Love is in the air - of course it is, it's St Valentine's Day!
The history of Valentine's Day--and the story of its patron saint--is shrouded in mystery. We do know that February has long been celebrated as a month of romance, and that St. Valentine's Day, as we know it today, contains vestiges of both Christian and ancient Roman tradition.


text:
There's No Elephant in This Mousehole: The Supreme Court Upholds State Court Jurisdiction Over Class Actions Brought Under the Securities Act of 1933
Caplin & Drysdale, Chartered
USA May 17 2018
On March 20, 2018, the Supreme Court issued a unanimous decision in Cyan, Inc. v. Beaver County Employees Retirement Fund,[1] a case concerning the Securities Litigation Uniform Standards Act of 1998 ("SLUSA"), which imposes certain limitations on class actions under the Securities Act of 1933 and the Securities Exchange Act of 1934. The Court resolved a split among state and federal courts[2] in affirming a California state court that held that SLUSA does not strip state courts of jurisdiction over class actions alleging only violations of the Securities Act of 1933.[3] The Court further held, after being asked by the United States appearing as amicus curiae, that SLUSA does not authorize defendants to remove such actions to federal court.[4]
Justice Kagan, writing for the court, first examined the interrelationship between the three relevant acts. The 1933 Act requires companies selling public securities to make disclosures and creates private rights of action, over which both federal and state courts have jurisdiction, for the failure to do so.[5] If brought in state court, such actions were not removeable to federal court by express provision of the Act.[6] The 1934 Act regulated the exchange of securities after their offering to the public, and could also be enforced through private right of action. However, in contrast to the 1933 Act, Congress gave federal courts exclusive jurisdiction over all actions arising from the 1934 Act.[7] Congress, in 1995, enacted the Private Securities Litigation Reform Act, which amended both the 1933 and the 1934 statutes.[8] The Reform Act modified certain procedures for litigating a securities action, but only when such an action was brought in federal court. Afterwards, more class actions were filed in state courts under state law.[9] In response, Congress enacted SLUSA, which "completely disallows (in both state and federal courts) sizable class actions that are founded on state law and allege dishonest practices respecting a nationally traded security's purchase or sale" and allows removal of such prohibited actions to federal court for dismissal.[10]
In the case before the Court, Cyan, a telecommunications company, and its officers were sued in California state court by three pension funds and an individual alleging that Cyan's initial securities offering was supported by false information, in violation of the 1933 Act.[11] Cyan moved to dismiss for lack of subject matter jurisdiction, arguing that SLUSA stripped the state courts of jurisdiction through its definition of a "covered class action," found in 15 U.S.C. §77p(f)(2).[12] The Court disagreed, holding that Cyan was attempting to read a major statutory alteration into a definitional provision, where such seismic alterations normally do not go, and in so doing was violating a well-established precept:
Congress does not make 'radical—but entirely implicit—change[s]' through 'technical and conforming amendments.' Director of Revenue of Mo. v. CoBank ACB, 531 U.S. 316, 324 (2001), 121 S. Ct. 941, 148 L. Ed. 2d 830 (2001) (internal quotation marks omitted). Or to use the more general (and snappier) formulation of that rule, relevant to all 'ancillary provisions,' Congress does not 'hide elephants in mouseholes.' Whitman v. Am. Trucking Assns., Inc., 531 U.S. 457, 468 (2001), 121 S. Ct. 903, 149 L. Ed. 2d 1 (2001).[13]
The Court noted that in the face of "recalcitrant statutory language, Cyan stakes much of its case on legislative purpose and history."[14] The Court countered Cyan's reasoning with the notion that the true aim of the reform act "does not depend on stripping state courts of jurisdiction over 1933 Act class suits."[15] Cyan's final argument that the "except clause would serve no purpose at all unless it works as Cyan says" was similarly rejected by the Court.[16] The Court noted that the "except" clause may have been intended as a failsafe provision, to ensure that state courts respected "SLUSA's new limitation on what they could decide."[17]
Finally, the Court responded to the Government's assertion as amicus curiae that any class action brought in state court under the 1933 Act could be removed to federal court.[18] "At bottom," the Court held, this argument "makes the same mistake as Cyan: It distorts SLUSA's text because it thinks Congress simply must have wanted 1933 Act class actions to be litigated in federal court."[19] The Court concluded that "SLUSA did nothing to strip state courts of their longstanding jurisdiction to adjudicate class actions alleging only 1933 Act violations" and likewise did not authorize removal of those suits to federal court.[20]
To view all formatting for this article (eg, tables, footnotes), please access the original here.
Caplin & Drysdale, Chartered - Todd E. Phillips, Kevin C. Maclay and Caroline E. Parke
Securities Act 1933 (USA)
IRS Takes Aim at Law Firms' Deferred Payment Arrangements for Contingency Cases *
Recent IRS activity focuses on nonqualified deferred compensation plans *
Recent Sixth Circuit Decision Clarifies Appealability of Bankruptcy Court Orders *
Recent Developments in FARA: Michael Cohen Hearing *
Foreign trusts: everything you wanted to know about the taxation of foreign trusts but were afraid to ask *
How-to guide How-to guide: Understanding key data protection definitions (UK)
Checklist Checklist: Delivery and acceptance of goods in a business-to-business sale of goods contract (USA) Recently updated
Checklist Checklist: Charitable and political donations (UK)

redpajama_set_name: RedPajamaCommonCrawl


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Sep 22 Drama Aside, It's OVER for Republicans on Defunding Obamacare
"Obamacare", "Tea Party", "establishment Republicans"
It was amusing to see the drama play out on today's Sunday morning TV shows. Needless to say, I enjoy few things more than seeing Republicans tying themselves into knots trying to insist that they don't want to shut down the government unless they can defund the Affordable Care, and attaching its defunding to the bill to keep the government running was just... I don't know, being cute, I guess. But for all the huffing and puffing from the Republicans and for all the pretense of "countdown to shutdown" from the media, the fight is over. Even Rand Paul (following his Tea Party nutty buddy Ted Cruz), has given up.
Paul, whose potential as presidential contender in 2016 has been a subject of much speculation, admitted that absent a sudden and total alignment behind an approach to tackling the health care law, his caucus' "disunity" means Republicans will most likely fall short of canceling, or even delaying, the law from going into effect.
"Leverage doesn't work unless people believe you'll actually do something," Paul said. "The fact that Democrats don't believe we'll do anything, in the end they'll get what they want and a bill will be cobbled together."
Sniffle, sniffle. The Republicans who would hold the line in the Senate on defunding Obamacare attached to a "must pass" bill have conceded defeat even before their body takes up the measure. House Republicans went on a limb to do it, all the while expecting exactly this outcome, of course. John Boehner and the House GOP leadership is not willing to fight this and take the hit for shutting down the government while their own party in the Senate abandons them at a time when there is little appetite in the country for defunding Obamacare.
Right now, there is absolute chaos in the Republican party and a fight between the far Right Tea Partiers and the GOP establishment that enabled it in the first place. The Tea Party base is incensed that the establishment is refusing to shut down the government over Obamacare, and the establishment is visibly distressed at the absolutism of the Tea Party base that they know is making the rest of the country absolutely despise the Republican party. But it's the establishment's fault, really. They were the ones that hatched the brilliant plan to win by obstructing Obama on everything and anything. Now they are complaining that their own base is upset they won't take it to the most absurd of the ends.
But I digress. My point is, the drama you see on TV is all for show. The government isn't going to shut down, and Obamacare is not going to be defunded. This isn't a fight that's "raging." It's already been conceded by not just the people who would have to stick by it to make it a real fight but by those who initiated it.
Obamacare is here to stay, and there is not a damn thing the Republicans or the Tea Party can do about it.
Sep 24 Timing is Everything: How Ted Cruz's Government Shutdown Gambit is Helping Democrats
Sep 21 First Syria, Now Iran: A Fair but Resolute President at the Center of Diplomatic Triumphs
Oct 5 "Slimdown?" "Slowdown?" Republicans in total disarray over #GOPshutdown
Aug 7 2010 vs. 2014: Obamacare and the Fall of the Tea Party
Jul 15 Republicans Squeal as Obama Again Challenges GOP on Grand Bargain


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Q: Firebase data structure: Attending school on date I am currently looking into my first application using Firebase as the backend.
I have 2 models, School and User. Each user can sign up for a date to attend the school, so I also need a Date.
A SQL table would look like this:
schools: id, name
users: id, name, email
schools_users: id, school_id, user_id, date
What would be the proper way of designing this data structure in Firebase?

A: Since you don't specify any requirements, I suggest starting with the most naive mapping at first:
root
    schools
        1: "name of school1"
        2: "name of school2"
    users:
        1: { "name": "Maeh", "email": "2523229@stackoverflow.com" }
        2: { "name": "Frank", "email": "209103@stackoverflow.com" }
    schools_users:
        1_1: "20141031"
        1_2: "20130102"

redpajama_set_name: RedPajamaStackExchange


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\section{Introduction and Preliminaries}
\label{one}
Subgroup depth $d_0(H,G)$ of a subgroup $H$ in a finite group $G$ is introduced in \cite{BKK} as the minimum depth of the induction-restriction table of irreducible characters of $H$ and $G$, a matrix
of nonnegative integers with nonzero rows and columns.  As the matrix of induction $K_0(\C H) \rightarrow K_0(\C G)$, the notion of depth also occurs in fields of topological algebra in various guises.  In ring theory, the minimum depth $d(B,A)$ of a subring $B \subseteq A$ is introduced in \cite{BDK} in terms of the natural bimodule ${}_BA_B$ and its tensor powers (and for even depth, tensored one more time by
the bimodule ${}_BA_A$ or ${}_AA_B$).  The depth $d_0(H,G)$ is recovered in \cite{BDK} by letting
$B$, $A$ be group algebras over a field of characteristic zero; indeed, \cite{BDK} shows that depth $d(kH,kG)$  depends only on the characteristic of the field $k$ \cite{BDK}.

The more general definition also allows one to consider depth of integral group ring extensions $\Z H \subseteq \Z G$ and their minimum depth $d_{\Z}(H,G)$.
In addition, a combinatorial depth $d_c(H,G)$ is introduced in \cite{BDK} by using G-set
analogues of balanced tensors and bimodules.  The following string of inequalities is from \cite[4.5]{BDK}:  $$d_0(H,G) \leq d_p(H,G) \leq d_{\Z}(H,G) \leq d_c(H,G) \leq 2|G: N_G(H) |.$$

In \cite{LK2011}, h-depth of a subring pair $B \subseteq A$ is introduced by the same process as in the definition of depth but focussing on the natural $A$-$A$-bimodules of the tensor powers of $A$ relative to $B$.   The minimum h-depth $d_h(B,A)$
is closely related to $d(B,A)$ by the inequality $| d(B,A) - d_h(B,A) | \leq 2$.  Its definition  in~\ref{def-depth} suggests the notion of h-depth is  natural and almost unavoidable when considering subgroup depth.  In this paper,  h-depth provides a natural transition (in Section~2) to depth of an $A$-coring, which is a notion of coalgebra generalized to $A$-bimodules
where the comultiplication and counits are $A$-bimodule morphisms.

In \cite{D} the subring depth of twisted group algebras of the permutation groups $\Sigma_n$
are computed as an intriguing contrast to  the untwisted case in \cite{BKK}, where it was shown that $d_0(\Sigma_n, \Sigma_{n+1}) = 2n-1$.
In \cite{D} it was shown that with $\alpha$ the nontrivial $2$-cocycle  (representing the nonzero element in  $H^2(\Sigma_n, \C^{\times})$), the twisted complex group algebra extensions have minimum depth
\begin{equation}
\label{eq: Danz}
d(\C_{\alpha}\Sigma_n, \C_{\alpha}\Sigma_{n+1}) = 2(n - \lceil \frac{\sqrt{8n+1} - 1}{2} \rceil) + 1.
\end{equation}

Note that $d(\C_{\alpha}\Sigma_n,\C_{\alpha}\Sigma_{n+1}) \leq d_0(\Sigma_n, \Sigma_{n+1})$
with a difference that goes to infinity as $n \rightarrow \infty$. The same is true of the alternating group series $A_n$, where  $d_0(A_n,A_{n+1}) = 2(n - \lceil \sqrt{n}\, \rceil) +1$ $\geq d(C_{\alpha}A_n, \C_{\alpha}A_{n+1})$, the last depth also equal to the right-hand side of Eq.~(\ref{eq: Danz}); cf.\ \cite[appendix]{BKK}.
Given a subgroup $H \leq G$, we show in this paper that  the crossed product of a twisted $G$-algebra $A$  with subalgebra
$A \#_{\sigma} H$ has  h-depth less than or equal to the h-depth of the corresponding group algebra extension:
i.e., we establish
$$d_h(A \#_{\sigma} H,A \#_{\sigma} G) \leq d_h(kH, kG)$$
in Eq.~(\ref{eq: hdepthineq}) in Section~4 below.
We will also extend an equality for h-depth of a Hopf subalgebra in terms of depth of its quotient module \cite{K2013} to the  equalition
for h-depth of a left coideal subalgebra of a Hopf algebra $H$ in  Corollary~\ref{cor-cue}.
Our method is to define depth of corings so that the depth of the Sweedler coring of a ring extension is its h-depth, and apply (Doi-Koppinen) entwining structures that are Galois corings.

 In Corollary~\ref{cor-core} below we show that $d_h(H,G) = d_h(G/N,H/N)$ where $N$ is the core of a subgroup $H$
in a finite group $G$ (over any ground field).  In Corollary~\ref{cor-cores} we note that subgroup depth  behaves precisely like combinatorial depth with repect to this subgroup $N$, after proving
that subring depth is preserved by quotienting of relatively nice ideals (Theorem~\ref{prop-sigma}), or
better yet, by
relatively nice Hopf ideals (Proposition~\ref{prop-crop}).  A final application is to a left coideal subalgebra $R$ of a finite-dimensional Hopf algebra $H$, which is left normal iff a nonzero right integral in $R$ is a normal element in $H$ (Theorem~\ref{th-leftrightleft}).

\subsection{Preliminaries on subalgebra depth} Let $A$ be a unital associative algebra over a field $k$. The category of  modules over $A$ will be denoted
by $\M_A$.  (For finite-dimensional $A$, the notation $\M_A$ denotes the category of finite-dimensional modules.)
Two modules $M_A$ and $N_A$ are \textit{similar} (or H-equivalent) if $M \oplus * \cong q \cdot N := N \oplus \cdots \oplus N$ ($q$ times) and $N \oplus * \cong r \cdot M$
for some $r,q \in \N$.  This is briefly denoted by $M \| q \cdot N$ and $N \| r \cdot M$ for some $q,r \in \N$  $\Leftrightarrow$ $M \sim N$. Recall that similar modules have Morita equivalent endomorphism rings.

 Let $B$ be a subalgebra of $A$ (always supposing $1_B = 1_A$).  Consider the natural bimodules ${}_AA_A$, ${}_BA_A$, ${}_AA_B$ and
${}_BA_B$ where the last is a restriction of the preceding, and so forth.  Denote the tensor powers
of ${}_BA_B$ by $A^{\otimes_B n} = A \otimes_B \cdots \otimes_B A$ ($n$ times $A$) for $n = 1,2,\ldots$, which is also a natural bimodule over  $B$ and $A$ in any one of four ways;     set $A^{\otimes_B 0} = B$ which is only a natural $B$-$B$-bimodule.

\begin{definition}
\label{def-depth}
Suppose that $A^{\otimes_B (n+1)}$ is similar to $A^{\otimes_B n}$ as natural $X$-$Y$-bimodules
for subrings $X$ and $Y$ of $A$ and $n \in \N$.  One says that subalgebra $B \subseteq A$ has
\begin{itemize}
\item \textbf{depth} $2n+1$ if $X = B = Y$ for $n \geq 0$;
\item \textbf{left depth} $2n$ if $X = B$ and $Y = A$,
\item \textbf{right depth} $2n$ if $X = A$ and $Y = B$,
\item \textbf{h-depth} $2n-1$ if $X = A = Y$,
\end{itemize}
 for $n \geq 1$.

Note that if $B \subseteq A$ has h-depth $2n-1$, the subalgebra has (left or right) depth $2n$ by restriction of modules.  Similarly, if $B \subseteq A$ has depth $2n$, it has depth $2n+1$.  If $B \subseteq A$ has depth $2n+1$, it has depth $2n+2$ by tensoring either $-\otimes_B A$ or $A \otimes_B -$ to $A^{\otimes_B (n+1)} \sim A^{\otimes_B n}$.     Similarly, if $B \subseteq A$ has left or right depth $2n$, it has h-depth $2n+1$.  Denote the \textbf{minimum depth}
of $B \subseteq A$  by $d(B,A)$ \cite{BDK}.  Denote the \textbf{minimum h-depth} of $B \subseteq A$ by $d_h(B,A)$ \cite{LK2011}.  Note that
$d(B,A) < \infty$ $\Leftrightarrow$ $d_h(B,A) < \infty$; if so, \cite{LK2011} shows that
\begin{equation}
-2 \leq  d(B,A) - d_h(B,A) \leq 1.
\end{equation}
\end{definition}

 For example, $B \subseteq A$ has depth $1$ iff ${}_BA_B$ and ${}_BB_B$ are similar \cite{BK2, LK2012}.  In this case, one deduces the following algebra isomorphism,
\begin{equation}
\label{eq: d1eq}
A \cong B \otimes_{Z(B)} A^B,
\end{equation}
 where
$Z(B), A^B$ denote the center of $B$ and centralizer of $B$ in $A$.

Another example is that $B \subset A$ has right depth $2$ iff ${}_AA_B$
and ${}_A A \otimes_B A_B$ are similar.  If $A = \C G$ is a group algebra of a  finite group $G$ and $B = \C H$ is a group algebra of a subgroup $H$ of $G$, then $B \subseteq A$ has right depth $2$ iff $H$ is a normal subgroup of $G$ iff $B \subseteq A$ has left depth $2$; a similar statement is true for a Hopf subalgebra $R \subseteq H$ of finite index and over any field \cite{BK}.

 Now let $H$ be a Hopf algebra with counit $\eps: H \rightarrow k$, antipode $S: H \rightarrow H$ and
coproduct $\cop: H \rightarrow H \otimes H, \cop(h) = h\1 \otimes h\2$ (Sweedler notation suppressing summations).  Let $A$ a \textit{right $H$-comodule algebra}, i.e., an $H$-comodule
with coaction $\rho: A \rightarrow A \otimes H, \rho(a) = a\0 \otimes a\1$ that is a unital algebra homomorphism.  The
coinvariants $\{ b \in A \, | \, \rho(b)  = b \otimes 1_H \}$ form a subalgebra
$B$; one says $A$ is an \textit{$H$-Galois extension} of $B$ if the Galois mapping $A \otimes_B A \rightarrow A \otimes H$ given by $x \otimes y \mapsto  x y\0 \otimes y\1$ is bijective.  Note that
the Galois mapping is an isomorphism of natural $A$-$B$-bimodules:  if $\dim H = n$,
then $A \otimes_B A \cong n \cdot {}_AA_B$, and $A \supseteq B$ has depth $2$.
There is the following remarkable converse growing out of Ocneanu's  ideas in subfactor theory with detailed papers by Szymanski, Longo, Nikshych-Vainerman and others:

  \begin{theorem}
Let $A \supseteq B$ be a Frobenius algebra extension with $1$-dimensional centralizer and surjective Frobenius homomorphism.  If $d(B,A) \leq 2$,
then $A$ is a Hopf-Galois extension of $B$.
\end{theorem}

The notion of Frobenius extension is defined in Example~\ref{example-Frobenius}; a short proof-with-references in \cite{LK2004}.  The surjectivity condition ensures
that $A_B$ is a generator (and conversely \cite{LK2012}).    This theorem requires particularly the use of the depth two condition for the construction of a Hopf algebra
structure on $\End {}_BA_B$ or its dual Hopf algebra structure on $\End {}_AA \otimes_B A_A \cong (A \otimes_B A)^B$ \cite{KS}.  (However,  if $d(B,A) = 1$, none of these difficulties arise, as Eq.~(\ref{eq: d1eq}) forces $A = B$ with one-dimensional center, the trivial Hopf algebra.)  The stringent condition on the centralizer $A^B$ may be relaxed if
one considers more general Hopf algebras and their Galois coactions (such as weak Hopf algebras and
Hopf algebroids) \cite{KS}.

Note that one always has
\begin{equation}
\label{eq: divides}
A^{\otimes_B n} \| A^{\otimes_B (n+1)}
\end{equation} as natural $A$-bimodules for all $n \geq 2$: one applies the unit mapping and multiplication to obtain a split monic (or split epi). For  $n =1$, though it holds for the other three of the bimodule structures, it
is not generally true  as $A$-$A$-bimodules,  $A \| A \otimes_B A$ being the separable extension condition on $B \subseteq A$.
Now $A \otimes_B A \| q \cdot A$ as $A$-$A$-bimodules for some $q \in \N$
is the H-separability condition and implies $A$ is a separable extension of $B$ by Hirata, cf.\ \cite[2.6]{NEFE}.  Somewhat similarly, ${}_BA_B \| q  \cdot {}_BB_B$ implies
${}_BB_B \| {}_BA_B$ \cite{LK2012}. It follows that subalgebra depth and h-depth may be equivalently defined by replacing the similarity bimodule conditions for depth and h-depth in Definition~\ref{def-depth} with the corresponding bimodules on
\begin{equation}
\label{eq: def}
A^{\otimes_B (n+1)} \| q \cdot A^{\otimes_B n}
\end{equation}
 for some positive integer $q$ \cite{BDK, LK2011, LK2012}.

For example, for the permutation groups $\Sigma_n < \Sigma_{n+1}$
and their corresponding group algebras over any
commutative ring $K$, one has depth $d_K(\Sigma_n,\Sigma_{n+1}) = 2n-1$ \cite{BDK}. Depths of subgroups in $PGL(2,q)$, Suzuki groups, twisted group algebra extensions and Young subgroups of $\Sigma_n$ are computed in \cite{F, HHP, D, FKR}.    If $B$ and $A$ are semisimple complex algebras, the minimum odd depth is computed from powers of an order $r$ symmetric matrix with nonnegative entries $\mathcal{S} := MM^T$ where $M$ is the inclusion matrix
$K_0(B) \rightarrow K_0(A)$ and $r$ is the number of irreducible representations of $B$ in a basic set of $K_0(B)$; the depth is $2n+1$ if $\mathcal{S}^n$ and $\mathcal{S}^{n+1}$ have an equal number of zero entries \cite{BKK}. It follows that the subalgebra pair of semisimple complex algebras $B \subseteq A$   always has finite depth.

Similarly, the minimum h-depth of $B\subseteq A$ is computed from
powers of an order $s$ symmetric matrix $\mathcal{T} = M^TM$, where $s$ is the rank of $K_0(A)$; the h-depth is $2n+1$ if $\mathcal{T}^n$ and $\mathcal{T}^{n+1}$ have an equal number of zero entries (equivalently, letting $\mathcal{T}^0 = I_{s \times s}$, one has
\begin{equation}
\label{matrixinequality}
\mathcal{T}^{n+1} \leq q\mathcal{T}^n
\end{equation}
 for some $q \in \N$) \cite[Section 3]{LK2012}.

\subsection{Depth of Hopf subalgebras, coideal subalgebras and modules in a tensor category $\mathcal{M}_H$}  Let $ H$ be a l Hopf algebra over an arbitrary field $k$. Let $R \subseteq H$ be a Hopf subalgebra, so that $\cop(R) \subseteq R \otimes R$ and the antipode satisfies $S(R) = R$.   It was shown in \cite[Prop.\ 3.6]{K2013} that the tensor powers  of $H$ over $R$, denoted by $H^{\otimes_R n}$,  reduce to tensor powers of the generalized quotient $Q := H/ R^+H$ as follows:
\begin{equation}
\label{eq: diagaction}
H^{\otimes_R n} \stackrel{\cong}{\longrightarrow} H_{\cdot} \otimes Q_{\cdot}^{\otimes (n-1)}
\end{equation} which
for $n = 2$ is given by $x \otimes_R y \mapsto xy\1 \otimes \overline{y\2}$; see \cite[Eq. (21)]{HKY} for the
straightforward extension of this to all $n$.
 The isomorphism  is an $H$-$H$-bimodule isomorphism where the left $H$-module structures are the natural endpoint actions (as well as the right action to the left), and the right $H$-module structure on $H \otimes Q \otimes \cdots \otimes Q$ is given by the diagonal action of $H$:
$$h'(y \otimes q_1 \otimes \cdots \otimes q_{n-1}) \cdot h = h'yh\1 \otimes q_1 h\2 \otimes \cdots \otimes q_{n-1} h_{(n)}.$$

The following proposition directly makes use of the isomorphism for each $n \geq 2$.  Given a subalgebra pair $U \supseteq T$, observe that the bimodule
${}_UU_T$ is projective iff the multiplication mapping $U \otimes T \rightarrow T$, $u \otimes t \mapsto ut$, is a split
epi of $U$-$T$-bimodules iff there is an element $e = e^1 \otimes e^2 \in U \otimes T$ such
that $e^1 e^2 = 1_U$ and $te = et$ for each $t \in T$ (a so-called ``right relative separable tower'' of algebras
$U \supseteq T \supseteq k1_U$).

\begin{prop}
Suppose $H$ is a finite-dimensional Hopf algebra with $R$ and $Q$ as above, with intermediate
Hopf subalgebras $H \supseteq U \supseteq T \supseteq R$.
If ${}_UU_T$ is a projective bimodule, then $d(R,H) < \infty$. In particular, the depth is finite if either $H$ or $R$ is semisimple \cite{LK2012, K2013}.
\end{prop}
\begin{proof}
In a finite tensor category, such as $\mathcal{M}_H$ with  the diagonal action, $P \otimes_k X$ is projective if  $P$ is projective and $X$ is any $H$-module \cite[Prop.\ 2.1]{EO} (its proof does not need $k$ to be algebraically closed).
The proof below will also require the notion of  tensor product Hopf algebra $H \otimes K$ of two Hopf algebras $H$, $K$, as well as the Hopf opposite algebra (with antipode $S^{-1}$) \cite{Ma}.    Then Eq.~(\ref{eq: diagaction}) shows (by restriction) that
each $H^{\otimes_R n}$ is a projective $U$-$T$-bimodule, equivalently projective $H \otimes T^{\rm op}$-module, for one  extends $Q$ to an  (left-sided trivial) $U$-$T$-bimodule via $uqt = \eps(u)qt$.  Since $R^e = R \otimes R^{\rm op}$ is a Hopf subalgebra in the finite-dimensional $U \otimes T^{\rm op}$, and therefore a free extension, it follows that each $H^{\otimes_R n}$ is projective
as a natural $R$-$R$-bimodule.
 The Krull-Schmidt Theorem, Eq.~(\ref{eq: divides}) and the fact that there are finitely many projective indecomposable isoclasses entail that $H^{\otimes_R (N+1)} \sim H^{\otimes_R (N+2)}$ for $N$ equal to this number (or the number of nonisomorphic simple $H$-$R$-bimodules).  Thus, $d(R,H) \leq 2N + 3$.

A semisimple Hopf algebra $H$ (or $R$) is a  separable algebra, separability   being characterized by the condition ${}_{H^e}H$ (or ${}_RR_R$)  is
projective.  The finite depth follows by letting $U = T = H$ (or $U = T = R$).
\end{proof}

 The bimodule isomorphism in Eq.~(\ref{eq: diagaction}) shows quite clearly
that the following definition applied to $Q$ will be of interest to computing $d(R,H)$.  Let $W$ be a right $H$-module and
$T_n(W) :=
 W \oplus W^{\otimes 2} \oplus \cdots \oplus W^{\otimes n}$.

\begin{definition}
A module $W$ over a Hopf algebra $H$ has \textbf{depth} $n$ if $T_{n+1}(W) \| t \cdot T_n(W) $
for some positive integer $t$,
and depth $0$ if $W$ is isomorphic to a direct sum of copies of $k_{\eps}$, where $\eps$ is the counit.
Note that this entails that $W$ also has depth $n+1$, $n+2$, $\ldots$.  Let $d(W, \M_H)$ denote
its \textbf{minimum depth}.  If $W$ has a finite depth, it is said to be an \textit{algebraic module}.
If $W$ is an $H$-module coalgebra, or $H$-module algebra, the condition of depth $n$ simplifies
to $W^{\otimes (n+1)} \| t \cdot W^{\otimes n}$ for some $t \in \Z_+$ and all $n \in \N$,  where
$W^{\otimes 0}$ denotes $k_{\eps}$.
  \end{definition}

\begin{lemma}
\label{lemma-modI}
Let  $I$ be a Hopf ideal in a Hopf algebra $H$.  Suppose $I$ is contained in the annihilator ideal of an $H$-module $W$.
Then depth of $W$ is the same over $H$ or $H/I$.
\end{lemma}
\begin{proof}
The lemma is proven by noting that a Hopf ideal $I$ in $\Ann_H W$ is contained in the annihilator ideal of each tensor power of $W$,  since $I$ is a coideal.  Additionally, split epis as in $T_{n+1}(W) \| t \cdot T_n(W) $ descend and lift along $H \rightarrow H/I$.
\end{proof}

Recall that the Green ring, or representation ring, of a finite-dimensional Hopf algebra $H$ over a field $k$, denoted by $A(H)$, is the free abelian group with basis consisting of  indecomposable (finite-dimensional) $H$-module isoclasses,
with addition given by direct sum, and the multiplication in its ring structure given by the tensor product.  For example,
$K_0(H)$ is a finite rank ideal in $A(H)$.
As shown in \cite{Fe}, a finite depth $H$-module $W$ satisfies a polynomial with integer coefficients in $A(H)$, and conversely.  Thus, an algebraic module has isoclass an algebraic element in the Green ring,
which explains the terminology.

The main theorem in \cite[5.1]{K2013} proves that Hopf subalgebra (minimum) depth and depth of its generalized quotient $Q$ are closely related by
\begin{equation}
\label{eq: inequalityfordepth}
2d(Q,\M_R) + 1 \leq d(R,H) \leq 2d(Q, \M_R) + 2.
\end{equation}
Here one restricts $Q$ to an $R$-module, in order to obtain the better result on depth. If $R$ is a left coideal subalgebra of $H$, it is not itself a Hopf algebra, and this as a result is unavailable: in this case, we prove below (Corollary~\ref{cor-cue}) that the minimum h-depth satisfies
\begin{equation}
\label{eq: hdepthandmoduledepth}
  d_h(R,H) = 2d(Q, \M_H) + 1.
\end{equation}

\begin{example}
\begin{rm}
For the reader who knows something about the Drinfeld double Hopf algebra $D(H)$ of a Hopf algebra $H$ (or group $G$),
we work an example using the notation of $h \in H$ and its dual Hopf subalgebra $f \in H^*$ as subalgebras of $fh \in D(H)$ subject to relations $hf = f(S^{-1}(h\3) ? h\1) h\2$ \cite{M,Ma}.  We compute the quotient $Q$ of (coopposite) $H^*$ as a Hopf subalgebra of $D(H)$ simply by $$Q = H^* H/{H^*}^+H \cong H$$
via $\overline{fh} \mapsto f(1_H)h$ with right $H^*$-action $\overline{h}f =  f(S^{-1}(h\3)h\1) h\2$.  This gives $\overline{h} f =  \overline{h} f(1_H)$ if $H$ is cocommutative.

Thus if $H$ is cocommutative, $d(Q, \M_{H^*}) = 0$, and by the inequality in (\ref{eq: inequalityfordepth}),
$$ d(H^*, D(H)) \leq 2,$$
i.e., $H^*$ is a normal Hopf subalgebra in $D(H)$ (recovering some folklore with a hands-off approach). Also by Eq.~(\ref{eq: hdepthandmoduledepth}), $d_h(H^*,D(H)) = 3$
since $Q_{D(H)} \cong  H_H$ the regular representation, which has depth $1$ if $\dim H > 1$ \cite{K2013}.  Minimum depth $d(G, D(G))$ is analyzed in \cite{HKY}.
\end{rm}
\end{example}

Finally we remark that if $H$ is semisimple, Eq.~(\ref{eq: hdepthandmoduledepth}) in principle computes the depth of the quotient module $Q$
in the finite tensor category $\M_H$ in terms of the symmetric matrix $\mathcal{T}$ in inequality~(\ref{matrixinequality}) (where $A = H$ and $B = R$ is semisimple
\cite[ch.\ 3]{M}): let
$d'(\mathcal{T})$ denote the least $n$ for which inequality~(\ref{matrixinequality}) holds,  then
\begin{equation}
d(Q, \M_H) = d'(\mathcal{T})
\end{equation}
Thus if
$M$ denotes the inclusion matrix $K_0(R) \rightarrow K_0(H)$, then  $2d'(\mathcal{T})+1 = d_{odd}(M^T)$ (cf.\ \cite[3.2]{LK2011}) in terms of the minimum (odd) depth of an irredundant nonnegative matrix $M$ (and its transpose $M^T$) defined in \cite[2.1]{BKK}.
\subsection{Depth of a group subalgebra pair compared with its corefree quotient pair}
Given a ring $A$ with subring $B$, say that an $A$-ideal is
\textit{relatively nice} if its intersection, the $B$-ideal $J = I \cap B$, satisfies $AJ = I$ or $JA = I$. Note that if both set equalities hold, such as when I is a relatively nice Hopf ideal in a finite-dimensional Hopf algebra with respect to a Hopf subalgebra, we are studying a type of normality condition related to the normality notion in  \cite[Section 4]{BKK}.
Noting the canonical inclusion $B/J \into A/I$, we extend   Theorem 3.6 in \cite{KY}, where  $I$ is fully contained in $B$, to relatively nice ideals.

\begin{theorem}
\label{prop-sigma}
Let $I$ be a relatively nice ideal in $A$, where $J$ is its intersection with $B$. Minimum depth satisfies $d(B/J, A/I) \leq d(B, A)$.
\end{theorem}
\begin{proof}
If $d(B,A) = \infty$, there is nothing to prove.
Assume that there is a split monomorphism $\sigma: A^{\otimes_B (n+1)} \into q \cdot A^{\otimes_B n}$ of (natural $B$- or $A$-) bimodules for some $q, n \in \N$,
where we use balanced multilinear notation for the arguments $a_i \in A$ ($i = 1, \ldots, n+1$).  Let $\pi$ denote the canonical surjection $A \rightarrow A/I$,
where $\pi(a_i) = \overline{a_i}$.
Define $$\overline{\sigma}(\overline{a_1},\ldots,\overline{a_{n+1}}) = q \cdot \pi^{\otimes n}\sigma(a_1,\ldots,a_{n+1}) = (\pi^{\otimes n}\sigma_i(a_1,\cdots,a_{n+1}))_{i=1}^q. $$
We must show that for any $i = 1,\ldots,q$ and $y \in I$, $$\pi^{\otimes n}\sigma_i(a_1, \cdots,y,\cdots,a_{n+1}) = 0$$ to see that $\overline{\sigma}$ is well-defined.
Suppose without loss of generality that $I = JA$, so there are finitely many $x_{j_{r+1}} \in J$ and $a_{j_{r+1}}\in A$ such that $y = \sum_{j_{r+1}} x_{j_{r+1}}a_{j_{r+1}}$, where we note  $$\sigma_i(a_1, \cdots,a_r, y,\ldots,a_{n+1}) =
\sum_{j_{r+1}}\sigma(a_1,\ldots, a_rx_{j_{r+1}},a_{j_{r+1}},a_{r+1}, \ldots,a_{n+1}),$$
but $a_r x_{j_{r+1}} \in I = JA$. Then this process may be repeated $r$ times, until we have the terms
$$\sigma_i(a_1,\ldots,y,\ldots,a_{n+1}) = $$
$$\ \ \ \ \ \sum_{j_1,\cdots,j_{r+1}}^N x_{j_1 \cdots j_{r+1}}\sigma(a_{j_1 \cdots j_{r+1}},\ldots,a_{j_r j_{r+1}}, a_{j_{r+1}},a_{r+1},\ldots,a_{n+1})$$
where $x_{j_1 \cdots j_{r+1}} \in J$, all terms mapping to zero under $\pi^{\otimes n}$.  Similarly a splitting for $\sigma$ descends to $q \cdot (A/I)^{\otimes_{B/J} n}
\rightarrow (A/I)^{\otimes_{B/J} (n+1)}$, a splitting for $\overline{\sigma}$.    The inequality of minimum depths follows as a consequence of the characterization of depth in (\ref{eq: def}).
\end{proof}

\begin{remark}
\begin{rm}
The proof of Theorem~\ref{prop-sigma} may be adapted slightly to show that relative Hochschild cochain groups of a subring pair
$B \subseteq A$ with coefficients in an $A$-bimodule $M$ \cite{H} with left and right annihilator containing a relatively nice ideal $I$
in $A$ satisfies $C^n(A,B;M) \cong C^n(A/I,B/J; M)$ induced by the canonical algebra epi $A \rightarrow A/I$,
a cochain isomorphism for all $n \geq 0$.
\end{rm}
\end{remark}

For the following corollary, assume  $I$ is a relatively nice Hopf ideal in
a Hopf algebra  $H$.
\begin{prop}
\label{prop-crop}
The minimum depth satisfies $$d(R/J, H/I) \leq d(R,H) \leq d(R/J, H/I) + 1.$$ Moreover, if $d(R/J,H/I)$
is even, the two minimum depths are equal.
\end{prop}
\begin{proof}
Note that $J = R \cap I$ is a Hopf ideal in $R$ and satisfies both $HJ = I = JH$ by an application of the antipode.
 Let $Q = H/R^+H$ be the quotient module of $R \subseteq H$, and
$Q'$ be the corresponding quotient module of $R/J \subseteq H/I$.  Since $J \subseteq R^+$ and so $I \subseteq R^+H$,
it follows from a Noether isomorphism theorem that $Q' \cong Q$ as $H$-modules.  Since $(H/R^+H)J = I/R^+H = 0$, it follows
from the lemma above that $d(Q, \M_R) = d(Q, \M_{R/J})$.  From Eq.~(\ref{eq: inequalityfordepth})
it follows that $$2d(Q,\M_R) + 1 \leq d(R,H), d(R/J, H/I) \leq 2d(Q, \M_R) + 2. $$ The proposition now
follows from Theorem~\ref{prop-sigma}.

If $d(R/J,H/I)$ is even, then $d(R/J,H/I) = 2 d(Q, \M_{R/J}) + 2$ from Eq.~(\ref{eq: inequalityfordepth}), but the Theorem and Eq.~(\ref{eq: inequalityfordepth}) imply that
$d(R,H) = d(R/J, H/I)$.
\end{proof}

\begin{example}
\label{example-8dim}
\begin{rm}
Let $H_8$ be the $8$-dimensional small quantum group (at the root-of-unity $q = i$).  This is generated as an algebra by $K,E,F$ such that $K^2 = 1$, $E^2 = 0 = F^2$,
$EF = FE$, $FK = - KF$, and $EK = - KE$.  Let $R_4$ be the $4$-dimensional Hopf subalgebra
generated by $K,F$, which is isomorphic to the Sweedler algebra. (Please refer to \cite[Example 4.9]{K2013} for the coalgebra structure and details related to depth.)
Consider the relatively nice Hopf ideal $I$ with basis $\{ F, FK, EF, EFK \}$.  Then $J$ is $\mbox{rad}\, R$ with basis $\{ F, FK \}$.  The quotient Hopf algebras $H_8/I \cong R_4$
and $R_4/J \cong \C \Z_2$ have depth $d( \C \Z_2, R_4) = 3$ computed in \cite[4.1]{Y}.  It follows from the proposition that
\begin{equation}
3 \leq d(R_4, H_8) \leq 4.
\end{equation}
\end{rm}
\end{example}

Recall that the core $\mbox{\rm Core}_G(H)$ of a subgroup pair of finite groups $H \leq G$ is the intersection of conjugate subgroups of $H$; equivalently, the largest normal subgroup contained in $H$.
Note that if $N = \mbox{\rm Core}_G(H)$, then $\mbox{\rm Core}_{G/N}(H/N)$ is the one-element group, i.e., $H/N \leq G/N$ is a \textit{corefree} subgroup.
\begin{cor}
\label{cor-cores}
Suppose $N$ is a normal subgroup of a finite group $G$ contained in a subgroup $H \leq G$. For any ground field $k$, ordinary depth satisfies the inequality,
$$ d_k(H/N, G/N) \leq d_k(H,G) \leq d_k(H/N, G/N) + 1. $$
Moreover, if $d_k(H/N, G/N)$ is even, the two minimum depths are equal.
\end{cor}
\begin{proof}
  Note that $I = kGkN^+$ is a relatively nice Hopf ideal in $kG$ (generated by $\{ g - gn \| \, g \in G, n \in N \}$). (And  conversely any Hopf ideal of the group algebra $kG$ is of this form \cite{PQ}.)
Also note $kG/I \cong k[G/N]$.  The corollary
now follows from the previous proposition.
\end{proof}

This result  improves \cite[Prop.\ 3.5]{HKY} to arbitrary characteristic.  Note too that combinatorial depth satisfies a completely analogous property  given in
\cite[Theorem 3.12(d)]{BDK}. The inequality is in a sense the best
result obtainable.
For example, let $G = \Sigma_4$, $H = D_8$, the dihedral group of
$8$ elements embedded in $G$, and $$N = \{ (1), (12)(34), (13)(24), (14)(23) \}.$$ It is noted after \cite[6.8]{BKK} that $G/N \cong \Sigma_3$,
$H/N \cong \Sigma_2$, and that minimum depth  satisfies  $d_0(G,H) = 4$ \cite[Example 2.6]{BKK}, $d_0(H/N, G/N) = 3$.

\section{Depth of corings}
\label{two}
  Let $A$ be a ring and $(\mathcal{C}, A, \cop, \eps)$ be an $A$-coring.  Recall that
$\mathcal{C}$ is an $A$-bimodule with coassociative coproduct $\cop: \mathcal{C} \rightarrow
\mathcal{C} \otimes_A \mathcal{C}$ and counit $\eps: \mathcal{C} \rightarrow A$, both $A$-bimodule morphisms satisfying $(\eps \otimes \id_{ \mathcal{C}})\cop = \id_{ \mathcal{C}} = (\id_{ \mathcal{C}} \otimes \eps) \cop$ \cite{BW}.  It follows
that $\mathcal{C}^{\otimes_A n} \| \mathcal{C}^{\otimes_A (n+1)}$ as $A$-bimodules for each $n \geq 1$.  For convenience in notation let $\mathcal{C}^{\otimes_A 0} = A$.  The reader may consult \cite{BW} for more on corings, their morphisms, their comodules and  useful examples, such as coalgebras over ground rings and others we bring up below.

\begin{definition}
An $A$-coring $\mathcal{C}$ has \textbf{depth} $2n+1$ if $\mathcal{C}^{\otimes_A n} \sim \mathcal{C}^{\otimes_A (n+1)} $, where $n$ is a nonnegative integer.  If $n$ is a positive integer,
$\mathcal{C}$ has depth $2n+1$ if $\mathcal{C}^{\otimes_A (n+1)} \| m \cdot \mathcal{C}^{\otimes_A n}$ for some $m \in \N$.  Note that $\mathcal{C}$ has depth
$2n+3$ if it has depth $2n+1$.   If $\mathcal{C}$ has a finite depth, let $d(\mathcal{C}, A)$ denote
its \textbf{minimum depth}.
  \end{definition}

\begin{example}
\begin{rm}
Given a ring extension $B \rightarrow A$, let $\mathcal{C}$ denote its Sweedler $A$-coring
$A \otimes_B A$ with coproduct $\cop: A \otimes_B A \rightarrow A \otimes_B A \otimes_B A$
($= A^{\otimes_B 3}$) given by $\cop(a \otimes_B a') = a \otimes 1 \otimes a'$
and counit $\mu: A \otimes_B A \rightarrow A$ by $\mu(a \otimes a') = aa'$ \cite{BW}.
Note that $\mathcal{C}^{\otimes_A n} \cong A^{\otimes_B (n+1)}$ for  integers $n \geq 0$.
 Therefore, comparing the tensor powers of $\mathcal{C}$ as natural $A$-bimodules is equivalent
to comparing the tensor powers $A^{\otimes_B n}$ as in the definition of h-depth in Definition~\ref{def-depth}.  It follows that
\begin{equation}
\label{eq: coringdepth=hdepth}
d(A \otimes_B A, A) = d_h(B,A)
\end{equation}
\end{rm}
\end{example}
An $A$-coring $\mathcal{C}$ has a \textit{grouplike} element $g \in \mathcal{C}$ if $\cop(g) = g \otimes_A g$
and $\eps(g) = 1_A$.  For example, the Sweedler $A$-coring $A \otimes_B A$ in the example
has a grouplike element $1_A \otimes_B 1_A$.  An $A$-coring $\mathcal{C}$ with grouplike $g$
has invariant subalgebra $A_g^{\mbox{co}\, \mathcal{C}} := \{ b \in A \| \, bg = gb \} := B$ (notation refers to coinvariants of $\mathcal{C}$-comodule $A$ determined by grouplike \cite[p.\ 278]{BW}).  Recall that
$\mathcal{C}$ is a \textit{Galois coring} if $A \otimes_B A \cong  \mathcal{C}$ via $a \otimes_B a' \mapsto
aga'$, a coring isomorphism of $\mathcal{C}$ with the Sweedler coring of $B \subseteq A$.
For example, the Sweedler coring of a faithfully flat ring extension $A \supseteq B$ is Galois, since $B =
\{a \in A \| \,  a \otimes_B 1_A = 1_A \otimes_B a \}$ follows from $A_B$ or ${}_BA$ being
faithfully flat (\cite[28.6]{BW}, or see Lemma~\ref{lemma-ff}).
\begin{example}
\label{example-Frobenius}
\begin{rm}
Recall that $A \supseteq B$ is a Frobenius extension  with $F: A \rightarrow B$ a ``Frobenius'' homomorphism and $e := \sum_i x_i \otimes_B y_i$ a ``dual bases tensor,'' satisfying  $ae= ea$ for every $a \in A$, if $A$
is a $B$-coring with coproduct $\cop:A \rightarrow A \otimes_B A, \ \cop(a) = ae$ and counit $F$, this coring being denoted by
$\mathcal{C}_{\mathrm{Frob}}$.  The more familiar conditions of the counit equations
characterize Frobenius extensions \cite{NEFE}.
   The tensor powers of this coring are now given
by natural $B$-bimodules $\mathcal{C}_{\mathrm{Frob}}^{\otimes_B n} = A^{\otimes_B n}$.
Using Definition~\ref{def-depth} for the definition of odd depth, we obtain
\begin{equation}
\label{eq: FrobeniusCoringDepth=OddDepth}
d(\mathcal{C}_{\mathrm{Frob}}, B) = d_{\mathrm{odd}}(B,A)
\end{equation}
in terms of the minimum odd depth.
\end{rm}
\end{example}
\section{Depth of Coalgebra-Galois Extensions}
\label{three}

In this section we define depth of a certain coalgebra-Galois extension and see that its minimum depth takes on at least as many interesting values as subgroup depth \cite{BK, BDK, BKK, D, F, FKR}.  In contrast, the minimum depth of a Hopf-Galois extension is one or two.

Let $k$ be a field; all unlabeled tensors are over $k$. We begin with a review of the entwining
structure of an  algebra $A$ and  coalgebra $C$.  The entwining (linear) mapping $\psi: C \otimes A \rightarrow A \otimes C$ satisfies two
commutative pentagons and two triangles (a bow-tie diagram on \cite[p.\ 324]{BW}).  Equivalently,
$(A \otimes C, \id_A \otimes \cop_C, \id_A \otimes \eps_C)$ is an $A$-coring with respect
to the $A$-bimodule structure $a(a' \otimes c){a''} = aa' \psi(c \otimes a'')$
(or conversely defining $\psi(c \otimes a) = (1_A \otimes c) a$).

An entwining structure mapping $\psi: C \otimes A \rightarrow A \otimes C$ takes
values that may be denoted by $\psi(c \otimes a) = a_{\alpha} \otimes c^{\alpha} = a_{\beta} \otimes c^{\beta}$, suppressing linear sums of rank one tensors, and satisfies the axioms: (for all $a,b \in A, c \in C$)
\begin{enumerate}
\item  $\psi(c \otimes ab) = a_{\alpha}b_{\beta} \otimes {c^{\alpha}}^{\beta}$;
\item $\psi(c \otimes 1_A) = 1_A \otimes c$;
\item $a_{\alpha} \otimes \cop_C(c^{\alpha}) = {a_{\alpha}}_{\beta} \otimes {c\1}^{\beta} \otimes {c\2}^{\alpha}$
\item $a_{\alpha} \eps_C(c^{\alpha}) = a \eps_C(c)$,
\end{enumerate}
which is equivalent to two commutative pentagons (for axioms 1 and 3) and two commutative
triangles (for axioms 2 and 4), in an exercise.

The following is  \cite[32.6]{BW} or \cite[Theorem 2.8.1]{CMZ}.

\begin{prop}
\label{prop-1-1}
Entwining structures $\psi: C \otimes A \rightarrow A \otimes C$ are in one-to-one correspondence with $A$-coring structures \newline
 $(A \otimes C, \id_A \otimes \cop_C, \id_A \otimes \eps_C)$.
\end{prop}
\begin{proof}
Given an entwining $\psi$, the obvious structure maps $(A \otimes C, \id_A \otimes \cop_C, \id_A \otimes \eps_C)$ form an $A$-coring with respect
to the $A$-bimodule structure $a(a' \otimes c){a''} = aa' \psi(c \otimes a'')$.
Conversely, given an $A$-coring $A \otimes C$ with
coproduct $\id_A \otimes \cop_C: A \otimes C \rightarrow A \otimes C \otimes C \cong
A \otimes C \otimes_A A \otimes C$ and counit $\id_A \otimes \eps_C: A \otimes C \rightarrow
A \otimes k \cong A$, one defines $\psi(c \otimes a) = (1_A \otimes c) a$ and checks
that $\psi$ is an entwining, and the other details, in an exercise.
\end{proof}

Our primary example in this section is $A = H$, a Hopf algebra with coproduct $\cop$, counit $\eps$
and antipode $S: H \rightarrow H$, and $C$ a right $H$-module coalgebra,
i.e. a coalgebra $(C, \cop_C, \eps_C)$ and module $C_H$ satisfying $\cop_C(ch) = c\1 h\1 \otimes c\2 h\2$
and $\eps_C (ch) = \eps_C(c) \eps(h)$ for each $c \in C, h \in H$. An entwining mapping $\psi: C \otimes H \rightarrow  H \otimes C$ is defined by $\psi(c \otimes h) = h\1 \otimes c h\2$.
The entwining axioms are checked in a more general setup in Section~\ref{four}.

The associated $H$-coring $H \otimes C$ has coproduct $\id_H \otimes \cop_C$ and counit
$\id_H \otimes \eps_C$ with $H$-bimodule structure: ($x,y,h \in H, c \in C$)
\begin{equation}
\label{eq: coring}
x(h \otimes c) y = xh y\1 \otimes c y\2
\end{equation}
Notice that this is the diagonal action from the right.

\begin{prop}
\label{prop-entwinedepth}
The depth of the $H$-coring $H \otimes C$ and the depth of the $H$-module $C$ are related by
$d(H \otimes C, H) = 2d(C, \mathcal{M}_H) + 1$.
\end{prop}

\begin{proof}
The $n$-fold tensor product of $H \otimes C$ over $H$ reduces  to the $H$-bimodule isomorphism
\begin{equation}
\label{eq: iso}
(H \otimes C)^{\otimes_H n} \cong H \otimes C^{\otimes n}
 \end{equation}
via the mapping
$$ \otimes_{i=1}^n h_i \otimes c_i \longmapsto h_1 {h_2}\1 \cdots {h_n}\1 \otimes_{i=1}^{n-1} c_i {h_{i+1}}\i+1 \cdots {h_n}\i+1 \otimes c_n $$
with inverse $h \otimes c_1 \otimes \cdots \otimes c_n \mapsto h \otimes c_1 \otimes_H 1_H \otimes c_2 \otimes_H \cdots \otimes_H 1_H \otimes c_n$,
where $H \otimes C^{\otimes n}$ has right $H$-module structure from the diagonal action by $H$:
$(h \otimes c_1 \otimes \cdots \otimes c_n)x = hx\1 \otimes c_1 x\2 \otimes \cdots \otimes c_n x_{(n+1)}$.  This follows  from Eq.~(\ref{eq: coring}), cancellations of the type $M \otimes_H H \cong M$ for modules $M_H$, and an induction on $n$.

Suppose $d(C, \mathcal{M}_H) = n$, so that $C^{\otimes n} \sim C^{\otimes (n+1)}$ as right
$H$-modules (in the finite tensor category $\mathcal{M}_H$).  Applying an additive functor,
it follows that $H \otimes C^{\otimes n} \sim H \otimes C^{\otimes (n+1)}$ as $H$-bimodules.
Applying the isomorphism~(\ref{eq: iso})  the coring depth satisfies $d(H \otimes C, H) \leq 2d(C, \mathcal{M}_H) + 1$.

Conversely, if $d(H \otimes C, H) = 2n+1$, so that $H \otimes C^{\otimes n} \sim H \otimes C^{\otimes (n+1)}$ from Eq.~(\ref{eq: iso}) again, we apply that additive functor $k \otimes_H -$ to the similarity
and obtain the similarity of right $H$-modules, $C^{\otimes n} \sim C^{\otimes (n+1)}$.  Thus $2d(C, \mathcal{M}_H) +1 \leq d(H \otimes C, H)$.
\end{proof}

Now suppose $R \subseteq H$ is a left coideal subalgebra of a finite-dimensional Hopf algebra;
i.e., $\cop(R)\subseteq  H \otimes R$.  Let $R^+$ denote the kernel of the counit restricted to $R$.
Then $R^+H$ is a right $H$-submodule of $H$ and a coideal by a short computation given
in \cite[34.2]{BW}.  Thus $Q := H/R^+H$ is a right $H$-module coalgebra (with a right $H$-module
coalgebra epimorphism $H \rightarrow Q$ given by $h \mapsto h + R^+H := \overline{h}$).
The $H$-coring $H \otimes Q$ has grouplike element $1_H \otimes \overline{1_H}$; in fact,
\cite[34.2]{BW} together with \cite{S} shows that this coring is Galois:
\begin{equation}
\label{eq: map}
H \otimes_R H \stackrel{\cong}{\longrightarrow} H \otimes Q
\end{equation}
via $x \otimes_R y \mapsto x y\1 \otimes \overline{y\2}$ (also noted in \cite{HKY} and in \cite{U}
for Hopf subalgebras).  That $H_R$ is faithfully flat follows from the result that $R$ is a Frobenius algebra and  $H_R$ is free \cite{S}.  Note that an inverse to (\ref{eq: map}) is given
by $x \otimes \overline{z} \mapsto xS(z\1) \otimes_R z\2$ for all $x,z \in H$.

From Proposition~\ref{prop-entwinedepth}, Eqs.~(\ref{eq: map}) and~(\ref{eq: coringdepth=hdepth}) we note the following.

\begin{cor}
\label{cor-cue}
For a left coideal subalgebra $R$ in a  Hopf algebra $H$, its h-depth
is related to the module depth of $Q$ by
\begin{equation}
\label{eq: cue}
d_h(R,H) = 2d(Q, \mathcal{M}_H) + 1.
\end{equation}
\end{cor}
\begin{proof}
The proof is sketched above for finite-dimensional $H$.  For infinite-dimensional $H$, note
that the $H$-bimodule isomorphism in Eq.~(\ref{eq: map}) remains valid, as does Proposition~\ref{prop-entwinedepth} and Eq.~(\ref{eq: coringdepth=hdepth}).
\end{proof}
Suppose $R$ is a Hopf subalgebra of $H$. Then $Q$ is an $R$-module coalgebra by restriction.
  A similar argument to the one above shows that $d(R,H)$ and $d(Q, \mathcal{M}_R)$
satisfy the inequalities in (\ref{eq: inequalityfordepth}).

Finally we recall that a \textit{$C$-Galois extension} $A \supseteq B$, where $C$ is a coalgebra and
$A$ a right $C$-comodule via coaction $\delta: A \rightarrow A \otimes C$,  the subalgebra of coinvariants is characterized by satisfying $B =
\{ b \in A \| \, \forall a \in A, \delta(ba) = b\delta(a) \}$ and $\beta: A \otimes_B A \mapsto
A \otimes C$ given by $\beta(a \otimes a') = a \delta(a')$ is bijective.  For example,
a left coideal subalgebra $R \subseteq H$ is a coalgebra-Galois extension with respect to
the ($H$-module) coalgebra $Q$, as sketched above (the details are in \cite[34.2]{BW}).
Of course, this applies to Hopf subalgebras and more particularly to finite group algebra extensions.
Then we see that coalgebra-Galois extensions have at least the range of values computed for
subgroup depth \cite{BDK, BKK, D, F, FKR, HHP}.

\subsection{A faithfully flat interlude} Let $C$ be a coalgebra and $H$-module quotient of $H$, with canonical epi $H \rightarrow C$ of right $H$-module coalgebras, and $A$ is a right $H$-comodule algebra, with the obvious $C$-comodule coaction $\delta: A \rightarrow A \otimes C$ given by $\delta(a) = a\0 \otimes \overline{a\1}$.  Define the subalgebra $B = \{ b \in A \, | \, \delta(b) = b \otimes \overline{1_H} \}$.  Now suppose that $D \subseteq B$ is a subalgebra for which the canonical (Galois) mapping $\beta: A \otimes_D A \stackrel{\cong}{\longrightarrow} A \otimes C$
given by  $a \otimes_D a' \mapsto a\delta(a') = a{a'}\0 \otimes \overline{ {a'}\1}$, is an isomorphism.
If either of the natural modules $A_D$ or ${}_DA$ is faithfully flat, then $D = B$.
This follows from noting that for each $b \in B$,  $\beta(1_A \otimes_D b) = b \otimes \overline{1_H} = \beta(b \otimes_D 1_A)$ and the next lemma.
\begin{lemma}
\label{lemma-ff}
If $b \otimes_D 1 = 1 \otimes_D b$ for $b \in A \supseteq D$ with $D$ a subring of $A$ such that
$A_D$ or ${}_DA$ is faithfully flat, then $b \in D$.
\end{lemma}
\begin{proof}
Recall that a flat module $A_D$ is faithfully flat iff for each module ${}_DN$ such that $A \otimes_D N = 0$,
it follows that $N = 0$.  Now form the module $N = B'/D$, where $B' = \{ b \in A \, | \,  b \otimes_D 1_A = 1_A \otimes b \}$.  By an exercise with a commutative square, $A \otimes_D N = 0$, whence $D = B'$.
\end{proof}

\subsection{A normal element characterization of left ad-stabity for left coideal subalgebras}
Suppose $R$ is left coideal subalgebra of a finite-dimensional Hopf algebra $H$.  Let $Q$ be its quotient
right $H$-module coalgebra defined above.  By Skryabin's Freeness Theorem \cite{S} $\dim R$ divides $\dim H$,
and $(R, \eps)$ is an augmented Frobenius algebra, so $R$ has a nonzero right integral $t_R$ unique up to scalar multiplication \cite[Ch. 6]{NEFE}.  Then one proves just as in \cite[Lemma 3.2]{K2013} that
\begin{equation}
\label{isomorph}
Q \stackrel{\cong}{\longrightarrow} t_R H
\end{equation}
via $h + R^+H \mapsto t_R h$.

Recall that a subalgebra $R$ in a Hopf algebra $H$ is stable under the left adjoint action of $H$
if $h\1 r S(h\2) \in R$ for all $r \in R$, $h \in H$.  One briefly says that $R$ is left ad-stable
(or left normal \cite{B2012}).  In section~3 of \cite{B2012}, Burciu shows how to prove the following.

\begin{lemma}(cf.\ \cite{B2012})
\label{lem-straightenedout}
 A left coideal subalgebra $R$ in a finite-dimensional Hopf algebra $H$ is left ad-stable if and only if  $HR^+ \subseteq  R^+H$ if and only if the subalgebra $R \subseteq H$ has right depth $2$.
\end{lemma}
\begin{proof} For the convenience of the reader, we sketch the proof.  If $R$ is left ad-stable in $H$,
$r \in R^+$ and $h \in H$, then $hr = h\1 r S(h\2) h\3 \in R^+H$.

If $HR^+ \subseteq R^+H$, the isomorphism $\beta: H \otimes_R H \stackrel{\cong}{\rightarrow}
H \otimes Q$ in Eq.~(\ref{eq: map}) is an $H$-$R$-bimodule morphism, since $Q_R$ is trivial.
Hence ${}_H H \otimes_R H_R \cong (\dim Q) \cdot {}_H H_R$ and $R$ has right depth $2$ in $H$.

If ${}_H H \otimes_R H \oplus * \cong n \cdot {}_H H_R$, then applying $k \otimes_H -$ yields
$Q_R \oplus * \cong n \cdot k_R$.  Since $R^+ = \Ann\, k_R$, it follows easily that $HR^+ \subseteq R^+H$.

If $HR^+ \subseteq R^+H$, then $R^+ H$ is a bi-ideal in $H$,  $Q$ is a bialgebra and has a natural $H$-bimodule structure  from multiplication in $H$.  Also $\beta$ defined above satisfies $\beta(1_H \otimes_R h) =
\beta(h \otimes_R 1_H)$ for $$h \in H^{\mbox{co}\, Q} = \{ x \in H | x\1 \otimes \overline{x\2} = x \otimes \overline{1} \}. $$
Of course, $R \subseteq H^{\mbox{co}\, Q}$.
Since $\beta$ is bijective, $h \otimes_R  1 = 1 \otimes_R h$, thus by Lemma~\ref{lemma-ff},
$R = H^{\mbox{co}\, Q}$.  But $H^{\mbox{co}\, Q}$ is  left ad-stable by an argument used in
\cite[3.4.2]{M} modified slightly by the remark in the first sentence of this paragraph.
\end{proof}

The following generalizes \cite[5.2]{K2013}.
\begin{theorem}
\label{th-leftrightleft}
A left coideal subalgebra $R$ in a finite-dimensional Hopf algebra $H$ is left ad-stable if and only if its right integral $t_R$ is a normal element in $H$.
\end{theorem}
\begin{proof}
($\Rightarrow$)  One argues as in \cite[5.2]{K2013}, using Eq.~(\ref{isomorph}) and $Q$ is under the hypothesis a trivial right $R$-module, that  $h\1 t_R Sh\2$
is a nonzero right integral in $R$ for any $h \in H$, so that $$ht_R = h\1 t_R S(h\2) h_3 \in t_R H.$$
Hence, $Ht_R \subseteq t_R H$.  For the opposite inclusion,  note that $$t_R S(h) = S(h\1) h\2 t_R S(h\3) \in Ht_R.$$

($\Leftarrow$) If $Ht_R = t_R H$, then $\Ann \, Q_R \supseteq R^+$.  Thus $HR^+ \subseteq  R^+H$.
We conclude that $R$ is left ad-stable in $H$ from Lemma~\ref{lem-straightenedout}.
\end{proof}
One says that a subalgebra $R$ of a Hopf algebra $H$ is right ad-stable if $S(h\1)rh\2 \in R$
for all $h \in H, r \in R$.
Applying the antipode anti-automorphism to the above theorem, one obtains (left as an exercise) the correct formulation and proof of the opposite (and equivalent) theorem.

\begin{cor}
\label{rightversion}
A right coideal
subalgebra is right ad-stable in a finite-dimensional Hopf algebra $H$ iff its left integral $t_L$ is a normal element in $H$.
\end{cor}
\begin{example}
\begin{rm}
Consider the $8$-dimensional small quantum group $H = H_8$ given as an algebra in Example~\ref{example-8dim}, with coalgebra
structure given by $\cop(K) = K \otimes K$, $\cop(E) = E \otimes 1 + K \otimes E$, and $\cop(F) = F \otimes K + 1 \otimes F$.  It follows that $S(K) = K$, $S(E) = -KE$ and $S(F) = - FK$.

Consider the $2$-dimensional Frobenius subalgebra $R$ generated by $E$ (the ``ring of dual numbers'').  Notice that $R$ is a left coideal subalgebra, since $\cop(E) \in H \otimes R$.
It is left ad-stable since $FEK + ES(F) = 0$.  Also $t_R = E$ ($= t_L$) and $HE = EH$ is the $4$-dimensional
vector subspace of $H$ with basis $\{ E, EF, EK, EFK \} $: equivalently, $HR^+ = R^+ H$.

However, $R$ is not right ad-stable, since $S(F)EK + EF = 2EF \not \in R$.  The Frobenius subalgebra
of dimension $2$ generated by $F$ in $H$ is however a right coideal subalgebra that is right ad-stable
with normal integral element, and provides an example of Corollary~\ref{rightversion}.   This example does not
contradict the correct  formulation of the opposite of Lemma~\ref{lem-straightenedout}:
\end{rm}
\end{example}
\begin{prop}[cf.\ \cite{B2012}]
A right coideal subalgebra $R$ of $H$
is right ad-stable iff $R^+H \subseteq HR^+$ iff $R$ has left depth $2$ in $H$.
\end{prop}
\section{The coring of an entwining structure of a comodule algebra and a module coalgebra}
\label{four}

Let $H$ be a Hopf algebra.  Suppose $A$ is a right $H$-comodule algebra, i.e., there is a coaction
$\rho: A \rightarrow A \otimes H$, denoted by $\rho(a) = a\0 \otimes a\1$ that is an algebra homomorphism and $(A, \rho)$ is a right $H$-comodule \cite{BW, CMZ, M}.  Moreover, let
$(C, \cop_C, \eps_C)$ be a right $H$-module coalgebra, i.e., a coalgebra in the tensor category
$\mathcal{M}_H$ introduced in more detail in Section~\ref{three}.

\begin{example}
\begin{rm}
The Hopf algebra $H$ is right $H$-comodule algebra over itself, where $\rho = \cop$.  Given a Hopf subalgebra
$R \subseteq H$ the quotient module $Q$ defined in Section~1 as $Q = H/ R^+H$ is a right $H$-module
coalgebra.

Note that $(H, \cop, \eps)$
is also a right $H$-module coalgebra.  The canonical epimorphism $H \rightarrow Q$ denoted by $h \mapsto \overline{h}$ is an epi of right $H$-module coalgebras, and module $Q_H$ has $\overline{1_H}$ as a cyclic generator.

Of course, if $H = k$ is the trivial one-dimensional Hopf algebra, $A$ may be any $k$-algebra and $C$
any $k$-coalgebra.
\end{rm}
\end{example}

The mapping $\psi: C \otimes A \rightarrow A \otimes C$ defined by
$\psi(c \otimes a) = a\0 \otimes c a\1$ is an entwining as the reader may easily check (the so-called Doi-Koppinen entwining
\cite[33.4]{BW}, \cite[2.1]{CMZ}, which includes the case considered in Section~\ref{three}).

From Proposition~\ref{prop-1-1} it follows that $A \otimes C$ has $A$-coring structure
\begin{equation}
\label{eq: comodulealgdiagonal}
a(a' \otimes c)a'' = aa' {a''}\0 \otimes c {a''}\1
\end{equation}
which defines the bimodule ${}_A(A \otimes C)_A$.  The coproduct is given by $\id_A \otimes \cop_C$
and the counit by $\id_A \otimes \eps_C$.

Suppose in this paragraph that $H$ is a finite-dimensional Hopf algebra over an algebraically closed field, in which case $\mathcal{M}_H$
is a finite tensor category \cite{EO}.
Notice in the equation above that the right $A$-module is given by a version of the diagonal action
in which the category $\mathcal{M}_A$ is a module category over $\mathcal{M}_H$ \cite{AM, EO}.

\begin{prop}
\label{prop-comodalgentwinedepth}
The depth of the $A$-coring $A \otimes C$ and the depth of the $H$-module $C$ are related by
$d(A \otimes C, A) \leq  2d(C, \mathcal{M}_H) + 1$.
\end{prop}
\begin{proof}
The $n$-fold tensor product of $A \otimes C$ over $A$ reduces  to the $A$-bimodule isomorphism
\begin{equation}
\label{eq: iso2}
(A \otimes C)^{\otimes_A n} \cong A \otimes C^{\otimes n}
 \end{equation}
via the mapping $$a \otimes c_1 \otimes \cdots \otimes c_n \mapsto a \otimes c_1 \otimes_A 1_A \otimes c_2 \otimes_A \cdots \otimes_A 1_A \otimes c_n,$$
where $A \otimes C^{\otimes n}$ has right $A$-module structure from the diagonal action by $A$:
$(a \otimes c_1 \otimes \cdots \otimes c_n)b = ab\0 \otimes c_1 b\1 \otimes \cdots \otimes c_n b_{(n)}$.  This follows  from Eq.~(\ref{eq: comodulealgdiagonal}), cancellations of the type $M \otimes_A A \cong M$ for modules $M_A$, and an induction on $n$.

Suppose $d(C, \mathcal{M}_H) = n$, so that $C^{\otimes n} \sim C^{\otimes (n+1)}$ as right
$H$-modules (in the finite tensor category $\mathcal{M}_H$).  Applying an additive functor,
it follows that $A \otimes C^{\otimes n} \sim A \otimes C^{\otimes (n+1)}$ as $A$-bimodules.
Then applying the isomorphism~(\ref{eq: iso2}) obtain $d(A \otimes C, A) \leq 2d(C, \mathcal{M}_H) + 1$.
\end{proof}

Suppose $C$ has grouplike $g'$.  Then $g = 1_A \otimes g'$ is a grouplike
element in $A \otimes C$.  Then we have from Eq.~(\ref{eq: coringdepth=hdepth}) and Proposition~\ref{prop-comodalgentwinedepth} the proof of the following.
\begin{lemma}
\label{lem-ineq}
Suppose $A \otimes C := \mathcal{C}$ is a Galois coring with $B = A_g^{\mbox{co}\, \mathcal{C}}$
(or just that $B$ is a subalgebra of $A$ such that $A \otimes_B A \cong \mathcal{C}$ as
natural and right-diagonal $A$-bimodules, respectively).
 Then the h-depth of the subalgebra $B \subseteq A$ satisfies the inequality, $d_h(B,A) \leq 2d(C, \mathcal{M}_H) +1$.
\end{lemma}

\subsection{Crossed products as comodule algebras}
Now let $A$ be an associative crossed product $D \#_{\sigma} H$ for some Hopf algebra algebra $H$ and twisted $H$-module algebra $D$ with $2$-cocycle $\sigma: D \otimes D \rightarrow H$ that
is convolution-invertible \cite[chapter 7]{M}.  Then $A$ is a right $H$-comodule algebra via
$\rho = \id_D \otimes \cop$, which is quite obvious from the formula for multiplication in
$D \#_{\sigma} H$: ($h,x,y \in H, d,d' \in D$)
\begin{equation}
\label{eq: crossprod}
(d \# h)(d' \# x) = d(h\1 \cdot d')\sigma(h\2, x\1) \# h\3 x\2
\end{equation}
Additionally, the formulas for $D$ a twisted right $H$-module and $\sigma$ a $2$-cocycle are useful:
\begin{equation}
\label{eq: twistedmodule}
h \cdot (x \cdot d) = \sigma(h\1, x\1) (h\2 x\2 \cdot d) \sigma^{-1}(h\3, x\3)
\end{equation}
\begin{equation}
\label{eq: twococycle}
(h\1 \cdot \sigma(x\1, y\1)) \sigma( h\2, x\2 y\2) = \sigma(h\1, x\1) \sigma( h\2 x\2, y)
\end{equation}

\begin{example}
\begin{rm}
For $H = kG$ a group algebra, one obtains from this the familiar conditions of the group crossed product, $A = D \# kG$,
where for $g,h, s \in G$ and $d, d' \in D$,
\begin{eqnarray}
g \cdot (h \cdot d) & =  & \sigma(g, h) (gh \cdot d)\sigma(g,h)^{-1} \label{action} \\
(g \cdot \sigma(h,s)) \sigma(g, hs) & =  &  \sigma(g,h) \sigma(gh, s) \label{cocycle}\\
(d \# g) (d' \# h) & = & d(g \cdot d') \sigma(g,h) \# gh
\end{eqnarray}
Note that $\sigma^{-1}(g,h)$ is interchangeable with $\sigma(g,h)^{-1}$.
\end{rm}
\end{example}
\begin{example}
\begin{rm}
If $\sigma$ is trivial, $\sigma = 1_D$, then the crossed product reduces to the skew group algebra
$D \ast G$, where $D$ is a $G$-module and multiplication is given by $(d \# g)(d' \# h) = d (g \cdot d') \# gh$; this is a well-known setup  in Galois theory of fields and commutative algebras.     More generally,
$D \#_{\sigma} H$ is clearly isomorphic  to the smash product $D \# H$ if $\sigma(x,y) = \eps(x)\eps(y)1_H$:
see Eq.~(\ref{eq: crossprod}).  Then $D$ is a left $H$-module algebra. (If the action is Galois,
then the endomorphism algebra of $D$ over its invariant subalgebra is isomorphic to the smash
product $D \# H$ \cite[8.3.3]{M}.)
\end{rm}
\end{example}

\begin{example}
\begin{rm}
If $\sigma$ is instead nontrivial, but the action of $G$ on $D$ is trivial, i.e. $g \cdot d = d$
for each $g \in G$ and $d \in D$, then $D \#_{\sigma} kG = D_{\sigma}[G]$, the twisted group
algebra \cite{P}, with multiplication given by $(d \# g)(d' \# h) = dd' \sigma(g,h) \# gh$.  For example,
the real quaternions are a twisted group algebra of $\R$ with $G \cong \Z_2 \oplus \Z_2$ and
$\sigma = \pm 1$.
\end{rm}
\end{example}

\begin{example}
\label{example-groupascrossedproduct}
\begin{rm}
For any group $G$ and normal subgroup $N$ of $G$, the group algebra is a crossed product of the quotient group algebra acting on
the subgroup $N$ as follows \cite[7.1.6]{M}. Let $Q$ denote the group $G/N$.  For each coset $q \in Q$, let $\gamma(q)$ denote a coset representative, choosing $\gamma(\overline{1_G}) = 1_G$.  It
is an exercise then to show that with $\sigma(\overline{x},\overline{y}) := \gamma(\overline{x}) \gamma(\overline{y}) \gamma(\overline{x}\overline{y})^{-1} \in N$ and action
of $Q$ on $kN$ given by $\overline{x}\cdot n = \gamma(\overline{x})n \gamma(\overline{x})^{-1}$, one has
$kG = kN \#_{\sigma} kQ$.
\end{rm}
\end{example}

Let $R \subseteq H$ be a Hopf subalgebra pair.  Again form the quotient module $Q = H / R^+H$,
which is a right $H$-module coalgebra with canonical epi of right $H$-module coalgebras
$\pi: H \rightarrow Q$, $h \mapsto \overline{h}$.  Then the crossed product
$A = D \#_{\sigma} H$, which is a right $H$-comodule algebra, is also a right $Q$-comodule
via $A \rightarrow A \otimes H \stackrel{A \otimes \pi}{\longrightarrow} A \otimes Q$, the composition $\rho$ being the coaction given by $\rho(d \# h) = d \# h\1 \otimes \overline{h\2}$.
Then one sees that $B : = D \#_{\sigma} R \subseteq A^{{\mbox co}\, Q} $, the $Q$-coinvariants defined as the subalgebra $\{ b \in A \| \, \forall a \in A, \rho(ba) = b\rho(a) \}$; the inclusion follows from noting $\overline{rh} = \eps(r) \overline{h}$ for all $r \in R, h \in H$.

Note  that $Q$ has the grouplike element $\overline{1_H}$, so that $g = 1_A \otimes \overline{1_H}$
is a grouplike element in the $A$-coring $\mathcal{C} = A \otimes Q$. From the previous observation and
Eq.~(\ref{eq: comodulealgdiagonal}), it follows that $B = D \# R \subseteq A_g^{\mbox{co}\, \mathcal{C}}$.

Suppose
that $A \otimes_B A \cong A \otimes Q$ as $A$-bimodules, or more strongly $\mathcal{C}$
is a Galois $A$-coring with respect to $B = A_g^{\mbox{co}\, \mathcal{C}}$.  Then putting together Lemma~\ref{lem-ineq} with Corollary~\ref{cor-cue} and Eq.~(\ref{eq: cue}),  we arrive at the inequality of minimum
h-depths,
\begin{equation}
\label{eq: mainresult}
d_h(D \#_{\sigma} R, D \#_{\sigma} H) \leq d_h(R, H),
\end{equation}
the main aim of this paper:  we next set about establishing this supposition for finite rank group algebra extensions and certain crossed product Hopf algebra extensions.
\begin{prop}
\label{prop-sch}
Consider  the $H$-comodule algebra $A = D \#_{\sigma} H$.  Let $B = D \#_{\sigma} R$ be
an $R$-comodule subalgebra of $A$, where $R$ is a Hopf subalgebra of $H$.  Suppose  the condition~(\ref{eq: cond}) below is satisfied. Then $A \otimes_B A \cong A \otimes Q$ as $A$-bimodules,
where $Q = H/ R^+H$.
 If $A_B$ is faithfully flat,  then $A$ is a coalgebra $Q$-Galois extension of $B$, i.e.,
$A \otimes Q$ is a Galois $A$-coring with $B$ the invariant subalgebra.
\end{prop}
\begin{proof}
We investigate if the canonical mapping $$\beta(a \otimes_B a') = aga' = a(d \# h\1) \otimes \overline{h\2}$$
(where $a' = d \# h \in D \# H = A$) is bijective, and the module $A_B$ is  faithfully flat.

The map $\beta: A \otimes_B A \stackrel{\cong}{\longrightarrow} A \otimes Q$ is well-defined:
suppose $d' \in D, r \in R$, we  check that $$\beta(a(d' \# r) \otimes_B  (d \# h)) = \beta(a \otimes_B (d'(r\1 \cdot d)\sigma(r\2,h\1) \# r\3 h\2))$$
i.e., $ a(d' \# r)(d \# h\1) \otimes \overline{h\2} = ad'(r\1 \cdot d)\sigma(r\2,h\1) \# r\3 h\2 \otimes \overline{r\4 h\3}$, which is clear since $\overline{rh} = \eps(r) \overline{h}$ for each $r \in R, h \in H$.

We note that $\beta^{-1}: A \otimes Q \rightarrow A \otimes_B A$ is given by
$$\beta^{-1}(a \otimes \overline{h}) = a(\sigma^{-1}(S(h\2), h\3) \# S(h\1)) \otimes_B (1_D \# h\4).$$
The computation $\beta^{-1} \circ \beta = \id_{A \otimes_B A}$ and the computation
$\beta \circ \beta^{-1} = \id_{A \otimes Q}$ follow from $\gamma: H \rightarrow A$,
defined by $\gamma(h) = 1_D \# h$, having convolution-inverse
$\mu: H \rightarrow A$, defined by $$\mu(h) = \sigma^{-1}(S(h\2),h\3) \# S(h\1),$$
on \cite[p.\ 109]{M}.  We are left with verifying that $\beta^{-1}$ is well-defined, i.e., vanishes when $h = rh'$
for some $r \in R^+, h' \in H$.  Dropping the prime, this becomes the condition
\begin{equation}
\label{eq: cond}
\sigma^{-1}(S(h\2) S(r\2), r\3 h\3) \# S(h\1) S(r\1) \otimes_B (1_D \# r\4 h\4) = 0
\end{equation}
for all $r \in R, h \in H$ such that $\eps(r) = 0$.

Finally, suppose either natural module, $A_B$ or ${}_BA$ is faithfully flat.  Then $A$ is a $Q$-Galois extension of $B$ (cf.\ \cite[34.10]{BW}, \cite[1.2, 3.1]{Sch}).
\end{proof}
\begin{cor}
Let $R \subseteq H$ be a finite-dimensional Hopf subalgebra pair, and $A$ a left $H$-module algebra.  Then
h-depth satisfies

\begin{equation}
\label{eq: smashproddepth}
d_h(A \# R, A \# H) \leq d_h(R,H).
\end{equation}
\end{cor}
\begin{proof}
If $\sigma(x,y) = \eps(x)\eps(y)1_D$ in Proposition~\ref{prop-sch}, then  the crossed products in the theorem are smash products. The corollary follows if Eq.~(\ref{eq: cond}) is satisfied with this choice of  $\sigma$.  But  the left-hand side
reduces  to $1 \# S(h\1) \otimes_B 1 \# S(r\1) r\2 h\2 $, indeed equal to zero for $r \in R^+$.
Finally, the  natural module ${}_RH$ is free by Nichols-Zoeller, so that $A \# H$ is easily shown to be  free as a natural left module
over $A \# R$.
\end{proof}
\begin{prop}
Suppose $G$ is a group and $S$ is a subgroup of $G$ such that $|G:S| < \infty$.  Let $H = kG$, $R = kS$ and $A$,$B$
be crossed products of the group algebras $H$ and $R$ with left twisted $H$-module algebra $D$ as above.  Then $A$ is a $Q$-Galois extension of $B$ (where $Q$ is the permutation module of right
cosets of $S$ in $G$).
\end{prop}
\begin{proof}
It suffice to check Eq.~(\ref{eq: cond}) for $h \in G$ and $r \in S$.  Note that $1 - r \in R^+$
and such elements form a $k$-basis.  Also note that $$\cop^{n-1}(1-r) = 1 \otimes \cdots \otimes 1
- r \otimes \cdots \otimes r$$ ($n$ $1$'s and $n$ $r$'s on the right-hand side of the equation).  Eq.~(\ref{eq: cond}) becomes
\begin{eqnarray}
\label{eq: simp}
\sigma^{-1}(h^{-1}r^{-1}, rh) \# h^{-1} r^{-1} \otimes_B (1_D \# rh) && \nonumber \\ &= & \sigma^{-1}(h^{-1},h) \# h^{-1} \otimes_B (1_D \# h)
\end{eqnarray}
The inverse of the $2$-cocycle Equation~(\ref{cocycle}) is ($\forall x,y,z \in G$)
\begin{equation}
\label{eq: twococycleinverse}
\sigma^{-1}(xy,z) \sigma^{-1}(x,y) = \sigma^{-1}(x,yz) [ x \cdot \sigma^{-1}(y,z)]
\end{equation}
Letting $x = h^{-1}, y = r^{-1}, z = rh$, the left-hand side of Eq.~(\ref{eq: simp}) becomes
$$ \sigma^{-1}(h^{-1},h)[h^{-1} \cdot \sigma^{-1}(r^{-1}, rh)] \sigma(h^{-1}, r^{-1}) \# h^{-1} r^{-1} \otimes_B (1_D \# rh) $$
$$ = (\sigma^{-1}(h^{-1}, h) \# h^{-1}) ( \sigma^{-1}(r^{-1}, rh) \# r^{-1}) \otimes_B (1_D \# rh)$$
$$ = (\sigma^{-1}(h^{-1}, h) \# h^{-1}) \otimes_B (\sigma^{-1}(r^{-1}, rh) \sigma(r^{-1}, rh) \# h)$$
which equals the right-hand side of Eq.~(\ref{eq: simp}).

The natural module ${}_BA$ is free by a short argument using coset representatives $g_1,\ldots,g_q$
giving a left $R$-basis for $H$.  I.e., $|G: S| = q$, $\sum_{i=1}^q Rg_i = H$ and $\sum_{i=1}^q r_i g_i = 0$
implies each $r_i = 0$.  Then $\sum_{i=1}^q (D \#_{\sigma} R)(1_D \# g_i) = \sum_i D\sigma(R, g_i) \# Rh_i = D \# H$, since $\sigma(x,y)$ is invertible in $D$ for all $x,y \in G$.
Suppose $$0= \sum_{i=1}^q (d_i \# r_i)(1 \# g_i) = \sum_i d_i \sigma({r_i}\1, g_i) \# {r_i}\2 g_i $$
where each $r_i$ is a linear combination of group elements $s_{ij}$ in $S$.   In this case each
$s_{ij} g_i \in Sg_i$ and it follows from the partition of $G$ into left cosets and the equation just above that each $d_i \# r_i = 0$.

Finally $Q = H / R^+H \cong k[S \setminus G]$ by noting that $R^+$ has $k$-basis $\{ 1 - s \, : \, s \in S \}$.
\end{proof}

\begin{theorem}
Let $G$ be a group with subgroup $H$ of finite index, $A$ a left twisted $G$-module $k$-algebra, and  $A \#_{\sigma} G$ an (associative) crossed product.  Then h-depth satisfies
\begin{equation}
\label{eq: hdepthineq}
d_h(A \#_{\sigma} H, A \#_{\sigma} G) \leq d_h(kH, kG).
\end{equation}
\begin{proof}
Follows from the previous proposition and  inequality (\ref{eq: mainresult}).
\end{proof}
\end{theorem}
\subsection{General results for Hopf algebra $H$} We may extend Eq.~(\ref{eq: hdepthineq})  to a finite-dimensional Hopf algebra extension and crossed product algebra extension as follows.
\begin{theorem}
Suppose $H$ is a finite-dimensional Hopf algebra, with $R$ a Hopf subalgebra, $A = D \#_{\sigma} H$, $B = D \#_{\sigma} R$ and $Q = H / R^+H$ as above.  Then
\begin{equation}
d_h(B,A) \leq d_h(R,H).
\end{equation}
\end{theorem}
\begin{proof}
The proof may be made to follow from \cite[3.6]{Sch} and Eq.~(\ref{eq: mainresult}), but we provide some more details as a convenience to the reader.
Note first  that $\beta: A \otimes_B A \rightarrow A \otimes Q$, $x \otimes y \mapsto xy\0 \otimes \overline{y\1}$ in the proof of Proposition~\ref{prop-sch} is shown to be surjective from the formal
inverse $\beta^{-1}$ defined there and the equation $\beta \circ \beta^{-1} = \id_{A \otimes Q}$.

That $\beta$ is injective follows from \cite{Sch} in the following way using norms of augmented Frobenius algebras, freeness of $H$ over $R$ and the augmentation of the convolution algebra $Q^*$ induced from the grouplike element $\overline{1_H} \in Q$.  Let $B$ denote $A^{\mbox{co}\, Q}$
for this argument.  Let $\lambda_H, \lambda_R$ be nonzero left integrals on $H$ and $R$, respectively.  Let $\Gamma'  \in R$ satisfy $\lambda_R \Gamma' = \eps$, so that $\Gamma'$ is
a nonzero right integral in $R$.  Define $\lambda: Q \rightarrow k$ by $\lambda(\overline{h}) = \lambda_H(\Gamma' h)$, and note the left integral property, $\overline{h\1} \lambda(\overline{h\2})$ for all $h \in H$
\cite[p.\ 307, (i)]{Sch}.  By \cite[p.\ 307, (ii)]{Sch}, there is $\Lambda \in H$ such that $\eps_Q =
\Lambda \lambda$,  which follows  from expressing a right norm $\Gamma$ of $\lambda_H$
as $\Gamma = h \Gamma'$, then applying a Nakayama automorphism $\alpha$ of $H$
to express $\Lambda = \alpha(h)$.  Next define as in \cite[p.\ 303, (1)]{Sch} $f: {}_BA \rightarrow {}_BB$ by $f(a) = a\0 \lambda(\overline{a\1})$ using the left integral property.

In order to continue,  note that $\mbox{can}: A \otimes A \rightarrow A \otimes H$, given
by $x \otimes y \mapsto xy\0 \otimes y\1$ is surjective, since given $a \otimes h \in A \otimes H$,
$\mbox{can}(a(\sigma^{-1}(S(h\2), h\3) \# S(h\1)) \otimes 1_D \# h\4) = a \otimes h$
by the computation on \cite[p.\ 109]{M}.  It follows from the bijectivity of the antipode
that $\mbox{can'}: A \otimes A \rightarrow A \otimes H$ given by $x \otimes y \mapsto x\0 y \otimes x\1$ is also surjective; cf. \cite[p.\ 124]{M}.  Let $\mbox{can'}(\sum_i r_i \otimes \ell_i) = 1_A \otimes \Lambda$.  Then  \cite[p.\ 303, (2)]{Sch} shows that $a = \sum_i f(ar_i)\ell_i$ for each $a \in A$.
Applying this projectivity equation to $\beta(\sum_k x_k \otimes_B y_k) = 0$, \cite[p.\ 303, (3)]{Sch} shows that
 $\sum_i x_k \otimes_B y_k = 0$.

Since $H$ is a free $R$-module, a faithfully flat descent along the crossed product extension shows
that $A \#_{\sigma} R = A^{\mbox{co}\, Q}$.
Since $\beta: A \otimes_B A \stackrel{\cong}{\longrightarrow} A \otimes Q$ as $A$-bimodules,
it follows from Eq.~(\ref{eq: mainresult}), and the proof preceding it, that the inequality in
the theorem holds.
\end{proof}

\begin{cor}
\label{cor-core}
Suppose $N$ is a normal subgroup of a finite group $G$ contained in a subgroup $H \leq G$.  Then the h-depth satisfies the equality,  $ d_h(kH,kG) = d_h(k[H/N], k[G/N])$.
\end{cor}
\begin{proof}
This follows from Corollary~\ref{cor-cue}, since in either case $Q \cong k[G \setminus H]$, while $N$ acts trivially
on this $G$-module.  Note that $I = kGkN^+$ is a Hopf ideal in $kG$ (generated by $\{ g - gn \| \, g \in G, n \in N \}$), which annihilates $Q$, and we apply Lemma~\ref{lemma-modI}.

A second proof is to apply the observation in  Example~\ref{example-groupascrossedproduct} that one has $kG = kN \#_{\sigma} k[G/N]$ and similarly $kH = kN \#_{\sigma} k[H/N]$.  Apply now the inequality~(\ref{eq: mainresult}) to obtain $d_h(kH,kG) \leq d_h(k[H/N], k[G/N])$.   The inequality $d_h(k[H/N], k[G/N]) \leq d_h(kH,kG)$ has a proof very similar to the  proof of Theorem~\ref{prop-sigma} using the relatively nice Hopf ideal $I = kG kN^+$.
\end{proof}

\subsection{Acknowledgements}  The authors  thank Dr.\ C. Young for stimulating conversations about depth-preserving algebra homomorphisms. Research for this paper was funded by the European Regional Development Fund through the programme {\tiny COMPETE}
and by the Portuguese Government through the FCT  under the project
\tiny{ PE-C/MAT/UI0144/2013.nts}.

redpajama_set_name: RedPajamaArXiv


About two years ago, my friends and I noticed a plethora of imitation Dr. Pepper drinks hitting the market. Most of these are store brands, but some well-known companies hit with their own versions as well. I was struck by how sudden all of these flooded the marketplace (or perhaps on how quickly I noticed this) and began keeping tabs on the names-after all, some of them were quite funny.
This one is also apparently quite old, although its origin seems slightly silly, since it is not very different from the Pepper, but costs more.
A slight variation on the Dr. moniker, this one is also widely available. If memory serves, it is also more red than the other sodas.
This is the first generic one I noticed, and it really struck me because it had a very funny name. It is made by Marquee Premium and available at Abco grocery stores.
This one also seemed very funny since I thought Slice would be too reputable name to rip off an actual soda (anything goes for the generics, of course). This is pretty widely available; I've even seen it in standard vending machines. The cans bill it as "the Wild One."
A Preferred Selection, we found this in New Mexico at a store I don't remember the name of. It is also available in Tucson, but I don't remember where.
Made by President's Choice and available at Smitty's, this one is billed as a "secret recipe soda." This one also went through a facelift, and I don't have the original can, as that was discontinued before I began my collection. Its new design is unusual because it discards the standard red coloring of most of these sodas.
An "American Premier" soda, available at Osco Drug.
The Big K variation, available at Fry's. The basketball allusion would be more clever if it had any depth; I'm sure it was just chance and keeping with the philosophy that many stores just adopt the word "Dr." and add it to whatever their sodas may already be called. This has a mostly white can, like all the Big K sodas.
Made by A+, the Albertson's brand.
Unusual syntax, but a similar drink, although this is the only one in my collection I have never had the pleasure to taste. It's made by Sunny Select, and I am indebted to my friend Tivon Jacobson for retrieving this from a store in California I no longer remember the name of.
A Blue Sky soda, a "natural" variation on the theme, available at the more health-oriented groceries, like Reay's Ranch Market and the Food Conspiracy.
Another healthy one, from Trader Joe's. Both of these are unusual in that they have no caffeine, showing that the appeal of Dr. Pepper-like sodas goes far beyond the "pep" the caffeine would give you.
The Furr's home brand, another one we found in New Mexico.
This one was just brought to me from a supermarket I don't remember the name of in Lubock, Texas, and is made by the Hy-Top company.
This one comes from a truckstop in Lordsburg, New Mexico, thanks to the Jacobson clan (mentioned below). A picture is forthcoming.
As you can see I need some updating here. When I get my pictures developed, these will be corrected and brought up to date.
That concludes the actual cans in my collection. Here is a list of sodas that are rumored to exist, or I can't find in cans. I'm very interested in any information that can confirm their existence, and would be really grateful if anyone could get me a can of them.
Quite frankly I have no idea where the rumor of the existence of this soda came from, but it is another stumper. My friend Josh Ostrander claims he has a bunch at his parent's house, but I have yet to see evidence of this. Does it exist, or is Josh a liar?
Available in New York somewhere, according to my friend Matt Buchholz. Unfortunately he was unable to find a can.
Unfortunately this one I think may only be available by bottle. I have seen it, though, so at least I am sure of this one's existence.
And then, finally, of course there are those sodas that are too good to join the medical practice, starting with Mr. Pibb (another old classic), but also including Skipper (from Safeway), Mr. Aah (or something like that, from Pennsylvania or something), and Southern Lightning (from Wal*Mart).
Well, if you've made it this far then you must be mildly interested in this topic at least (or amazed at how anyone could waste this much time) and will contribute to our search for the elusive remaining doctors yet to be uncovered. If you can contribute, please do, send me an email immediately leading to the conquest of this strange beast at lars@u.arizona.edu. Your assistance will be greatly appreciated.
Credits for this page: Natan Jacobson, pictured at left, helped me take all these pictures with his family's digital camera. Thanks to all the Jacobsons. As far as people helping me with the hunt, these are the people I can name who have helped me: Keith Turausky, Russell Wiener, Tivon Jacobson, Daniel Brito, John Salwin, Shira Jacobson, Michael Rhodes, Matt Buchholz, and probably others. Mail me if you think you belong on this list; you probably do.
This page and collection managed by Lars Christensen.


We see every design as a unique challenge. That's why we approach each project as a fresh start and an exciting adventure. Strong marketing and design is about telling a great story, and that's where we use our passion, strategy and problem solving to make your ideas come to life. We're The Shft and we love working with our clients near and far.
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Chelsea Academy: The money and pressure in the game has turned the Academy into a money making machine.
It's been a subject for debate for good knows how many seasons but it's becoming clear that Chelsea's Academy players are becoming more valuable for differing reasons.
The sale of Nathan Ake has for me re-iterated just how much the game has changed in this country. As more and more money is ploughed into our game, expectations increase and with that comes pressure to succeed in the Premier League. For clubs such as Chelsea who are expected to challenge for every domestic and European honour in the game season in, season out, they simply cannot afford to fall behind their challengers and continue to invest big money in new players who come in and replace those we currently have or ones previously brought in to then be sold.
Just look at the staggering amount of money that Chelsea have made from player sales since 2014 to the current day - A whopping £400 million!. It's fair to say that the majority of those players were squad members or players who had fallen out of favour with the current Manager at the time.
The players from our Academy are highlighted in bold. The figure of sales of just those players is £69 Million (which could rise if an independent tribunal deems that Liverpool have to pay Chelsea more than the £3m quoted), it also does not take into account Izzy Brown's impending move to Huddersfield for a possible £8m either!
People moan and groan about our loan system with Chelsea's insistence that our young players get a feel for competitive football in and around the loan leagues. Those that have impressed in the realms of League One and the Championship are also then given the chance to prove what they can do in the Premier League.
However, to this day it's only really been Nathan Ake who came through our Academy that had made an impact playing regular first team football on loan in the Premier League with others such as Patrick Bamford being unable to prove himself at that level.
That's the problem here. These players are not given enough chances at their parent clubs with Chelsea being a prime example. That's because of the demands of the Premier League, the massive and quite simply ridiculous monies on offer for avoiding relegation let alone to qualify in the top four. Each Premier League manager up and down the county knows exactly what is required from their board of directors, owners and most importantly the fans. They cannot afford to play a younger player week in, week out because if it doesn't work out and they lose games it could prove to be detrimental to their goals for the season which could then have a knock on effect.
It's such a frustrating situation at Chelsea because just look at how good our youth teams are. We have some real proven quality within our youth ranks but they have to be nurtured away from the club plying their trade elsewhere! OK, Chelsea has a dedicated team of people that work with each of our loan players throughout the season but that's not the point.
If you look at the flip side of things and see the potential player values of our Academy players such as Ryan Bertrand and Nathan Ake then you will see that the whole situation has changed. Just think that all those Academy player sales since 2014 will have more or less paid for the signatures of Tiemoue Bakayoko and Antonio Rudiger will Willy Caballero thrown in! It's crazy!
Unless the rule makers in the Premier League step in and implement salary caps, limit the number of foreign players in squad/subsequent match day eleven and ensure that at least one player from the Academy is implemented into a squad every year playing a minimum of ten games, things will never change!
As I said, the money in the game now and the pressures of expectation has changed our game forever. Academy players are now seen as potential moneymakers rather than future first team players with only the exceptional ones given a possible chance.


text:
Intersectionality at Denver's Pride Liberation March
By Annie Burky
Jun 16, 2020 | News | 0 comments
//Porsha DeMarco-Douglas stands on the steps of the Colorado State Capital Building on June 14 during the Denver Pride Liberation March. Photos by Esteban Fernandez | sovereign73811b@gmail.com
The alliance between the LGBTQ+ and black community was on full display at the Pride Liberation March in Denver on June 14.
The bond between the two minority groups can be traced back to the inception of the pride movement. Marsha P. Johnson, queer activist, touched off the Stonewall Riots in June 1969 by throwing a brick through the window of a cop car. It was her memory that activists evoked repeatedly at Sunday's Pride Liberation March in Denver.
"If one of us isn't free none of us are free. Don't forget, Pride started as a riot, lead by Queer women of color," said State Rep. Leslie Herod, the first Black LGBTQ+ person elected into office in Colorado.
The event was a collaboration between the Denver Dyke Rally and Black Lives Matter 5280. Both organizations share the common goal of liberation, which they imbued the march with as a whole.
//State Rep. Leslie Herod speaks at Denver's Pride Liberation March at Cheesman Park on June 14.
In former years, Pride month would be marked by parades, drag shows and countless other events. This year, most events have been moved online in the name of public safety due to the pandemic.
Despite this fear, protests following the death of George Floyd erupted in Denver on the night of May 28. The first few days of protests were punctuated by somewhat violent riots during the nights. Now, with less police presence and more organization, the demonstrations have remained widely peaceful as they have continued across the city.
It is this coalescence that brought crowds out despite the continued fear of COVID-19.
"There will ultimately be a cure for COVID but this is the cure for a lack of liberation," said marcher Matthew H.
These realities and events led to an afternoon of intersectionality marked by high attendance.
"I hope today's march achieves a sense of community between the LGBTQ+ community and black lives. They held space for us to have our revolution. The least we can do is hold space for them in a time like this," said Onyx Steel, the Safety and March Marshal of the event.
Not only were the two causes aligned in the goal for liberation but also in how to accomplish it.
"The time has come for us to stop being silent and to speak up," said Tyrell Rae, also known as drag artist Miss Zarah.
Organizer Kia designed the march to start at Cheesman park with the feeling of a Pride event. Various members of the community spoke, such as RTD Board Member Shontel M. Lewis, a group of sex workers, and Colorado's first transgender lawmaker State Rep. Brianna Titone.
The crowd marched down Colfax, chanting "silence is not golden" and "black trans lives matter," in a call to action after the recent deaths of two black trans women in other parts of the country. Riah Milton, from Ohio, was killed during a robbery on June 9. Dominique "Rem'Mie" Fells, from Pennsylvania, was found dead on June 8. Her death was ruled a homicide, but the Philidelphia police investigation is ongoing.
At the Capitol building, the focus shifted toward race. Members of the Choctaw Nation performed a "Blessing of the Four Directions" before several individuals aligned with the Black Lives Matter movement spoke about the complicated nature of intersectionality in America.
"I want to see all the community come out and support our black kings and queens. The only reason we have rights is because Marsha P. threw a brick through a cop's car," said Alex Johnson, an organizer with The Q Crew and bartender at Denver's lesbian bar Blush and Blue.
Despite invoking the same anger that ignited riots over police brutality and the Pride movement, Sunday's march ended with a quiet sunset on the Capitol steps and a candlelit vigil for those lost.
University of Denver commemorates MLK Day amidst voting rights battle in Congress
by Esteban Fernandez | Jan 19, 2022
The University of Denver commemorated MLK Day on Monday, with speakers calling attention to the ongoing struggle for voting rights.
How to help the victims of the Denver-Lakewood mass shooting
by Madison Lauterbach | Dec 30, 2021
Here are the links to fundraisers, campaigns and events that will benefit the victims and their families of the Dec. 27 shootings.
The local effects of craft beer's #MeToo reckoning, six months later
by Deborah Cameron | Nov 18, 2021
In May, Instagram posts sparked a major #MeToo movement within the craft beer industry. Six months later, what's changed in Colorado?


I did not grow up a fan of the manga or the anime source material, so had a clean slate of expectations when I went in to see the movie. That James Cameron worked on the script was a good sign, as he has some sort of magic wand to make people love otherwise forgettable stories. Robert Rodriguez as director has been hit or miss for me in the past (Sin City-Yeah! Shark Boy and Lave Girl-Dafuq!).
So I sat down in the theater with an open mind and…I loved it.
The movie was beautiful. Iron City felt real. I wanted to go to Zalem. Alita's giant anime eyes didn't bother me. The acting was solid.
I love stories about soldiers cast off after a war, and Alita had that going for her. Just how she came to realize who she really was, and how she was going to fit in, made for a great time. Bravo!
The supporting cast was mostly excellent. Dr. Ido was compelling. You get why he did what he did and why he was an obstacle to Alita a few times. Christof Waltz was great as always.
I thought Chiren deserved more screen time. She vanished from the film for too long and popped up towards the end with a character revelation that wasn't earned.
Hugo proved to be an interesting surprise with his character, though every time he showed up before the end of the movie was a great time to go to the bathroom.
What was with Nurse Gerhad? She was basically a prop in the movie but had way too many reaction shocks. Que the 'dull surprise' reaction from you MST3K die hards.
There were a few weak points. Alita's speech in the bounty hunter bar was just cringy. And the ending was blatant sequel bait.
I wanted to know more about the URM and more about Zalem. We got too much tease and not enough pay off for those plot threads, but sequel, amiright?
I had a great time at that movie and want to see the next one. Definitely not for the under 13 crowd.


Preheat the oven to 350 degrees. Coat a baking sheet pan with cooking spray.
Beat 4 of the eggs lightly with a fork, stir in ¼ teaspoon salt and the pepper, and add to the matzo. Toss the mixture well, and then occasionally for the next 5 minutes to allow some of the egg to be absorbed.
Meanwhile, toss the arugula with 1 tablespoon of the oil, the vinegar, and the remaining 1/8 teaspoon salt.
Heat 1 tablespoon of the oil in a large nonstick skillet over medium heat. Add the matzo mixture and press it lightly into an even disc in the pan. Cook until the bottom is browned and the mixture holds together well when you shake the pan, about 5 minutes. Place the sheet pan (sprayed side down) over the skillet and invert the rosti onto the pan. (The browned side will face up.) Spread the top evenly with the ricotta and place in the oven for 5 minutes.
Meanwhile, heat the remaining 1 tablespoon olive oil in the skillet. Crack the remaining 4 eggs in the skillet and cook them, sunny side up, until the whites are just cooked through and the yolks still runny, about 3 minutes.
Cut the rosti into four wedges and place each on a plate. Top with the arugula and then an egg; sprinkle with the feta and serve immediately.


text:
La rue Edmond-About est une voie du  de Paris, en France.

Situation et accès
La rue Edmond-About est une voie publique située dans le  de Paris. Longue de , elle commence 17, rue de Siam et 1, rue Guy-de-Maupassant et finit 46, boulevard Émile-Augier.

Non loti, le pont qui surplombe la tranchée ferroviaire en friche comprise entre le boulevard Émile-Augier et le boulevard Jules-Sandeau fait partie de l'emprise de la rue. La rue Edmond-About est poursuivie, à l'ouest, par la rue Édouard-Fournier.

Le quartier est desservi par la ligne C du RER, à la gare de l'avenue Henri-Martin, par la ligne 9 du métro, à la station Rue de la Pompe, et par les lignes de bus RATP 52 et 63.

Origine du nom

Elle a été nommée en l'honneur du littérateur français Edmond François Valentin About (1828-1885).

Historique
Cette voie est ouverte sous sa dénomination actuelle par un arrêté du . Elle est percée sur le site de l'ancien jardin fleuriste de la Muette. Les rues voisines Édouard-Fournier, Eugène-Labiche, Guy-de-Maupassant et Octave-Feuillet sont percées au même moment, dans le cadre du réaménagement du quartier.

En 1906, des maçons en grève, après une réunion à la bourse du travail, tentent de débaucher les ouvriers travaillant sur des chantiers de construction situés dans le quartier, boulevard Émile-Augier, rue Edmond-About, rue de Franqueville, etc. En arrivant rue Edmond-About, ils se heurtent à une compagnie d'infanterie et à un escadron de dragons. Des violences s'ensuivent et une trentaine d'individus sont arrêtés.

Notes et références

Articles connexes
 Liste des voies du  arrondissement de Paris
 Liste des voies de Paris

Edmond-About
Edmond-About

redpajama_set_name: RedPajamaWikipedia


Goodness Gracious! I made it. The February Letters Challenge has been pretty awesome. Thanks to everyone that's written back so far! Expect one more comprehensive post about the whole experience, and then it'll be back to our regularly scheduled programming.
Looking forward to hearing about the entire experience. Whooo-hooo, nice job.


Time And Attendance Systems - Can You Afford Not To?
One employee wasting just ten minutes a day could cost you over £200 each year. If you have ten employees that's over £2,000. With Clocking-In-Machines from £140 and Self Installed Swipe and Fingerprint Systems From Only £395 can you afford NOT to have automated time and attendance?
Over payment of wages; having too many staff for a project or too few; payroll errors - all these cost may be hidden to you but they can be significant.
Introducing a time and attendance system can result in considerable benefits for a business.
Improved punctuality - on average 2hrs can be lost through lateness, early departure and long lunches etc. You could save an average of £220 per employee.
Eliminating payroll errors - on average 2-3% of payroll costs is accounted for by errors. These could be eliminated with an automated time and attendance system saving around £160 per employee.
Automated time-sheets - removing manual time sheets and consequently stopping hours being rounded-up when calculated could save an average of £260 per employee.
Reduced absence and sickness - workers are less likely to take unauthorised time off if they know they're monitored.
Removing inefficient work activity - using time and attendance systems you can see which activities are productive and which aren't. In manufacturing industry on average 600 hours are lost to unproductive practises. Removing these has clear financial benefits for any business.
Why not make your staff more effective - maximise their time and save your money.


text:
Q: $1-\phi$ is surjective for a $\phi$-module? Let $A$ be a ring which is complete for the $p$-adic topology, in another word, there is an isomorphism between the two topological rings $A\rightarrow \lim_{\longleftarrow_{n \in \mathbb N_+}}  A/P^nA$. Assume $A/pA=k$ with $k$ is an algebraic closed field.(it is easy to see the characteristic of $k$ is $p$), so $A$ is endowed with a frobenius $\sigma$ lifting the map $x\rightarrow x^p$ in $k$.
Define $V$ is a free $A$ module of rank $n$ and with a map $\phi: V\rightarrow V$ which satisifies $\phi(k_1v_1+k_2v_2)=\sigma(k_1)\phi(v_1)+\sigma(k_2)\phi(v_2)$ and there exists a basis such that $Mat(\phi)\in GL_n(A)$.
Then the question is how to prove $1-\phi$ is surjective. This is an important fact metioned in Laurent Berger's note about $p$-adic Galois representations(page 19, after the proof of corollary 8.7).
Thanks!