings gaming and entertainment.
Dragon Age 4 Game Awards Announcement On The Way, But The Game's Not Dropping Anytime Soon
Splinter Cell Announcement Possibly Teased For The Game Awards (VIDEO)
The Last Of Us 2 Won't Make An Appearance At The Game Awards This Year
PlayStatio n4
The Game Awards 2018
PlayStation Classic Datamining Reveals List Of Rejected Games
This Elder Scrolls Online Cosplay Is Insanely Perfect
The Division 2 Director Is Gauging Interest For A Single Player Spin-Off
Wolfenstein: Youngblood System Requirements Revealed

redpajama_set_name: RedPajamaCommonCrawl


text:
from datetime import timedelta
from test.support import captured_stderr

from django.core.urlresolvers import reverse
from django.test import TestCase
from django.utils import timezone

from .models import Entry, Event
from .sitemaps import WeblogSitemap

class DateTimeMixin(object):
    def setUp(self):
        self.now = timezone.now()
        self.yesterday = self.now - timedelta(days=1)
        self.tomorrow = self.now + timedelta(days=1)

class EntryTestCase(DateTimeMixin, TestCase):
    def test_manager_active(self):
        """
        Make sure that the Entry manager's `active` method works
        """
        Entry.objects.create(pub_date=self.now, is_active=False, headline='inactive', slug='a')
        Entry.objects.create(pub_date=self.now, is_active=True, headline='active', slug='b')

        self.assertQuerysetEqual(Entry.objects.published(), ['active'], transform=lambda entry: entry.headline)

    def test_manager_published(self):
        """
        Make sure that the Entry manager's `published` method works
        """
        Entry.objects.create(pub_date=self.yesterday, is_active=False, headline='past inactive', slug='a')
        Entry.objects.create(pub_date=self.yesterday, is_active=True, headline='past active', slug='b')
        Entry.objects.create(pub_date=self.tomorrow, is_active=False, headline='future inactive', slug='c')
        Entry.objects.create(pub_date=self.tomorrow, is_active=True, headline='future active', slug='d')

        self.assertQuerysetEqual(Entry.objects.published(), ['past active'], transform=lambda entry: entry.headline)

    def test_docutils_safe(self):
        """
        Make sure docutils' file inclusion directives are disabled by default.
        """
        with captured_stderr() as self.docutils_stderr:
            entry = Entry.objects.create(
                pub_date=self.now, is_active=True, headline='active', content_format='reST',
                body='.. raw:: html\n    :file: somefile\n', slug='a',
            )
        self.assertIn('<p>"raw" directive disabled.</p>', entry.body_html)
        self.assertIn('.. raw:: html\n    :file: somefile', entry.body_html)

class EventTestCase(DateTimeMixin, TestCase):
    def test_manager_past_future(self):
        """
        Make sure that the Event manager's `past` and `future` methods works
        """
        Event.objects.create(date=self.yesterday, pub_date=self.now, headline='past')
        Event.objects.create(date=self.tomorrow, pub_date=self.now, headline='future')

        self.assertQuerysetEqual(Event.objects.future(), ['future'], transform=lambda event: event.headline)
        self.assertQuerysetEqual(Event.objects.past(), ['past'], transform=lambda event: event.headline)

    def test_manager_past_future_include_today(self):
        """
        Make sure that both .future() and .past() include today's events.
        """
        Event.objects.create(date=self.now, pub_date=self.now, headline='today')

        self.assertQuerysetEqual(Event.objects.future(), ['today'], transform=lambda event: event.headline)
        self.assertQuerysetEqual(Event.objects.past(), ['today'], transform=lambda event: event.headline)

    def test_past_future_ordering(self):
        """
        Make sure the that .future() and .past() use the actual date for ordering
        (and not the pub_date).
        """
        D = timedelta(days=1)
        Event.objects.create(date=self.yesterday - D, pub_date=self.yesterday - D, headline='a')
        Event.objects.create(date=self.yesterday, pub_date=self.yesterday, headline='b')

        Event.objects.create(date=self.tomorrow, pub_date=self.tomorrow, headline='c')
        Event.objects.create(date=self.tomorrow + D, pub_date=self.tomorrow + D, headline='d')

        self.assertQuerysetEqual(Event.objects.future(), ['c', 'd'], transform=lambda event: event.headline)
        self.assertQuerysetEqual(Event.objects.past(), ['b', 'a'], transform=lambda event: event.headline)

class ViewsTestCase(DateTimeMixin, TestCase):
    def test_no_past_upcoming_events(self):
        """
        Make sure there are no past event in the "upcoming events" sidebar (#399)
        """
        # We need a published entry on the index page so that it doesn't return a 404
        Entry.objects.create(pub_date=self.yesterday, is_active=True, slug='a')
        Event.objects.create(date=self.yesterday, pub_date=self.now, is_active=True, headline='Jezdezcon')
        response = self.client.get(reverse('weblog:index'))
        self.assertEqual(response.status_code, 200)
        self.assertQuerysetEqual(response.context['events'], [])

class SitemapTests(DateTimeMixin, TestCase):

    def test_sitemap(self):
        entry = Entry.objects.create(pub_date=self.yesterday, is_active=True, headline='foo', slug='foo')
        sitemap = WeblogSitemap()
        urls = sitemap.get_urls()
        self.assertEqual(len(urls), 1)
        url_info = urls[0]
        self.assertEqual(url_info['location'], entry.get_absolute_url())

redpajama_set_name: RedPajamaGithub


A highly stylish range of Mardome dome rooflights. The sleek aluminium frame hides all the fixings and can be powder coated to any standard RAL colour. This dome has a secret lock system, making it a maximum security rooflight.
Measure overall dimensions of the opening (length x width). Standard roof opening sizes / dome sizes are detailed below. All domes in the Mardome Ultra range can also be specified to be fixed directly to an existing builders upstand, and where necessary these are supplied with a 'direct fix' adaptor to enable this. An optional Mardome kerb adaptor allows standard size domes to be fitted to non-standard size upstands.
Mardome Ultra is available in single, double and triple skin polycarbonation glazed units. Options are available to further improve thermal efficiency include glass inner glazing. An average rooflight area of 2.2 W/m²K is required to satisfy Part L Building Regulations. To assist compliance with Part L Ariel Plastics recommend installing no less than triple glazed Mardome rooflights which achieve a U-value of 1.8 W/m²K.
Mardome Ultra has tamper proof security fixings which are concealed under the aluminium frame for maximum security and complete peace of mind. A secret lock system locks the dome into place providing the highest possible security level and can only be opened with the Mardome security key.
The white PVC kerb is designed to be thermally insulating and all cabling is enclosed within on all mains powered opening Mardome Ultras. The robust kerb has an integral clamp to help hold roofing membrane in place for a tidy roof covering finish.


text:
Q: mysql select result based on multiple matching criteria of 2 or more columns My table is like this:
CREATE TABLE candidate_tbl  (
  `name` VARCHAR(1),
  `degree` VARCHAR(41),
  `doneMasters` VARCHAR(6)
);

INSERT INTO candidate_tbl
  (`name`, `degree`, `doneMasters`)
VALUES
  ('A', 'MBA', 'true'),
  ('A', 'MS', 'false'),
  ('B', 'MBA', 'true'),
  ('B', 'MS', 'true'),
  ('C', 'MBA', 'false'),
  ('C', 'MS', 'false'),
  ('D', 'MBA', 'false'),
  ('D', 'MS','true'),
  ('E', 'MBA', 'false'),
  ('E', 'MS', 'false'),
  ('F', 'MBA', 'false'),
  ('F', 'MS', 'true'),
  ('G', 'MBA', 'false'),
  ('G', 'MS', 'false'),
  ('H', 'MS', 'true'),
  ('H', 'MS', 'true');

I want the list of candidates who neither has MS nor MBA, which is C,E and G
I tried these 4 ways, Fiddle given here
Method-1
SELECT name,count(*) FROM candidate_tbl
WHERE doneMasters = 'false'
AND doneMasters = 'false'
GROUP BY name
HAVING count(*) = 2;

Method-2
SELECT DISTINCT name FROM candidate_tbl ct1 WHERE NOT EXISTS
(
SELECT name FROM candidate_tbl ct2 WHERE ct1.name = ct2.name and doneMasters = 'true'
);

Method-3
SELECT name FROM (SELECT name,group_concat(doneMasters) as dmf FROM candidate_tbl
GROUP BY name) dd
WHERE dmf = 'false,false';

Method-4
SELECT name FROM (SELECT name, group_concat(doneMasters) as dmf FROM candidate_tbl
GROUP BY name) dd
WHERE dmf = 'false,false';

I want to know which is the best solution in terms of performance and accuracy, or other alternate solutions.

A: Your data model doesn't really make sense to me.  Why store negative information in a table, for instance?
The best way would be to normalize the data model to have a separate table of candidates and their degrees.  Then use not exists:
select c.*
from candidates c
where not exists (select 1
                  from candidate_degrees cd
                  where cd.candidate_id = c.candidate_id and
                        cd.degree in ('MS', 'MBA')
                 );

This would be a rather dramatic change to your data model:

*

*The degrees table would only have degrees that exist (and perhaps other information such as the year, school, and so on).

*The candidates would be identified by an id that can be used for joining.

A: Try this.
SELECT name FROM candidate_tbl
WHERE doneMasters = 'false'
AND degree in ('MBA', 'MS')
GROUP BY name HAVING count(1) = 2;

DEMO HERE: DB-FIDDLE

A: the solution is simply
SELECT name FROM candidate_tbl
WHERE degree in ('MBA', 'MS') AND doneMasters = 'false'
GROUP BY name;

it will return a unique name from list who has not done MBA or MS.

A: If for each name there are always exactly 2 rows with values 'MBA' and 'MS' in the column degree, then there is no reason to check or filter the column degree.
The simplest way is to group by name, filter out the rows with true in the column doneMasters and check in the HAVING clause if in the results there are 2 rows (these 2 rows will contain 'false' in the column doneMasters):
SELECT name
FROM candidate_tbl
WHERE doneMasters = 'false'
GROUP BY name
HAVING COUNT(*) = 2

See the demo.
Results:
> | name |
> | :--- |
> | C    |
> | E    |
> | G    |

Note that a flag column like doneMasters would be better (for clarity, coding simplicity, less storage space and better performance) defined as BOOLEAN.
See the demo.

redpajama_set_name: RedPajamaStackExchange


text:
About Canada: Poverty
EPUB $17.99
Ebook via Retailer
For a country as wealthy as Canada, poverty is utterly unnecessary. In About Canada: Poverty, Jim Silver illustrates that poverty is about more than a shortage of money: it is complex and multifaceted and can profoundly damage the human spirit. At the centre of this analysis are Canada's neoliberal economic policies, which have created conditions that make a growing number of people vulnerable to low income, vanishing public services and poor physical health. Silver also highlights the ways in which poverty is intimately connected to colonialism and racial and gender discrimination, and finds that the political and economic policies enacted by the Canadian government serve only a powerful minority, while producing a range of negative outcomes for the rest of us, especially the poor. Silver points out that the costs of poverty — relating to health care, crime, education and unemployment — are higher than the costs of solving poverty, and he lays out an achievable strategy for its dramatic reduction in Canada. When poverty is understood as resulting from political choices, its elimination requires putting pressure on governments to ensure that different choices are made.
Jim Silver's impressive new book tells us about the complexity of why people are poor in Canada, a country so rich in resources. Canadians who are poor face many social challenges such as inadequate housing, poor health, racism, opportunities for employment, and low levels of education. Jim Silver's conclusion is challenging Canadians to demand our governments "to invest in poverty reduction solutions."
— Kathy Mallet, long-time community development activist in Winnipeg
"Jim Silver offers a frank and at times troubling picture of deepening poverty and rising inequality in Canada. His significant contribution is to motivate all citizens to demand available solutions to this national tragedy."
— Paul Moist, National President, Canadian Union of Public Employees (CUPE).
"Comprehensive and often provocative, this book represents a major contribution to our understanding of poverty in Canada. It explains the nature of poverty as variegated and complex, and acknowledges that poverty is gendered and racialized and that it has a spatial dimension."
— Grace-Edward Galabuzi, Department of Politics and Public Administration, Ryerson University
Canadian Politics, Public Policy, Ebooks
For sale worldwide
Forms of Poverty
Poverty by the Numbers
Neoliberalism and its Effects
Complex Poverty
The Costs of Poverty
Solutions that Work
Jim Silver is a professor in the Department of Urban and Inner-City Studies at the University of Winnipeg. His research interests are in inner-city, poverty-related, and community development issues. His most recent book is Solving Poverty: Innovative Solutions from Winnipeg's Inner City (Fernwood 2016). Jim is a member of the Manitoba Research Alliance and the leader of the Housing and Neighbourhood Revitalization stream of its SSHRC Partnership project, "Partnering for Change: Community-Based Solutions for Aboriginal and Inner-City Poverty."
Restless Ideas
Contemporary Social Theory in an Anxious Age
By Tony Simmons
In Defense of Julian Assange
Edited by Tariq Ali and Margaret Kunstler
Frontline Farmers
How the National Farmers Union Resists Agribusiness and Creates Our New Food Future
Edited by Annette Aurélie Desmarais
Beyond Market Dystopia
New Ways of Living
Edited by Leo Panitch and Greg Albo
The Desirable Sister
By Taslim Burkowicz
The Daughter Who Walked Away
By Kimia Eslah
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text:
tshirts4ukraine@gmail.com
Tshirts4Ukraine
I'm a 17-year-old High School Junior. My family and I are ordinary Americans from Colorado who are deeply troubled by Russia's brutal, unprovoked war on Ukraine. We couldn't stand by and do nothing while Russia's dictator Vladimir Putin tries to destroy a free and peaceful country and its people.
We can't go to Ukraine to fight. We can't stop countries from buying Russian oil. But we can sell t-shirts and donate 100% of net proceeds to the Bank of Ukraine. So that's what we're going to do.
Please feel free to watch the video below and thanks in advance.
CLICK THE IMAGE OF THE T-SHIRT BELOW TO PURCHASE
Home: Video
The shirts are high-quality, 100% Cotton, unisex Ts hand-screened in Fort Collins Colorado. Prices are $25 for adults. Students can use the discount code "student" for a $5 discount. Honor system on student discounts, please.
100% of net proceeds will go to one of two official fundraising accounts operated by the National Bank of Ukraine. One account is for humanitarian assistance, like food and medicine, and the other is for defensive supplies, including weapons and ammunition. You can choose which account you'd like to support with your t-shirt purchase. To choose the account, simply write "Humanitarian Aid" or "Defense" in the Notes section at Checkout. If you don't choose an account, the default will be to support the Ukrainian defense.
For me, what to put on the front of the t-shirt was easy. You might remember, on the first day of the war, February 24th, 2022 a Russian warship shelled a Border Guard outpost on Ukraine's tiny "Snake Island" in the Black Sea. The captain of the Russian ship radioed the 13 Ukrainian defenders and ordered them to surrender, or they would be destroyed.
The Ukrainians replied, in Russian, "Русский военный корабль, иди нахуй", or "Russian Warship, go fuck yourself".
If that doesn't tell you all you need to know about the courage and determination of the Ukrainian people, nothing can.
Below that legendary Ukrainian reply, also on the front of the t-shirt, is a map of Ukraine in the national blue and yellow colors.
The reverse of the shirt has the country's Coat of Arms and their national salute, written in Ukrainian, "Слава Україні" – "Glory to Ukraine".
Please consider supporting this fundraiser or giving directly to the Bank of Ukraine (See links below)
The hour has come and we Americans need to help Ukraine.
Official Bank of Ukraine Donation Portals:
For more information / media / bulk pricing, please contact us at: tshirts4ukraine@gmail.com
Transparency: We'll regularly update this website with sales and donation information. We're learning as we go here; this is our first website, let alone online store, so please bear with us as we "learn by doing" and iron out the kinks in our processes.
Please feel free to share this fundraiser with friends and family, and thanks for looking.
Glory to Ukraine
Ukraine Fundraising T-shirt
©2022 by Tshirts4Ukraine


text:
IAAPA 2021 Focused on Growth and Recovery From COVID Rollercoaster
TSNN News
Following an incredibly challenging time for the $55 billion global attractions industry, IAAPA Expo 2021, the industry's premier event, recently reconvened for the first time since 2019. Held Nov. 15-19 at the Orange County Convention Center in Orlando, Fla., the festive trade show and conference opened its doors to more than 28,000 global attractions industry professionals seeking to network, learn and buy.
Hosted by IAAPA, the largest international trade association for permanently located attractions, the event featured more than 870 exhibiting companies from around the world touting new rides, technology, products and services across more than 375,000 net square feet of indoor and outdoor expo space.
During the event, IAAPA officials voiced their appreciation for being able to bring the industry together face-to-face safely and successfully.
"[Our community is] thrilled to be back together in-person," said Hal McEvoy, president and CEO of IAAPA, whose members include professionals from amusement parks, theme parks, attractions, water parks, resorts, family entertainment centers, zoos, aquariums, science centers, museums, cruise lines, manufacturers and suppliers.
He continued, "The attractions industry is working tirelessly to come back stronger than ever, and we're excited to reflect and showcase that here. We have an outstanding lineup of special events, speakers and educational programming along with our amazing, diverse trade show floor. It's going to be a fantastic week."
From advanced ride engineering and virtual reality experiences to ticketing and security, the five-day show was packed with new products and technologies representing 25 product categories, while the event's robust education conference offered more than 90 educational experiences that included classroom sessions led by industry experts and behind-the-scenes tours at local Orlando attractions.
Attendees were also able to take part in more than 50 special events aimed at member constituencies and tracks, including the International Reception, Water Park Social and Family Entertainment Centers Reception.
Other highlights included:
News from Merlin Entertainments that the world's first Peppa Pig Theme Park (opening Feb. 24, 2022, in Florida) will be designed with all children in mind as a Certified Autism Center. The company also revealed one of its attraction's ride vehicles, designed specifically for increased accessibility for children with wheelchairs.
Fun Spot America's largest investment in any ride or attraction ever at its Atlanta location, with the addition of Arie Force One. On this steel roller coaster, riders will soar over 3,400 feet of track, towering 15 stories and "flying" at speeds up to 64 miles per hour.
A "hair-raising," jaw-dropping moment as the president of Ripley's Believe It or Not! shaved his "pandemic beard" and added it to the Guinness World Records contender for Largest Ball of Human Hair, while raising more than $34,000 for Give Kids The World (GKTW) Village.
The sold-out GM and Owners' Breakfast Nov. 17, featuring Richard A. Zimmerman, president and CEO of Cedar Fair Entertainment Company, and the Opening Reception Nov. 16, during which the 2021 IAAPA Brass Ring Award recipients and IAAPA Award winners, including IAAPA Hall of Fame inductees, IAAPA Service Awards and the IAAPA Young Professional of the Year, were celebrated.
A "Lunch and Learn" featuring Josh D'Amaro, chairman of Disney Parks, Experiences and Products, and Game Changer–From Spectator to Participant: The New Paradigm of Immersive Entertainment featuring Winston Fisher, CEO of AREA15 and partner, Fisher Brothers.
The event was estimated to bring approximately $79.6 million in economic impact to the Orlando community, according to OCCC officials.
"The successful turnout for IAAPA Expo 2021 is a testament to the resilience and continued growth and strength of the global attractions industry," McEvoy said. "It's been a phenomenal week, bringing together the innovators who are moving our industry forward and building the future of attractions."
IAAPA Expo will return to the OCCC Nov. 14-18, 2022. According to IAAPA officials, more than 200 exhibit contracts have already been signed for next year's show.
Check out video highlights from the show here.
Don't miss any event-related news: Sign up for our weekly e-newsletter HERE and engage with us on Twitter, Facebook, LinkedIn and Instagram!
CSR Ideas to Borrow from Las Vegas Market, Orbus, FPSA and IAAPA
The summer months have been anything but slow and leisurely when it comes to CSR activities in the trade show industry. Here's a roundup of event companies, associations and…
8 Trends from PCMA Survey of COVID-19 Impact on Event Professionals
A few weeks ago, PCMA sent out an email invitation to event planners and suppliers asking for participation in a survey on how they were reacting to the COVID-19 pandemic and…
Events Industry Council Names COVID-19 Business Recovery Task Force Members
Just over a week after announcing the creation of the APEX COVID-19 Business Recovery Task Force, the Events Industry Council has released a full list of task force members…
Global Exhibitions Day 2021 Celebrates Resiliency and Recovery
Global Exhibitions Day, set for June 2, offers an opportunity for the industry to remark about its uncanny resiliency and value, perhaps never more demonstrated than in the…
6 Ways to Build Leadership Skills to Emerge from COVID-19 Stronger Than Ever
Obviously, times are tough for the events industry right now. It's something we're all experiencing, feeling and grappling with as we wake up each morning. So many article …
UFI Report Projects Growth for Trade Shows in 2021
A new report from UFI, the Global Association of the Exhibition Industry, predicts that a comeback is in the offing for trade shows. Having polled 457 companies in 64…


text:
Posts Tagged 'saved the whales'
The Sequel Strikes Again
Recently, it was reported on Empire Online that Will Smith and Tommy Lee Jones are confirmed for Men in Black III and it will be a 3D film. The expected release date is May 30, 2011.
Since the late 1970's, there have been sequels galore. Notable examples are the Star Wars films and Star Trek films. Yet, it wasn't always like this. Up until the 1970's a film was made, it was a success or failure and that was that. The characters were never revisited and the story was not continued. Of course, the exception was the James Bond films. The movie going public in years past were willing to accept that Elsa never saw Rick again, Dorothy never returned to the land of Oz and while Scarlett said that she was going to get Rhett back, it was seen as wishful thinking on her part.
My theory is the establishment of episodic television shows lead a desire in the movie going public to know what happens next. It wasn't enough to have the main characters ride off into the sunset. People wanted to know if Luke Skywalker became a Jedi and defeated Darth Vader or if the crew of the Enterprise found Spock and saved the whales.[1]
I like sequels. There I said it. The Star Wars films are among my favorite movies. I saw Shrek and Shrek 2 and liked them both. I like the Star Trek films with the original television cast, the Next Generation cast and the J.J. Abrams reboot. Also, The Dark Knight was a blockbuster film that I felt was more thought provoking than most blockbuster films.
Of course, not every movie benefits from a sequel and after a while, the story gets old. The two Tim Burton Batman movies were great. Burton and Michael Keaton in the title role got the darkness and the tortured soul of Batman/Bruce Wayne right. When Joel Schumacher took over the franchise, the movies were just about some rich guy who dresses up and chases bad guys. Also, as many Trekkies know, some Star Trek films are better than others.
Still, the powers that be in Hollywood don't want to mess with a good formula. So, if a movie made lots of money, more often than not a sequel is in order. As the engineers say, "If it works, it works."
http://www.empireonline.com/news/feed.asp?NID=27639
[1] Star Wars Episodes IV-VI and Star Trek II & III respectively.
Tags:3D, Batman, blockbuster films, Bruce Wayne, Darth Vader, Dorothy, Elsa, Empire Online, Enterprise, episodic television shows, Hollywood, J.J. Abrams, James Bond films, Jedi, Joel Schumacher, Late 1970's, Luke Skywalker, Men in Black III, Michael Keaton, Next Generation, Oz, Rhett, Rick, saved the whales, Scarlett, Shrek, Shrek 2, Spock, Star Trek films, Star Wars films, The Dark Knight, Tim Burton, Tommy Lee Jones, Trekkies, Will Smith