text:
\subsection{#1}\label{#2}}

\begin{document}

\title[Lefschetz and Hirzebruch-Riemann-Roch formulas]{Lefschetz and Hirzebruch-Riemann-Roch formulas \\via noncommutative motives}

\author{Denis-Charles Cisinski}
\address{Universit\'e Paul Sabatier\\
Institut de Math\'ematiques de Toulouse\\
118 route de Narbonne\\
F-31062 Toulouse Cedex 9}
\email{denis-charles.cisinski@math.univ-toulouse.fr
\urladdr{http://www.math.univ-toulouse.fr/~dcisinsk/}

\author{Gon{\c c}alo~Tabuada}

\address{Gon{\c c}alo Tabuada, Department of Mathematics, MIT, Cambridge, MA 02139, USA}
\email{tabuada@math.mit.edu}
\urladdr{http://math.mit.edu/~tabuada/}

\subjclass[2000]{14A22, 14C20, 14F25, 14F40, 18D20, 19L10}
\date{\today}

\thanks{Gon{\c c}alo Tabuada was partially supported by the NEC Award-2742738.}

\abstract{V.~Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai
functors and dg algebras. In the same vein, D.~Shklyarov introduced the noncommutative analogue
of the Hirzebruch-Riemman-Roch theorem. In this short article, we see how these constructions
and computations formally stem from their motivic counterparts.}
}
\maketitle
\vskip-\baselineskip
\vskip-\baselineskip
\section{Introduction: traces, pairings, and Hochschild homology}
Let $k$ be a field. Lunts' results on the noncommutative Lefschetz formula
can be summarised as follows:
\begin{theorem}{(see~\cite[Thms 1.1, 1.2 and 1.4]{Lunts})}\label{thm:Lunts1}
Let $X$ be a smooth and proper $k$-scheme and ${\mathcal E}$ a bounded
complex of coherent ${\mathcal O}_{X\times X}$-modules. Then, the following equality holds
\begin{equation}\label{eq:equality1}
\sum_i (-1)^i \, \mathrm{dim}\,  \HH_i({\mathcal E}) = \sum_j (-1)^j \,
\mathrm{Tr}\, \HH_j(\Phi_{\mathcal E})\,,
\end{equation}
where $\Phi_{\mathcal E}$ is the associated Fourier-Mukai functor and
$\HH({\mathcal E})=\HH(X,{\mathcal E})$ the Hochschild homology of $X$ with coefficients in ${\mathcal E}$.
Moreover, when $k=\mathbf{C}$ we have
\begin{equation}\label{eq:equality2}
\sum_i (-1)^i \, \mathrm{dim} \, \HH_i({\mathcal E}) =
\mathrm{Tr}\,  H^{\mathrm{even}}(\Phi_{\mathcal E}) - \mathrm{Tr} \, H^{\mathrm{odd}}(\Phi_{\mathcal E})\,,
\end{equation}
where $H^{\mathrm{even}}$ and $H^{\mathrm{odd}}$ are the even and odd parts of Betti cohomology
with rational coefficients and $H^\ast(\Phi_{\mathcal E}):H^\ast(X)\to H^\ast(X)$ the correspondence associated to the class $\mathit{ch}({\mathcal E})\cdot\sqrt{\todd}_X \in H^\ast(X\times X)$.

Let $A$ be a smooth and proper dg $k$-algebra and $M$ a perfect dg $A$-bimodule. Then, the following equality holds
\begin{equation}\label{eq:equality3}
\sum_i (-1)^i \, \mathrm{dim} \, \HH_i(A;M) =
\sum_j (-1)^j\, \mathrm{Tr}\, \HH_j(\Phi_M)\,.
\end{equation}
\end{theorem}
Shklyarov's noncommutative analogue of the Hirzebruch-Riemman-Roch theorem
can be stated as follows.
\begin{theorem}{(see~\cite[Thm.~3]{Shklyarov})}\label{thm:HRR}
 Let $A$ be a proper dg $k$-algebra. Then, there is a canonical pairing
$$ \langle -,- \rangle: \HH(A)\otimes_k\HH(A^\mathsf{op})\longrightarrow k$$
such that for any two perfect dg $A$-modules $M$ and $N$
the following equality holds
\begin{equation}\label{eq:HRR}
\chi(M,N) = \langle \chi(N), \chi(DM)\rangle\,,
\end{equation}
where $DM$ is the formal dual of $M$, $\chi$ is the Euler character
(also known as Dennis trace map), and
$\chi(M,N)=\chi(\mathbf{R}\mathit{Hom}_A(M,N))$ is the Euler character of the derived Hom
of maps from $M$ to $N$.
\end{theorem}
In this short article we prove the following general motivic results (over a base commutative ring $k$) and show how the above equalities~\eqref{eq:equality1}-\eqref{eq:equality3} and \eqref{eq:HRR} can be recovered from them.
\begin{theorem}[Noncommutative Lefschetz]\label{thm:main}
Let ${\mathcal A}$ be a smooth and proper dg category (see \S\ref{sec:dg}),
$M$ a perfect dg ${\mathcal A}$-bimodule, and $L:\mathsf{dgcat}(k) \to \mathsf{D}$ a
symmetric monoidal\footnote{As far as monoidal categories and monoidal functors
are concerned, we shall use the terminology of MacLane's book~\cite{ML}.}
additive invariant (see Definition~\ref{def:tensor-additive2})
which becomes strongly monoidal when restricted to smooth and proper dg categories.
Assume that the base ring $k$ is local (or more generally that $K_0(k)=\mathbf{Z}$).
Then, the following equality holds in the ring of endomorphims of the unit object
of $\mathsf{D}$:
\begin{equation}\label{eq:equality-main}
\sum_i(-1)^i \, \mathrm{rk}\,  \HH_i({\mathcal A};M) = \mathrm{Tr}\, L(\Phi_M)\,.
\end{equation}
\end{theorem}
\begin{theorem}\label{thm:main2}
Let $k$ be a field, $X$ a smooth and proper $k$-scheme, and ${\mathcal E}$ a bounded complex of coherent ${\mathcal O}_{X\times X}$-modules.
Then, for every Weil cohomology $H^\ast$ defined on smooth and proper
$k$-schemes, we have the equality
\begin{equation}\label{eq:equality-main2}
\sum_i (-1)^i \, \mathrm{dim} \, \HH_i({\mathcal E})
= \mathrm{Tr}\,  H^{\mathrm{even}}(\Phi_{\mathcal E}) - \mathrm{Tr}\,  H^{\mathrm{odd}}(\Phi_{\mathcal E})\, ,
\end{equation}
where $H^{\mathrm{even}}$ and $H^{\mathrm{odd}}$ are the
the even and odd parts of $H^\ast$.
\end{theorem}

\begin{theorem}[Noncommutative Hirzebruch-Riemman-Roch]\label{thm:HRR1}
Let $L:\mathsf{dgcat}(k)\to \mathsf{D}$ be
a strongly symmetric monoidal additive invariant (see Definition~\ref{def:tensor-additive2}),
and $\mathbf{1}$ the unit object of $\mathsf{D}$.
For any dg category ${\mathcal A}$, there is a canonical Chern character map
$$\mathit{ch}_L:K_0({\mathcal A})\longrightarrow \mathrm{Hom}_\mathsf{D}(\mathbf{1},L({\mathcal A}))\, .$$
Moreover, in the case where ${\mathcal A}$ is proper, there is a canonical pairing in $\mathsf{D}$
$$L({\mathcal A})\otimes L({\mathcal A}^\mathsf{op})\longrightarrow\mathbf{1}$$
and consequently a canonical pairing of abelian groups
$$\langle -,- \rangle:\mathrm{Hom}_\mathsf{D}(\mathbf{1},L({\mathcal A}))\otimes
\mathrm{Hom}_\mathsf{D}(\mathbf{1},L({\mathcal A}^\mathsf{op}))\to\mathrm{End}_\mathsf{D}({\bf 1})\,.$$
Furthermore, for any two perfect dg ${\mathcal A}$-modules $M$ and $N$, the following equality
\begin{equation}\label{eq:HRR1}
ch_L(\mathbf{R}\mathit{Hom}_{\mathcal A}(M,N))=\langle ch_L(N), ch_L(DM)\rangle
\end{equation}
holds in $\mathrm{End}_{\mathsf{D}}({\bf 1})$.
\end{theorem}
Consult Sections \ref{section:nclefschetz} and \ref{sec:HRR}
for the proofs of these three statements. The proofs of Theorems \ref{thm:main} and \ref{thm:HRR1} rely on formal arguments, namely on the fact that symmetric monoidal additive invariants correspond to
symmetric monoidal functors from an adequate category of noncommutative motives, whose
Hom's can be understood as $K_0$-groups; see Proposition \ref{prop:universalmonoidal}.
The proof of Theorem \ref{thm:main2}
is also quite formal, once we have related the classical category of Chow motives
with the aforementioned category of noncommutative motives. This is
just a categorical reformulation of the Grothendieck-Riemann-Roch theorem; see
Theorem \ref{thm:categoricalGRR}. The proof of the equalities
\eqref{eq:equality-main}, \eqref{eq:equality-main2} and \eqref{eq:HRR1} follows also the same pattern: the left-hand-sides are numbers obtained from a motivic construction (trace or pairing
in a suitable category of motives) which one manages to compute using
the functorialities of the theory of dg categories. The right-hand-sides are similarly obtained from the realisation of the original data. In conclusion, these equalities follow simply from the fact that realizations of motives are strongly symmetric monoidal, \textsl{i.e.}\  they satisfy K\"unneth formulas.

\subsection*{Acknowledgments} The authors are very grateful to the anonymous referee for all his/her
comments that made us improve this short article.

\subsection*{Notations}\label{sec:notations}
The letter $k$ will always stand for a commutative ring with unit $k$.
Given a quasi-compact separated $k$-scheme $X$, we will denote by $\mathsf{perf}(X)$ the dg category
of perfect complexes of ${\mathcal O}_X$-modules, and by ${\mathcal D}_\mathsf{perf}(X)$ the corresponding
triangulated category. Recall that if $X$ is regular then every bounded complex of coherent sheaves on
$X$ is perfect (up to quasi-isomorphism).

Given an essentially small category ${\mathcal C}$, we will write $\mathrm{Iso}\, {\mathcal C}$ for the
set of isomorphism classes of objects of ${\mathcal C}$.

\section{Differential graded categories}\label{sec:dg}
Let $k$ be a base commutative ring and ${\mathcal C}(k)$ the category of dg $k$-modules. A {\em differential graded (=dg) category} ${\mathcal A}$ is a category enriched over ${\mathcal C}(k)$ (morphism sets ${\mathcal A}(x,y)$ are dg $k$-modules) in such a way that composition fulfills the Leibniz rule: $d(f\circ g) = d(f) \circ g + (-1)^{\mathrm{deg}(f)}f \circ d(g)$. A {\em dg functor} $F:{\mathcal A} \to {\mathcal B}$ is a functor enriched over ${\mathcal C}(k)$; consult Keller's ICM address~\cite[\S2.2-2.3]{ICM}. In what follows we will write $\mathsf{dgcat}(k)$ for the category of (small) dg categories and dg functors.

Let ${\mathcal A}$ be a dg category. Its {\em opposite} dg category ${\mathcal A}^\mathsf{op}$ has the same objects and dg
$k$-modules of morphisms given by ${\mathcal A}^\mathsf{op}(x,y):={\mathcal A}(y,x)$. A (right) dg {\em ${\mathcal A}$-module} is a
dg functor $M:{\mathcal A}^\mathsf{op} \to {\mathcal C}^\mathsf{dg}(k)$ with values in the dg category ${\mathcal C}^\mathsf{dg}(k)$ of dg $k$-modules.
Let us denote by $\widehat{{\mathcal A}}$ the category of dg ${\mathcal A}$-modules; see \cite[\S2.3]{ICM}.
Recall from \cite[\S3.2]{ICM} that the {\em derived category ${\mathcal D}({\mathcal A})$ of ${\mathcal A}$} is the
localization of $\widehat{{\mathcal A}}$ with respect to the class of objectwise quasi-isomorphisms.
Its full subcategory of compact objects (see \cite[Def.~4.2.7]{Neeman}) will be denoted by
${\mathcal D}_c({\mathcal A})$. A dg functor $F:{\mathcal A} \to {\mathcal B}$ is called a {\em Morita equivalence} if the restriction
of scalars functor ${\mathcal D}({\mathcal B}) \stackrel{\sim}{\to} {\mathcal D}({\mathcal A})$ is an equivalence of triangulated
categories; see \cite[\S4.6]{ICM}.

Let $\Hmo(k)$ be the localization of $\mathsf{dgcat}(k)$ with respect to the class of Morita equivalences.
The tensor product of dg categories can be naturally derived $-\otimes^\mathbf{L}-$ giving thus rise to a
symmetric monoidal structure on $\Hmo(k)$; see \cite[Remark~5.11]{IMRN}.
A {\em ${\mathcal A}\text{-}{\mathcal B}$-bimodule $M$} is a dg functor $M:{\mathcal A} \otimes^\mathbf{L} {\mathcal B}^\mathsf{op}\to {\mathcal C}^\mathsf{dg}(k)$,
\textsl{i.e.}\  a dg $({\mathcal A}^\mathsf{op} \otimes^\mathbf{L} {\mathcal B})$-module.
Given any two dg categories
${\mathcal A}$ and ${\mathcal B}$, we have a bijection
\begin{eqnarray}\label{eq:bij}
\mathrm{Iso}\,\mathsf{rep}({\mathcal A},{\mathcal B}) \stackrel{\sim}{\longrightarrow} \mathrm{Hom}_{\Hmo(k)}({\mathcal A},{\mathcal B}) &&
M \longmapsto \left(x\longmapsto {\mathcal A}(-,x)\otimes^\mathbf{L}_{\mathcal A} M \right)\,,
\end{eqnarray}
where $\mathsf{rep}({\mathcal A},{\mathcal B})$ is the full triangulated subcategory of ${\mathcal D}({\mathcal A}^\mathsf{op} \otimes^\mathbf{L}{\mathcal B})$ spanned by the dg
${\mathcal A}\text{-}{\mathcal B}$-bimodules $M$ such that for every object $x \in {\mathcal A}$ the associated dg ${\mathcal B}$-module $M(-,x)$ belongs
to ${\mathcal D}_c({\mathcal B})$; consult \cite[Cor.~5.10]{IMRN} for further details. Moreover, under the above bijection \eqref{eq:bij}, the composition law in $\Hmo(k)$ corresponds to the (derived) tensor product of bimodules.

Following Kontsevich~\cite{IAS,ENS}, a dg category ${\mathcal A}$ is called  {\em smooth} if it is perfect as a
dg bimodule over itself. A dg category ${\mathcal A}$ is said to be {\em proper} if for every ordered pair of objects $(x,y)$ the dg $k$-module ${\mathcal A}(x,y)$ is perfect\footnote{If the
base ring is not a field, this notion of properness really belongs to the realm of
derived geometry; perfect complexes over flat and proper $k$-schemes still give
examples, though.}.
If ${\mathcal A}$ is smooth and proper we have an equivalence of triangulated categories
$\mathsf{rep}({\mathcal A},{\mathcal B})\simeq {\mathcal D}_c({\mathcal A}^\mathsf{op} \otimes^\mathbf{L} {\mathcal B})$.
Consequently, \eqref{eq:bij} reduces to the bijection
\begin{eqnarray*
\mathrm{Iso}\, {\mathcal D}_c({\mathcal A}^\mathsf{op} \otimes^\mathbf{L} {\mathcal B}) \stackrel{\sim}{\longrightarrow} \mathrm{Hom}_{\Hmo(k)}({\mathcal A},{\mathcal B})&&
M\longmapsto \Phi_M:=\left(x\longmapsto {\mathcal A}(-,x)\otimes^\mathbf{L}_{\mathcal A} M \right) \,.
\end{eqnarray*}
In fact, smooth and proper dg categories are precisely the
rigid objects in $\Hmo(k)$; see \cite[Theorem~4.8]{CT1}. More precisely, if ${\mathcal A}$ is smooth and proper then ${\mathcal A}^\mathsf{op}$ is its dual and for any two dg categories ${\mathcal B}$ and ${\mathcal C}$ one has
$$\mathrm{Hom}_{\Hmo(k)}({\mathcal A}\otimes^\mathbf{L}{\mathcal B},{\mathcal C})\simeq
\mathrm{Hom}_{\Hmo(k)}({\mathcal B},{\mathcal A}^\mathsf{op}\otimes^\mathbf{L}{\mathcal C})\, .$$
For instance, for any smooth and proper $k$-scheme $X$, the dg category $\mathsf{perf}(X)$
is smooth and proper (this follows immediately from To\"en's
beautiful results \cite[Lemma 8.11 and Theorem 8.15]{Toen}).
\section{Category of integral kernels}\label{sec:category-kernels}
The category $\mathbf{P}(k)$ of {\em integral kernels} is the full subcategory of $\Hmo(k)$
whose objects are the dg categories isomorphic to $\mathsf{perf}(X)$ for any smooth and proper
$k$-scheme $X$. Let us choose, for each smooth and proper $k$-scheme $X$,
and equivalence
\begin{eqnarray}\label{eq:isom0}
\mathsf{perf}(X)^\mathsf{op}\simeq\mathsf{perf}(X)
\end{eqnarray}
(that is, let us identify $\mathsf{perf}(X)$ with its dual). We have then, for any two
smooth and proper $k$-schemes $X$ and $Y$, an identification
of the form
\begin{eqnarray}\label{eq:isom00}
\mathsf{perf}(X)^\mathsf{op}\otimes^\mathbf{L}\mathsf{perf}(Y)\simeq\mathsf{perf}(X)\otimes^\mathbf{L}\mathsf{perf}(Y)
\simeq\mathsf{perf}(X\times Y)
\end{eqnarray}
where the latter isomorphism follows automatically from \cite[Lemma 8.11]{Toen}
for instance.
Using \eqref{eq:bij}, we thus get a natural bijection
\begin{eqnarray}\label{eq:isom000}
\mathrm{Hom}_{\Hmo(k)}(\mathsf{perf}(X),\mathsf{perf}(Y))\simeq\mathrm{Iso}\,{\mathcal D}_\mathsf{perf}(X\times Y)\, .
\end{eqnarray}
In the case where \eqref{eq:isom0} is induced by the
duality functor ${\mathcal E}\mapsto \mathbf{R}\mathit{Hom}({\mathcal E},{\mathcal O}_X)$,
the bijection \eqref{eq:isom000} can be described explicitely as
\begin{eqnarray}\label{eq:isom1}
\mathrm{Iso}\,{\mathcal D}_\mathsf{perf}(X\times Y) \stackrel{\sim}{\longrightarrow}
\mathrm{Hom}_{\Hmo(k)}(\mathsf{perf}(X),\mathsf{perf}(Y)) && {\mathcal F} \mapsto \Phi_{\mathcal F}\,,
\end{eqnarray}
where $\Phi_{\mathcal F}$ is the {\em Fourier-Mukai dg functor}
\begin{eqnarray}\label{eq:isom2}
\Phi_{\mathcal F}: \mathsf{perf}(X) \longrightarrow \mathsf{perf}(Y) && {\mathcal G} \mapsto \mathbf{R}(\pi_Y)_\ast(\pi^\ast_X({\mathcal G})\otimes^\mathbf{L} {\mathcal F})
\end{eqnarray}
(see To\"en's proof of \cite[Theorem 8.15]{Toen}).
If one prefers to define \eqref{eq:isom0} using
Grothendieck-Serre duality, that is via the functor ${\mathcal E}\mapsto \mathbf{R}\mathit{Hom}({\mathcal E},{\mathcal K}_X)$ with ${\mathcal K}_X$ the dualizing complex, then the bijection \eqref{eq:isom000}
gives the description of the category $\mathbf{P}(k)$ considered by
Caldararu and Willerton in \cite{Cald} for instance\footnote{The
category $\mathbf{P}(k)$ is in fact the truncation of the bicategory
studied in \cite{Cald} (in the case where $k$ is a field).}.
Note however that the choice of the isomorphisms \eqref{eq:isom0} does not really matters,
as any two duals of $\mathsf{perf}(X)$ are uniquely isomorphic (as duals),
so that the corresponding
theories of pairings and trace maps cannot be seriously affected by any such
choices\footnote{For the reader who prefers to have explicit computations, at least
through the lens of Hochschild homology, we refer to the work of Ramadoss \cite{Ramadoss}.}.
In what follows, we will thus
work using the explicit identification \eqref{eq:isom1}, which corresponds to the convention
adopted in the work of To\"en~\cite{Toen} and Lunts~\cite{Lunts}. In this case, we can understand
the structure of monoidal category of $\mathbf{P}(k)$ as follows.
Given perfect complexes ${\mathcal F}\in {\mathcal D}_\mathsf{perf}(X\times Y)$ and ${\mathcal G} \in {\mathcal D}_\mathsf{perf}(Y\times Z)$,
their composition is the perfect complex
$$\mathbf{R}(\pi_{XZ})_\ast (\pi^\ast_{XY}({\mathcal F}) \otimes^\mathbf{L} \pi^\ast_{YZ}({\mathcal G}))\in {\mathcal D}_\mathsf{perf}(X\times Z)\, ,$$
(where $\pi_{ST}$ denotes the canonical projection from $X\times Y\times Z$ to $S\times T$).
The identity of an object $\mathsf{perf}(X)$ of $\mathbf{P}(k)$ is given by the structure
sheaf $\mathbf{R}\Delta_\ast({\mathcal O}_X) \in {\mathcal D}_\mathsf{perf}(X\times X)$ of the diagonal $\Delta$ of $X\times X$.
Given perfect complexes ${\mathcal E} \in {\mathcal D}_\mathsf{perf}(X\times W)$ and ${\mathcal F} \in {\mathcal D}_\mathsf{perf}(Y\times Z)$,
their product is
$$ {\mathcal E} \boxtimes{\mathcal F} \in {\mathcal D}_\mathsf{perf}(X\times Y \times W \times Z)\,.$$
This explicit description of $\mathbf{P}(k)$ will be mainly used
to make the link between noncommutative Chow motives and classical intersection
theory; see Theorem \ref{thm:categoricalGRR}.
\section{Noncommutative (Chow) motives}\label{sec:Chow}
Let $\Hmo_0(k)$ be the additive category with the same objects as $\Hmo(k)$ and with morphisms given by $\mathrm{Hom}_{\Hmo_0(k)}({\mathcal A},{\mathcal B}):= K_0 \mathsf{rep}({\mathcal A},{\mathcal B})$ where $K_0\mathsf{rep}({\mathcal A},{\mathcal B})$ is the Grothendieck group of the triangulated category $\mathsf{rep}({\mathcal A},{\mathcal B})$. The composition law is induced from the one on $\Hmo(k)$; consult \cite[\S6]{IMRN}. The derived tensor product on $\Hmo(k)$ extends by linearity to $\Hmo_0(k)$ giving rise to a symmetric monoidal structure. Moreover, there is a natural sequence of symmetric monoidal functors
\begin{equation}\label{eq:universal}
{\mathcal U}: \mathsf{dgcat}(k) \longrightarrow \Hmo(k) \stackrel{[-]}{\longrightarrow} \Hmo_0(k)\,.
\end{equation}
The first one is the identity on objects and sends a dg functor $F:{\mathcal A} \to {\mathcal B}$ to the corresponding ${\mathcal A}\text{-}{\mathcal B}$-bimodule. The functor $[-]$ is also the identity on objects and sends a ${\mathcal A}\text{-}{\mathcal B}$-bimodule $M$ to its class $[M]$ in the Grothendieck group $K_0\mathsf{rep}({\mathcal A},{\mathcal B})$.

The category $\NChow(k)$ of {\em noncommutative (Chow) motives} is by definition the idempotent completion of full subcategory of $\Hmo_0(k)$ of the smooth and proper dg categories. Note that we have the following description of its morphism
sets
$$\mathrm{Hom}_{\NChow(k)}({\mathcal A},{\mathcal B})=K_0({\mathcal D}_c({\mathcal A}^\mathsf{op} \otimes^\mathbf{L}{\mathcal B}))=K_0({\mathcal A}^\mathsf{op} \otimes^\mathbf{L} {\mathcal B})\,.$$
Note also that since the smooth and proper dg categories are stable under (derived) tensor product, the category $\NChow(k)$ is symmetric monoidal. Since the rigid objects in the symmetric monoidal category $\Hmo(k)$ are precisely the smooth and proper dg categories, $\NChow(k)$ is a rigid tensor category.
\section{Symmetric monoidal additive invariants}\label{sec:symmetric}
Let ${\mathcal A}$ be a dg category. Consider the dg category $T({\mathcal A})$ whose objects are the pairs $(i,x)$,
with $i \in \{1,2\}$ and $x$ an object of ${\mathcal A}$. The dg $k$-module
$T({\mathcal A})((i,x),(i',x'))$ is equal to ${\mathcal A}(x,x')$ if $i'\geq i$ and is $0$ otherwise.
Composition is induced from ${\mathcal A}$; consult \cite[\S4]{IMRN} for details.
Note that we have two natural inclusion dg functors
\begin{eqnarray*}
\iota_1 : {\mathcal A} \longrightarrow T({\mathcal A}) && \iota_2: {\mathcal A} \longrightarrow T({\mathcal A})\,.
\end{eqnarray*}

\begin{definition}\label{def:tensor-additive}
Let $L:\mathsf{dgcat}(k) \to \mathsf{D}$ be a functor with values in an additive category. We say that $L$ is an {\em additive invariant} if it satisfies the following two conditions:
\begin{itemize}
\item[(i)] it sends Morita equivalences to isomorphisms;
\item[(ii)] the above inclusion dg functors $\iota_1$ and $\iota_2$ induce an isomorphism\footnote{Condition (ii) can be equivalently formulated in terms of semi-orthogonal decompositions in the sense of Bondal-Orlov; see \cite[Thm.~6.3(4)]{IMRN}.}
$$ [L(\iota_1)\,\, L(\iota_2)]: L({\mathcal A}) \oplus L({\mathcal A}) \stackrel{\sim}{\longrightarrow} L(T({\mathcal A}))\,.$$
\end{itemize}
\end{definition}
\begin{theorem}{(\cite[Thms. 5.3 and 6.3]{IMRN})}\label{thm:universal}
The above functor \eqref{eq:universal} is the {\em universal additive invariant}, \textsl{i.e.}\  given any additive category $\mathsf{D}$ we have an induced equivalence of categories
\begin{equation}\label{eq:induced}
{\mathcal U}^\ast: \mathsf{Fun}_{\mathsf{add}}(\Hmo_0(k),\mathsf{D}) \stackrel{\sim}{\longrightarrow} \mathsf{Fun}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D})\,,
\end{equation}
where the left-hand-side denotes the category of additive functors from $\Hmo_0(k)$ to $\mathsf{D}$ and the right-hand-side the category of additive invariants with values in $\mathsf{D}$.
\end{theorem}

Note that $\Hmo_0(k)$ is a symmetric monoidal category such that the
canonical functor $\Hmo(k)\to\Hmo_0(k)$ is strictly symmetric monoidal.
The preceding theorem has a monoidal version as follows.
We will say that a dg category ${\mathcal A}$ is \emph{flat} if, for any
objects $x$ and $y$ of ${\mathcal A}$, the dg $k$-module ${\mathcal A}(x,y)$
is a flat $k$-module in each degree. We will write $\mathsf{dgcat}_f(k)$
for the full subcategory of $\mathsf{dgcat}(k)$ whose objects are the flat
dg categories.

\begin{definition}\label{def:tensor-additive2}
A \emph{symmetric monoidal additive invariant} is an additive invariant
$L:\mathsf{dgcat}(k)\to\mathsf{D}$ with values in a symmetric monoidal category $\mathsf{D}$
together with a structure of symmetric monoidal functor on the restriction of $L$
to the subcategory $\mathsf{dgcat}_f(k)$. Such an additive invaiant $L$ is said
to be \emph{strongly symmetric monoidal when restricted to
smooth and proper dg categories} if its restriction to the
full subcategory of $\mathsf{dgcat}_f(k)$ which consists of flat smooth and
proper dg categories is strongly monoidal in the sense of MacLane \cite{ML}.
\end{definition}

\begin{proposition}\label{prop:universalmonoidal}
Let $L:\mathsf{dgcat}(k)\to\mathsf{D}$ be a symmetric monoidal additive invariant.
Then, the corresponding additive functor $\underline{L}:\Hmo_0(k)\to\mathsf{D}$
has a canonical structure of symmetric monoidal functor.
In particular, if $L$ is moreover strongly symmetric monoidal when restricted to
smooth and proper dg categories, then the restriction of
$\underline{L}$ to the category $\NChow(k)$ is strongly monoidal.
\end{proposition}

\begin{proof}
Let $n\geq 1$ be an integer and $\mathsf{D}$ an additive category.
We claim that the composing with the functor ${\mathcal U}^n:\mathsf{dgcat}(k)^n\to\Hmo_0(k)$
induces an equivalence of categories
\begin{equation}\label{eq:multiinduced}
\mathsf{Fun}^n_{\mathsf{add}}(\Hmo_0(k),\mathsf{D}) \stackrel{\sim}{\longrightarrow}
\mathsf{Fun}^n_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D})\, ,
\end{equation}
where the left-hand-side denotes the category of functors $\Hmo_0(k)^n\to\mathsf{D}$ which
are additive in each variable, and the right-hand-side the category of functors $\mathsf{dgcat}(k)^n\to\mathsf{D}$ which are additive invariants
in each variable.
We claim also that the inclusion $\mathsf{dgcat}_f(k)\subset\mathsf{dgcat}$ induces equivalences of categories
\begin{equation}\label{eq:multiinduced2}
\mathsf{Fun}^n_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D}) \stackrel{\sim}{\longrightarrow}
\mathsf{Fun}^n_{\mathsf{A}}(\mathsf{dgcat}_f(k),\mathsf{D})\, ,
\end{equation}
where $\mathsf{Fun}^n_{\mathsf{A}}(\mathsf{dgcat}_f(k),\mathsf{D})$ denotes the category of
functors $\mathsf{dgcat}_f(k)^n\to\mathsf{D}$ which satisfy conditions (i)-(ii)
of Definition \ref{def:tensor-additive} (this makes sense because $T({\mathcal A})$ is flat
whenever ${\mathcal A}$ is flat). Note that the proof of Proposition~\ref{prop:universalmonoidal} follows automatically from the equivalences \eqref{eq:multiinduced}-\eqref{eq:multiinduced2}. The equivalences \eqref{eq:multiinduced2}
follow immediately from the compatibility of the functor $T$ with Morita equivalences
and from the fact that the inclusion functor
$\mathsf{dgcat}_f(k)\subset\mathsf{dgcat}(k)$ induces an equivalence after localization by Morita
equivalences since every cofibrant dg category in the sense of the
model category structures of \cite{CRAS,IMRN} is flat. In order to finish the proof it thus remains only to check that \eqref{eq:multiinduced} are equivalences of categories. We proceed by induction on $n$. The case where $n=1$ is provided by Theorem \ref{thm:universal}.
If $n>1$, then we have the following obvious equivalences of categories.
$$\begin{aligned}
\mathsf{Fun}^n_{\mathsf{add}}(\Hmo_0(k),\mathsf{D})
&\simeq\mathsf{Fun}_{\mathsf{add}}(\Hmo_0(k),\mathsf{Fun}^{n-1}_{\mathsf{add}}(\Hmo_0(k),\mathsf{D}))\\
\mathsf{Fun}^n_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D})
&\simeq\mathsf{Fun}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{Fun}^{n-1}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D}))\, .
\end{aligned}$$
On the other hand, by induction (and by applying again Theorem \ref{thm:universal})
we have
$$\begin{aligned}
\mathsf{Fun}_{\mathsf{add}}(\Hmo_0(k),\mathsf{Fun}^{n-1}_{\mathsf{add}}(\Hmo_0(k),\mathsf{D}))
&\simeq\mathsf{Fun}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{Fun}^{n-1}_{\mathsf{add}}(\Hmo_0(k),\mathsf{D}))\\
&\simeq\mathsf{Fun}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{Fun}^{n-1}_{\mathsf{A}}(\mathsf{dgcat}(k),\mathsf{D}))\,.
\end{aligned}$$
This achieves the proof.
\end{proof}
\begin{remark}
Note that the proof of the Proposition~\ref{prop:universalmonoidal} is also the proof
of the existence of the canonical symmetric monoidal structure
on $\Hmo_0(k)$.
\end{remark}
Let us now describe some examples of additive invariants which are moreover
symmetric monoidal.
\begin{example}[Hoschchild homology]\label{example:HH}
Let ${\mathcal D}(k)$ be the derived category of the base commutative ring $k$. By construction this category is additive and symmetric monoidal. As explained in \cite[\S6.1]{IMRN}, the Hochschild homology functor
\begin{equation*
\HH: \mathsf{dgcat}(k) \longrightarrow {\mathcal D}(k)
\end{equation*}
is an additive invariant. Thanks to \cite[Example~7.9]{CT1},
$\HH$ is furthermore symmetric monoidal and hence defines a
symmetric monoidal functor
\begin{equation*
\HH: \Hmo_0(k) \longrightarrow {\mathcal D}(k)\, .
\end{equation*}
In particular, for any smooth and proper dg category ${\mathcal A}$ (= rigid object of $\Hmo_0$ with dual ${\mathcal A}^\mathsf{op}$), $\HH({\mathcal A})$ is a rigid object of ${\mathcal D}(k)$ (\textsl{i.e.}\  a perfect complex) and the pairing considered in Theorem \ref{thm:HRR}
is non degenerate.

For a $k$-scheme $X$, we will write
$\HH(X)=\HH(\mathsf{perf}(X))$. The restriction of the functor $\HH$
to the category $\mathbf{P}(k)$
gives the usual Hochschild homology of smooth and proper $k$-schemes, whatever
version the reader might prefer (the agreement with Weibel's version of Hochschild
homology of schemes (with coefficients) is proved by Keller in \cite{Ringed},
while, as explained in~\cite[\S4.2]{Cald}, Weibel's definition is isomorphic to
Caldararu-Willerton's version of Hochschild homology).
\end{example}
\begin{example}[Mixed complexes]\label{example:mix}
Following Kassel \cite[\S1]{Kassel}, a {\em mixed complex} $(N,b,B)$ is a $\mathbf{Z}$-graded $k$-module $\{N_n\}_{n \in \mathbf{Z}}$ endowed with a degree $+1$ endomorphism $b$ and a degree $-1$ endomorphism $B$ satisfying the relations $b^2=B^2=Bb +bB=0$. Equivalently, a mixed complex is a right dg module over the dg algebra $\Lambda:=k[\epsilon]/\epsilon^2$, where $\epsilon$ is of degree $-1$ and $d(\epsilon)=0$. Let ${\mathcal D}(\Lambda)$ be the derived category of mixed complexes. By construction, ${\mathcal D}(\Lambda)$ is additive and as explained in \cite[\S6.1]{IMRN} the mixed complex functor
\begin{equation}\label{eq:C}
C: \mathsf{dgcat}(k) \longrightarrow {\mathcal D}(\Lambda)
\end{equation}
is an additive invariant. Moreover, ${\mathcal D}(\Lambda)$ carries a symmetric monoidal structure defined on the underlying dg $k$-modules under which $C$ is symmetric monoidal; see \cite[Example~7.10]{CT1}.
\end{example}
\begin{example}[Periodic cyclic homology]\label{Example:HP}
Assume that $k$ is a field. Using the work of Kassel~\cite{Kassel},
one can understand periodic cyclic homology as follows (we refer
to \cite[Examples 8.11 and 9.11]{CT1} for a slightly more detailed exposition).
Let $k[u]$ be the cocommutative Hopf algebra
of polynomials in one variable $u$ in degree $2$. To a mixed complex
(that is a $\Lambda$-module) $M$, one associates its periodization $P(M)$. This is
the $k[u]$-comodule whose underlying complex is $k\otimes^\mathbf{L}_\Lambda M$ and whose
comultiplication by $u$ is induced by Connes' map
$$S:k\otimes^\mathbf{L}_\Lambda M[-2]\longrightarrow k\otimes^\mathbf{L}_\Lambda M\, .$$
If ${\mathcal D}(k[u]\text{-}\mathrm{Comod})$ denotes the homotopy category of
$k[u]$-comodules, endowed with its symmetric tensor product induced
by the cotensor product of comodules, this defines
a symmetric monoidal triangulated functor
\begin{equation*
P: {\mathcal D}(\Lambda)\longrightarrow{\mathcal D}(k[u]\text{-}\mathrm{Comod})\, .
\end{equation*}
The unit object of ${\mathcal D}(k[u]\text{-}\mathrm{Comod})$ is $k[u]$ and for any
dg category ${\mathcal A}$ we have a canonical isomorphism in the derived
category of $k$-modules
\begin{equation*
\HP({\mathcal A})\simeq\mathbf{R}\mathrm{Hom}(k[u],P(C({\mathcal A})))\, ,
\end{equation*}
where $\HP({\mathcal A})$ is the periodic cyclic homology complex of ${\mathcal A}$.
In particular, we have a natural isomorphism of $k$-vector spaces
\begin{equation*
\HP_n({\mathcal A})\simeq\mathrm{Hom}_{{\mathcal D}(k[u]\text{-}\mathrm{Comod})}(k[u],P(C({\mathcal A})[-n]))\,.
\end{equation*}
Note that for any $k[u]$-comodule $M$ we have
$$\mathrm{Hom}_{{\mathcal D}(k[u]\text{-}\mathrm{Comod})}(k[u],M[n])=
\begin{cases}
\mathrm{Hom}_{{\mathcal D}(k[u]\text{-}\mathrm{Comod})}(k[u],M)&\text{if $n$ is even,}\\
\mathrm{Hom}_{{\mathcal D}(k[u]\text{-}\mathrm{Comod})}(k[u],M[1])&\text{if $n$ is odd.}
\end{cases}$$
We thus have a canonical symmetric monoidal functor
from ${\mathcal D}(k[u]\text{-}\mathrm{Comod})$ to $\mathrm{Vect}_{\mathbf{Z}/2}(k)$. By pre-composing it with the mixed complex functor $C$ we thus get
a symmetric monoidal functor
\begin{equation}\label{eq:P4}
\HP_\ast:\mathsf{dgcat}(k)\longrightarrow \mathrm{Vect}_{\mathbf{Z}/2}(k)\,.
\end{equation}
Thanks to Kassel's work, \eqref{eq:P4} becomes
strongly symmetric monoidal when restricted to smooth and proper dg categories;
see~\cite[Theorem~7.2]{Galois}.
\end{example}
\section{Perioditization of classical Chow motives}\label{sec:periodizationChow}
Let ${\mathcal C}$ be an additive symmetric monoidal category ${\mathcal C}$ endowed with a $\otimes$-invertible
object $L$ (\textsl{i.e.}\  such that the functor $X\mapsto L\otimes X$ is an equivalence of categories),
such that the twist $\tau : L\otimes L\to L\otimes L$ is the identity. Out of this data we define the category ${\mathcal C}^\mathrm{per}$ as follows. The objects are the same as those of ${\mathcal C}$ and the maps are given by the formula
\begin{equation*
\mathrm{Hom}_{{\mathcal C}^\mathrm{per}}(X,Y)=\bigoplus_{n\in\mathbf{Z}}\mathrm{Hom}_{\mathcal C}(X,L^{\otimes n}\otimes Y)\, .
\end{equation*}
The composition of two maps $u:X\to L^{\otimes m}\otimes Y$ and $v:Y\to L^{\otimes n}\otimes Z$ of ${\mathcal C}^\mathrm{per}$ is by definition
$$X\xrightarrow{ \ u \ }L^{\otimes m}\otimes Y\xrightarrow{L^{\otimes n}\otimes v}
L^{\otimes m}\otimes L^{\otimes n}\otimes Z
\simeq L^{\otimes m+n}\otimes Z\,.$$
Note that we have a canonical isomorphism $\mathbf{1}\simeq L$ in ${\mathcal C}^\mathrm{per}$ given by the
identity of the unit object $\mathbf{1}$ in ${\mathcal C}$. The category ${\mathcal C}^\mathrm{per}$ has a symmetric
monoidal structure (this is where we use the assumption that the twist $\tau$ is the identity) which is uniquely determined by the fact that
we have a strict symmetric monoidal functor
\begin{equation}\label{eq:mapsper2}
c:{\mathcal C}\longrightarrow{\mathcal C}^\mathrm{per}
\end{equation}
defined as the identity on objects and by the canonical inclusions
$$\mathrm{Hom}_{\mathcal C}(X,Y)=\mathrm{Hom}_{\mathcal C}(X,L^{\otimes 0}\otimes Y)\subset
\bigoplus_{n\in\mathbf{Z}}\mathrm{Hom}_{\mathcal C}(X,L^{\otimes n}\otimes Y)=\mathrm{Hom}_{{\mathcal C}^\mathrm{per}}(X,Y)\,.$$
It is easy to check that the functor \eqref{eq:mapsper2} has the following universal property:
given any symmetric monoidal category ${\mathcal D}$, composing with $c$ induces
an equivalence of categories between the category of strong symmetric
monoidal functors from ${\mathcal C}^\mathrm{per}$ to ${\mathcal D}$ and the category of pairs $(F,i)$,
where $F:{\mathcal C}\to{\mathcal D}$ is a strong symmetric monoidal functor and $i:\mathbf{1}\stackrel{\sim}{\to} F(L)$
is an isomorphism.
\begin{example}\label{example:periodisationofgradedvectorspaces}
In the case where ${\mathcal C}=\mathrm{Vect}_\mathbf{Z}(k)$ is the category of $\mathbf{Z}$-graded vector spaces over some field $k$, and $L$ is a graded
vector space of rank $1$ concentrated in degree $2$, ${\mathcal C}^\mathrm{per}$ is (canonically equivalent to) the category $\mathrm{Vect}_{\mathbf{Z}/2}(k)$
of super vector spaces (once we choose a generator of $L$).
\end{example}
Given a field $k$, let $\Chow(k)_\mathbf{Q}$ be the $\mathbf{Q}$-linear
category of \emph{Chow motives}; see \cite[Chapter~4]{andre}, \cite[Chapter~16]{Fulton} or \cite{Manin} for instance).
This category is defined as the idempotent completion of the
category whose objects are the pairs $(X,m)$, with $X$ a smooth and proper
$k$-scheme, and whose morphisms are given by
$$\mathrm{Hom}_{\Chow(k)_\mathbf{Q}}((X,m),(Y,n))=\prod_{i\in I}\CH^{d_i-m+n}(X_i\times Y)\,.$$
Here, $(X_i)_{i\in I}$, is the family of connected components of $X$, $d_i$ the dimension of $X_i$, and $\CH^c(W)_\mathbf{Q}$ the $\mathbf{Q}$-vector
space of algebraic cycles of codimension $c$ in $W$ modulo rational
equivalence\footnote{The category of Chow motives is often made out of
smooth and projective $k$-schemes, but this gives equivalent categories, at least
in the case where the ground field $k$ is perfect. In general, the theory of weights
tells us that the good theory of Chow motives should be constructed out of proper
and regular $k$-schemes (as opposed to proper and smooth ones),
but we won't go in this direction here.}.
The composition law is the usual composition of algebraic correspondences.
We will denote by ${\mathcal P}(k)$ the category of smooth and proper $k$-schemes.
We have a natural functor
\begin{equation}\label{eq:chowmotiveofschemes}
M:{\mathcal P}(k)^\mathsf{op}\longrightarrow\Chow(k)_\mathbf{Q} \quad X\longmapsto M(X)=(X,0)
\end{equation}
which sends a map $f:X\to Y$ to the cycle of its graph.
There is a unique symmetric monoidal structure on $\Chow_\mathbf{Q}(k)$
such that the functor $M$ is strictly symmetric monoidal with respect to the
cartesian product in ${\mathcal P}(k)$.
Let us write $L$ for the Lefschetz
motive (so that the motive $M(\mathbf{P}^1)$ is canonically
isomorphic to $\mathbf{Q}\oplus L$, where $\mathbf{Q}$ stands for the
motive of $\mathrm{Spec}(k)$). For any smooth and proper $k$-scheme $X$
and any integer $m$, we have a natural isomorphism $(X,m)\otimes L\simeq (X,m+1)$; this is a reformulation of the projective bundle formula.
We have also a universal perioditization functor with respect to $L$
\begin{equation}\label{eq:mapsper3}
\Chow(k)_\mathbf{Q}\longrightarrow\Chow(k)^\mathrm{per}_\mathbf{Q}
\end{equation}
which is the identity on objects. Since this functor is symmetric
monoidal and every object of $\Chow(k)$ is rigid, all the objects of
$\Chow(k)_\mathbf{Q}^\mathrm{per}$ are rigid as well.

Recall that a \emph{Weil cohomology} is an additive
strong symmetric monoidal functor
\begin{equation}\label{eq:weilcoh}
H^\ast:\Chow(k)_\mathbf{Q}\longrightarrow \mathrm{Vect}_\mathbf{Z}(K)
\end{equation}
such that $H^\ast(L)$ is concentrated in degree $2$
(for some field of characteristic zero $K$); see \cite[Proposition 4.2.5.1]{andre}.
Note that this implies that the vector space $H^\ast(L)$ is of dimension $1$.
In particular, by considering the perioditization of $\mathrm{Vect}_\mathbf{Z}(K)$
with respect to $H^\ast(L)$, we get a canonical commutative diagram
of strong symmetric monoidal functors
\begin{equation}\label{eq:periodizeWeilcoh}
\begin{split}
\xymatrix{
\Chow(k)_\mathbf{Q}\ar[r]^{H^\ast}\ar[d]&\mathrm{Vect}_\mathbf{Z}(K)\ar[d]\\
\Chow(k)^\mathrm{per}_\mathbf{Q} \ar[r]^{H^\mathrm{per}}&\mathrm{Vect}_{\mathbf{Z}/2}(K)
}
\end{split}
\end{equation}
in which $H^\mathrm{per}$ consists of the even and the odd part of $H^\ast$, \textsl{i.e.}\
$H^\mathrm{per}(X)=(H^\mathrm{even}(X),H^\mathrm{odd}(X))$ with
$$H^\mathrm{even}(X)=\bigoplus_{\text{$n$ even}}H^n(X)\quad \text{and} \quad
H^\mathrm{odd}(X)=\bigoplus_{\text{$n$ odd}}H^n(X) \, .$$
The known examples of Weil cohomologies include \'etale $\ell$-adic cohomology for any
prime number $\ell$ distinct from the characteristic of $k$,
crystalline cohomology (if $\mathit{char}(k)>0$), algebraic de Rham
cohomology (if $\mathit{char}(k)=0$), and Betti (or singular) cohomology
(if $k\subset\mathbf{C}$).

The category $\Chow(k)^\mathrm{per}_\mathbf{Q}$ is strongly related with the category
$\Hmo_0(k)$ as follows. Let $\Hmo_{0,\mathbf{Q}}(k)$ be the idempotent
completion of the universal $\mathbf{Q}$-linear additive category obtained from $\Hmo_0(k)$.
In other words, $\Hmo_{0,\mathbf{Q}}(k)$ is the idempotent completion of the
category $\Hmo_{0}(k)\otimes\mathbf{Q}$ whose objects are those of $\Hmo_0(k)$
and whose groups of morphisms are given by the formula
$$\mathrm{Hom}_{\Hmo_0(k)\otimes\mathbf{Q}}({\mathcal A},{\mathcal B})=\mathrm{Hom}_{\Hmo_0(k)}({\mathcal A},{\mathcal B})\otimes\mathbf{Q}\, .$$

Given a smooth and proper morphism $X\to S$, let us denote by $\mathrm{Td}_{X/S}$
(or simply by $\mathrm{Td}_X$ if the base $S$ is fixed) the
Todd class of the tangent bundle of $X/S$.
Given a commutative $\mathbf{Q}$-algebra $A$ and a power series of the form
$$\varphi=1+\sum_{n\geq 1}a_n X^n\in A[[X]]\, ,$$
its square root $\sqrt{\varphi}$ is defined as $\exp\Big(\frac{1}{2}\log(\varphi)\Big)$. In particular, this defines the square root of a Todd class.

A categorical reformulation of the Grothendieck-Riemann-Roch theorem
for proper morphisms between smooth and proper $k$-schemes
(using the explicit description of the category $\mathbf{P}(k)$
given at the end of Section \ref{sec:category-kernels}) is the following.
\begin{theorem}\label{thm:categoricalGRR}
There is a unique strong symmetric monoidal fully faithful functor
\begin{equation}\label{eq:embedChowinNChow}
\iota:\Chow(k)^\mathrm{per}_\mathbf{Q}\longrightarrow \Hmo_{0,\mathbf{Q}}(k)
\end{equation}
with the following three properties:
\begin{itemize}
\item[(i)] it sends the motive $M(X)$ of a smooth and proper $k$-scheme $X$ to
the dg category $\mathsf{perf}(X)$;
\item[(ii)] for any two smooth and projective $k$-schemes $X$ and $Y$, the
monoidal constraint
$$\iota(M(X))\otimes\iota(M(Y))=\mathsf{perf}(X)\otimes\mathsf{perf}(Y)\longrightarrow\mathsf{perf}(X\times Y)=\iota(M(X)\otimes M(Y))$$
is the map induced by the canonical isomorphism \eqref{eq:isom00};
\item[(iii)] on maps between motives of smooth and projective $k$-schemes, the above functor is given by the inverse of the isomorphism
$$\mathit{ch}\cdot\sqrt{\todd}_{X\times Y}: K_0(X\times Y)\otimes\mathbf{Q}
\xrightarrow{ \sim }\bigoplus_{n\in\mathbf{Z}}\CH^n(X\times Y)_\mathbf{Q}\,.$$
\end{itemize}
\end{theorem}
\begin{proof}
Given a smooth and proper $k$-scheme $X$, let us write
$$\CH^\ast(X)=\bigoplus_{n\in\mathbf{Z}}\CH^n(X)_\mathbf{Q}=\prod_{n\in\mathbf{Z}}\CH_\mathbf{Q}^n(X)\, .$$
Consider now a smooth and proper $k$-scheme $S$. Given two smooth and proper $S$-schemes $X$ and $Y$, let us denote by
$p:X\times_S Y\to X$ and $q:X\times_S Y\to Y$ the canonical projections.
For any ${\mathcal E}\in K_0(X\times_S Y)$
we then have the following commutative diagram of abelian groups
(in which $\mathrm{Td}_q=p^\ast(\mathrm{Td}_X)$
denotes the Todd class of the tangent bundle of $q$)
\begin{equation}\label{eq:proofcatGRR1}\begin{split}\xymatrix{
K_0(X)\ar[r]^(.45){p^\ast}\ar[d]^{\mathit{ch}\cdot\sqrt{\todd}_X}
&K_0(X\times_S Y)\ar[r]^{\cdot{\mathcal E}}\ar[d]^{\mathit{ch}\cdot p^\ast(\sqrt{\todd}_{X})}
&K_0(X\times_S Y)\ar[r]^(.55){q_\ast}\ar[d]^{\mathit{ch}\cdot\mathrm{Td}_q\cdot q^\ast(\sqrt{\todd}_{Y})}
&K_0(Y)\ar[d]^{\mathit{ch}\cdot\sqrt{\todd}_Y}\\
\CH^\ast(X)\ar[r]_(.45){p^\ast}
&\CH^\ast(X\times_S Y)\ar[r]_{\cdot\mathit{ch}({\mathcal E})\cdot \sqrt{\todd}_{X\times_S Y}}
&\CH^\ast(X\times_S Y)\ar[r]_(.55){q_\ast}
&\CH^\ast(Y)
}\end{split}\end{equation}
The commutavity of the left-hand and middle squares is obvious, while
the commutativity of the right-hand-side square stems from the Grothendieck-Riemann-Roch
theorem (see \cite[Theorem 15.2]{Fulton}, for instance),
and from the projection formula $q_\ast(a)\cdot b=q_\ast(a\cdot q^\ast(b))$.

Now, let ${\mathcal C}$ denote the category whose objects are the smooth and proper $k$-schemes, whose morphisms are defined by the formula
$$\mathrm{Hom}_{\mathcal C}(X,Y)=\CH^\ast(X\times Y)\,,$$
and whose composition law is defined through the usual formulas
for correspondences. The above arguments imply that there is a unique functor
\begin{equation}\label{eq:proofcatGRR2}
\iota:{\mathcal C}\longrightarrow\Hmo_{0,\mathbf{Q}}(k) \quad X\longmapsto\mathsf{perf}(X)
\end{equation}
which is defined on maps by the inverse of the isomorphisms $\mathit{ch}\cdot\sqrt{\todd}_{X\times Y}$.
Indeed, the compatibility with the composition of maps has already been checked
above in disguise: let $X$ and $Y$ be smooth and proper $k$-schemes and ${\mathcal E}\in K_0(X\times Y)$. If $\pi:X\times Y\times Z\to X\times Y$ denotes the canonical projection,
one just has to put $S=Z$ and to replace $X$ by $X\times Z$, $Y$ by $Y\times Z$ and ${\mathcal E}$ by $\pi^\ast({\mathcal E})$ in the commutative diagram \eqref{eq:proofcatGRR1}. The proof of this claim follows from the description of the composition law
of maps between objects of the form $\mathsf{perf}(X)$ in $\Hmo_{0,\mathbf{Q}}(k)$, from the
explicit description of the category of integral kernels (see Section~\ref{sec:category-kernels}),
and from the Yoneda lemma. It is clear that the functor \eqref{eq:proofcatGRR2}
is strongly symmetric monoidal, fully faithful, and uniquely determined by
the isomorphisms $\mathit{ch}\cdot\sqrt{\todd}_{X\times Y}$.
Since one can describe $\Chow^\mathrm{per}_\mathbf{Q}(k)$ as the idempotent completion of ${\mathcal C}$
and since $\Hmo_{0,\mathbf{Q}}(k)$ is idempotent complete, the proof of the theorem now follows immediately from pure ``abstract nonsense''.
\end{proof}

Let $\mathbf{P}_{0,\mathbf{Q}}(k)$ be the full subcategory of $\Hmo_{0,\mathbf{Q}}(k)$
whose objects are those isomorphic to a direct factor of a dg category
of the form $\mathsf{perf}(X)$, with $X$ a proper and smooth $k$-scheme.
Theorem~\ref{thm:categoricalGRR} might be equivalently formulated as an explicit
equivalence of symmetric monoidal categories from $\Chow^\mathrm{per}_\mathbf{Q}(k)$
to $\mathbf{P}_{0,\mathbf{Q}}(k)$. In particular, for any Weil cohomology $H^\ast$, its
perioditization $H^\mathrm{per}$ may as well be seen as a symmetric monoidal additive
functor
\begin{equation}\label{eq:defFMChowper0}
H^\mathrm{per}:\mathbf{P}_{0,\mathbf{Q}}(k)\to\mathrm{Vect}_{\mathbf{Z}/2}(K)\, .
\end{equation}
Moreover, the explicit description of the category $\mathbf{P}(k)$
of integral kernels (Section \ref{sec:category-kernels})
implies that the category $\mathbf{P}_{0,\mathbf{Q}}(k)$
is canonically equivalent to the idempotent completion of the
category whose objects are smooth and proper $k$-schemes, and whose maps are given by the groups $K_0(X\times Y)\otimes\mathbf{Q}$
(with the usual composition law for correspondences, using pullback
and pushforward maps in $K$-theory).

The explicit description of the functor \eqref{eq:defFMChowper0} is the following. For any two proper and smooth $k$-schemes $X$ and $Y$,
given an element ${\mathcal E}\in K_0(X\times Y)$, we will write
\begin{equation}\label{eq:defFMChowper1}
\Phi_{\mathcal E}=\mathit{ch}({\mathcal E})\cdot\sqrt{\todd}_{X\times Y}\in\bigoplus_{n\in\mathbf{Z}}\CH^n(X\times Y)_\mathbf{Q}\, .
\end{equation}
Given a Weil cohomology $H^\ast$ as above, as its perioditization $H^\mathrm{per}$
defines a functor from $\Chow^\mathrm{per}_\mathbf{Q}(k)$ to the category of super vector spaces,
we obtain a super map
\begin{equation}\label{eq:defFMChowper2}
H^\mathrm{per}(\Phi_{\mathcal E}):H^\mathrm{per}(X)\longrightarrow H^\mathrm{per}(Y)\,.
\end{equation}
Its trace is computed by the formula
\begin{equation}\label{eq:defFMChowper3}
\mathrm{Tr}\, H^\mathrm{per}(\Phi_{\mathcal E})=
\mathrm{Tr}\,  H^{\mathrm{even}}(\Phi_{\mathcal E}) - \mathrm{Tr} \, H^{\mathrm{odd}}(\Phi_{\mathcal E})
\in K=\mathrm{End}_{\mathrm{Vect}_{\mathbf{Z}/2}}(K)\, .
\end{equation}
\section{Noncommutative Lefschetz Theorem}\label{section:nclefschetz}
\begin{lemma}\label{lemma:abstracttraces}
Let ${\mathcal C}$ be symmetric monoidal additive category with unit object $\mathbf{1}$
such that $\mathrm{End}_{\mathcal C}(\mathbf{1})\subset\mathbf{Q}$.
Consider a strongly symmetric monoidal additive functor $F:{\mathcal C}\to\mathsf{D}$.
Then, for any rigid object $X$ of ${\mathcal C}$, the object $L(X)$ is also rigid, and
for any map $f:X\to X$ of ${\mathcal C}$, the trace $\mathrm{Tr}(L(f))$ is equal to the image
of the trace $\mathrm{Tr}(f)$ under the morphism of rings $\mathrm{End}_{\mathcal C}(\mathbf{1})\to \mathrm{End}_\mathsf{D}(\mathbf{1})$ induced by $L$. In particular, $\mathrm{Tr}(L(f))$ is the image of a rational number
(more precisely, of an element of $\mathrm{End}_{\mathcal C}(\mathbf{1})$) which does
not depend on the functor $L$.
\end{lemma}
The proof of the preceding lemma is straightforward.

\begin{proposition}\label{prop:absolutetracencmot}
Let ${\mathcal A}$ be a smooth and proper dg category. Consider
a perfect dg ${\mathcal A}$-bimodule $M$ (\textsl{i.e.}\  $M \in {\mathcal D}_c({\mathcal A}^\mathsf{op}\otimes^\mathbf{L} {\mathcal A})$), and
let us denote by $\Phi_{[M]}: {\mathcal A} \to {\mathcal A}$ the associated endomorphism
of ${\mathcal A}$ in $\NChow(k)$. Then we have the formula
$$\mathrm{Tr}(\Phi_{[M]}) = [\HH({\mathcal A};M)] \in K_0({\mathcal D}_c(k))\,.$$
In particular if $k$ is a local ring we have the identification $K_0(k)=K_0({\mathcal D}_c(k))\simeq\mathbf{Z}$ and consequently the formula
$$ \mathrm{Tr}(\Phi_{[M]}) = \sum_i (-1)^i \mathrm{rk}\, \HH_i({\mathcal A};M)\,.$$
\end{proposition}

\begin{proof}
See \cite[Prop.~4.3]{Semi}.
\end{proof}
\begin{corollary}\label{cor:absolutetracencmotcomm}
Given a smooth and proper $k$-scheme $X$ and a bounded complex ${\mathcal E}$ of
coherent ${\mathcal O}_{X\times X}$-modules, let us denote by
$\Phi_{\mathcal E}:\mathsf{perf}(X)\to\mathsf{perf}(X)$ the Fourier-Mukai morphism in $\Hmo_0(k)$
induced by ${\mathcal E}$. Then, the following equality holds
$$\mathrm{Tr}(\Phi_{{\mathcal E}}) = [\HH({\mathcal E})] \in K_0({\mathcal D}_c(k))\,.$$
\end{corollary}
\begin{proof}
This follows automatically from the combination of Proposition~\ref{prop:absolutetracencmot} with
the identification $\HH(\mathsf{perf}(X);M)\simeq\HH({\mathcal E})$
where $M$ denotes the $\mathsf{perf}(X)$-bimodule defined by ${\mathcal E}$
via the isomorphism \eqref{eq:isom00}; recall from Example \ref{example:HH} that thanks to the work of Keller all versions of Hochschild homology of schemes agree (including
with coefficients).
\end{proof}

\subsection*{Proof of Theorem~\ref{thm:main}}\label{proof:thm:main}
Let $L:\mathsf{dgcat}\to\mathsf{D}$ be a symmetric monoidal invariant and $\underline L:\Hmo_0\to\mathsf{D}$
the corresponding symmetric monoidal additive functor; see Proposition \ref{prop:universalmonoidal}.
Theorem~\ref{thm:main} follows immediate from Lemma \ref{lemma:abstracttraces} applied to $\underline L$
and from Proposition \ref{prop:absolutetracencmot}.\qed

\subsection*{Proof of Theorem~\ref{thm:main2}}\label{proof:thm:main2}
Assume that $k$ is a field and consider a Weil cohomology
$$H^\ast:\Chow_\mathbf{Q}(k)\longrightarrow\mathrm{Vect}_\mathbf{Z}(K)\, .$$
Recall from \eqref{eq:periodizeWeilcoh} the construction of the perioditization functor
\begin{equation}\label{eq:H-per}
H^\mathrm{per}:\Chow_\mathbf{Q}^\mathrm{per}(k)\longrightarrow\mathrm{Vect}_{\mathbf{Z}/2}(K)\, .
\end{equation}
Let ${\mathcal E}$ be a bounded complex of coherent ${\mathcal O}_{X\times X}$-modules and $\Phi_{\mathcal E}$ be the corresponding Fourier-Mukai
endomorphism in the category $\Chow^\mathrm{per}_\mathbf{Q}(k)$; see \eqref{eq:defFMChowper1}.
Thanks to Theorem \ref{thm:categoricalGRR} and
Corollary \ref{cor:absolutetracencmotcomm} the
trace of $\Phi_{\mathcal E}$ in $\Chow^\mathrm{per}_\mathbf{Q}(k)$ is equal to the Euler
character of the Hochschild homology complex $\HH({\mathcal E})$. The proof follows then from the application of Lemma \ref{lemma:abstracttraces} to \eqref{eq:H-per} and from the Formula \eqref{eq:defFMChowper3}.\qed

\subsection*{Proof of Lunts' Theorem \ref{thm:Lunts1}}
If $k$ is a field, by applying Theorem~\ref{thm:main} for $L=\HH$, the Hochschild
homology functor (Example~\ref{example:HH}), we see that
equality \eqref{eq:equality3} is a particular case of
Theorem \ref{thm:main}. Similarly, by applying Theorem \ref{thm:main2} in the
case of Betti cohomology with rational (or complex) coefficients,
we get Equality \eqref{eq:equality2}.
Equality \eqref{eq:equality1}
follows immediately from Equality \eqref{eq:equality3}
(and from Corollary \ref{cor:absolutetracencmotcomm}).\qed

\begin{remark}
Lunts derives Formula \eqref{eq:equality2}
from Formula \eqref{eq:equality1} using the work of Macri-Stellari \cite[Theorem~1.2]{MS} on the  Hochschild-Kostant-Rosenberg
isomorphism; see \cite[Theorem~1.4]{Lunts}. In other words, Lunts' proof of \eqref{eq:equality2} first considers Betti cohomology as a strong symmetric monoidal
functor from the category of Hochschild correspondences between smooth
and projective $k$-schemes (instead of the category of rational
$K$-theory correspondences) and then applies some form of Lemma \ref{lemma:abstracttraces}.
This argument requires to check that
the HKR isomorphism is functorial with respect to Fourier-Mukai functors, which is precisely the content of \cite[Theorem~1.2]{MS}. Nervertheless, note that Lunts' argument uses some
weak form of Theorem \ref{thm:categoricalGRR} though, at least to define
the functoriality with respect to Fourier-Mukai functors for (periodic) Betti
cohomology.
\end{remark}
\section{Noncommutative Hirzebruch-Riemman-Roch}\label{sec:HRR}

\subsection*{Proof of Theorem~\ref{thm:HRR1}}\label{proof:thm:HRR1}
Given a proper dg category ${\mathcal A}$, we have a well-defined pairing in $\Hmo(k)$:
\begin{equation}\label{eq:absolutepairing0}
{\mathcal A}\otimes^\mathbf{L} {\mathcal A}^\mathsf{op} \to \mathsf{perf}(k)\simeq k \ , \quad
(x,y) \mapsto {\mathcal A}(y,x)\,.
\end{equation}
By applying the symmetric monoidal
additive functor $\mathrm{Hom}_{\Hmo(k)}(k,-)$, and using the identifications $K_0({\mathcal A})=\mathrm{Hom}_{\Hmo(k)}(k,{\mathcal A})$, we then get a pairing of abelian groups:
\begin{equation}\label{eq:absolutepairing1}
\langle - , - \rangle:K_0({\mathcal A})\otimes K_0({\mathcal A})\to K_0(k)\, .
\end{equation}
Now, let $L:\mathsf{dgcat}(k) \to \mathsf{D}$ be
a strong symmetric monoidal additive invariant.
Thanks to Proposition \ref{prop:universalmonoidal} the functor $L$
canonically defines a strong symmetric monoidal functor $\underline{L}:\Hmo_0(k)\to\mathsf{D}$.
For a dg category ${\mathcal C}$, we will write
\begin{equation*
H({\mathcal C})=\mathrm{Hom}_\mathsf{D}(\mathbf{1},L({\mathcal C}))=\mathrm{Hom}_\mathsf{D}(\mathbf{1},\underline{L}({\mathcal C}))\, .
\end{equation*}
Since the functor $\underline{L}$ is strongly unital it induces
functorial maps
\begin{equation}\label{eq:abtractChern}
\mathit{ch}_L:K_0({\mathcal C})\longrightarrow H({\mathcal C})
\end{equation}
which we call the \emph{Chern characters}.

By applying the strong tensor functor $\underline{L}$ to \eqref{eq:absolutepairing0},
we obtain a pairing
\begin{equation}\label{eq:pair}
\underline{L}(\underline{{\mathcal A}}): L({\mathcal A}) \otimes L({\mathcal A}^\mathsf{op}) \longrightarrow {\bf 1}
\end{equation}
and consequently, by further applying the symmetric monoidal additive functor $\mathrm{Hom}_\mathsf{D}(\mathbf{1},-)$, a canonical pairing of abelian groups
\begin{equation}\label{eq:pair2}
\langle - , - \rangle: H({\mathcal A}) \otimes H({\mathcal A}^\mathsf{op}) \longrightarrow H(k)=\mathrm{End}_\mathsf{D}(\mathbf{1})\,.
\end{equation}
The fact that the functor $\underline{L}$ is strongly symmetric monoidal implies that the pairings \eqref{eq:absolutepairing1}
and \eqref{eq:pair2} are compatible with the Chern character \eqref{eq:abtractChern}.
In other words, we have the following commutative diagram:
\begin{equation}\label{eq:formalncHRR}
\begin{split}
\xymatrix{
K_0({\mathcal A})\otimes K_0({\mathcal A}^\mathsf{op})\ar[r]^(.65){\langle - , - \rangle}
\ar[d]_{\mathit{ch}_L\otimes\mathit{ch}_L}& K_0(k)\ar[d]^{\mathit{ch}_L}\\
H({\mathcal A}) \otimes H({\mathcal A}^\mathsf{op})\ar[r]^(.65){\langle - , - \rangle}& H(k)
}\end{split}
\end{equation}
Given a perfect ${\mathcal A}$-module $M$, which we can see as a map $M:k\simeq\mathsf{perf}(k)\to{\mathcal A}$ in $\Hmo(k)$,
we will still denote by $M$
its class in $K_0({\mathcal A})$. This can (and should) be interpreted
as a map from $k\simeq\mathsf{perf}(k)$ to ${\mathcal A}$ in $\Hmo_0(k)$.
By composing the duality functor $\mathsf{perf}(k)^\mathsf{op}\to\mathsf{perf}(k)$
with $M:\mathsf{perf}(k)\to{\mathcal A}$ we obtain
a map $\mathsf{perf}(k)^\mathsf{op}\to{\mathcal A}$ and consequently a map
$$DM:k\simeq\mathsf{perf}(k)\longrightarrow {\mathcal A}^\mathsf{op}$$
(in other words, $DM$ is a perfect ${\mathcal A}^\mathsf{op}$-module).
Note that, as we work up to Morita equivalence, we may always
suppose that $M$ is quasi-isomorphic to an ${\mathcal A}$-module of the
form ${\mathcal A}(-,x)$, with $x$ an object of ${\mathcal A}$, in which case $DM$
simply corresponds to the ${\mathcal A}^\mathsf{op}$-module ${\mathcal A}(x,-)={\mathcal A}^\mathsf{op}(-,x)$. In particular,
the pairing \eqref{eq:absolutepairing0} applied to the pair $(N,DM)$, with $M$ and $N$ any two perfect ${\mathcal A}$-modules, gives the perfect $k$-module
$\mathbf{R}\mathit{Hom}_{\mathcal A}(M,N)$.
We then have two maps $N:k\to {\mathcal A}$ and $DM:k\to{\mathcal A}^\mathsf{op}$ in $\Hmo_0(k)$,
and the commutativity of the square \eqref{eq:formalncHRR} gives the formula
$$\mathit{ch}_L(\langle N , DM \rangle)=\langle \mathit{ch}_L(N) , \mathit{ch}_L(DM) \rangle\, .$$
But, by definition, $\langle N , DM \rangle$ is the class of the complex
obtained by applying the pairing \eqref{eq:absolutepairing0} to $N$ and $DM$.
We then obtain Equality \eqref{eq:HRR1}.\qed
\subsection*{Proof of Theorem~\ref{thm:HRR}}\label{proof:thm:HRR}
In the case where $L=\HH$ is Hochschild homology, the
Chern characters \eqref{eq:abtractChern} agree with Dennis trace maps; see \cite[Theorem~2.8]{Prods}.
Therefore our constructions are compatible with those of Shklyarov \cite{Shklyarov}: Theorem \ref{thm:HRR}
readily follows from Theorem \ref{thm:HRR1}.\qed

\begin{example}
When $L$ is the mixed complex functor $C$ of Example~\ref{example:mix}
and $A$ is a proper dg algebra $A$, equality \eqref{eq:HRR1} reduces to
$$ \mathit{ch}^-(\mathbf{R}\mathit{Hom}_{\mathcal A}(M,N)) = \langle \mathit{ch}^-(N), \mathit{ch}^-(DM)\rangle\,,$$
where $\mathit{ch}^-$ is the classical Chern character with values in negative cyclic homology; see \cite[Theorem~2.8]{Prods}.
\end{example}
\begin{remark}
Given a proper dg category ${\mathcal A}$ and two perfect dg ${\mathcal A}$-modules $M$ and $N$, the following equality holds (essentially by definition)
\begin{equation}\label{eq:last1}
\HH({\mathcal A};DM\otimes^\mathbf{L}_k N)=\mathbf{R}\mathit{Hom}_{\mathcal A}(M,N)\,.
\end{equation}
Hence, whenever ${\mathcal A}$ is smooth
and proper, Shklyarov's Theorem~\ref{thm:HRR} is a direct consequence of
Lunts' Formula \eqref{eq:equality3}.
\end{remark}

redpajama_set_name: RedPajamaArXiv


I've done reports on many of the big international field and 3D championships in recent years. Basically they've been a compilation of photographs, scores and general information from various sources. For this year's World Archery Field Championships that hasn't happened. The reason is that I was a little more intimately involved in this one as it took place on home turf.
Consequently this is a very personal account and the photos aren't to the usual standard. Dark woods and bright sunshine don't help an amateur photographer using his phone as a camera.
A light into bright sunlight shot.
My role in this undertaking was as volunteer field crew, a small role. The bulk of the organisation and work took place above my pay grade. And what a job that was! You can't really comprehend all that goes into an event of that stature until you're in the middle of it.
Transporting electronic equipment for the World Archery team- more prestigious than moving toilets in the dark.
To state the obvious, field courses had to be designed, built, rebuilt, taken down and returned to storage. The field crew consisted of three teams designated, green, white and gold (funny that!). Each team was assigned to a course of the same name.
Field crew briefing from the Competition Manager.
The team consisted of a Course Designer, an Assistant Course Designer and three Field Crew working under the Competition Manager who oversaw all three courses and everything to do with the running of the competition.
The first three courses for the Qualification rounds were built in the rain the week before the competition started and by the Day of Official Practice were ready just to put target faces out. The target butts used were from the manufacturer Eleven. They are made of a hard grey foam square with softer, replaceable, yellow cores. They are designed specifically for each type of target face with yellow cores in the appropriate locations.
The target faces were pasted with PVA adhesive onto pre-cut, sized cardboard a few weeks earlier.The pins were nails and washers painted black. They were hard to shove in and harder to pull out. The faces were in no danger of coming off!
The courses were designed from the start to reduce the work load during the event. The Unmarked course required very little work to set up for the marked round. Archers moved around the three courses and never shot the same target twice.
On our course we hosted the compound archers on the first (unmarked distance) day and the recurves on the second (marked distance) day. We then hosted the barebow archers for the first and second elimination rounds.
Between the first two days all that needed to done was move the red pegs around and put distance stickers on them. This also required repositioning the number plate. The top 12 targets of the 24 target course with a few butts repositioned formed the first Elimination round course. This took place on our course in the 'Delta Force' paint ball arena, adding something a little different to the shots.
Courses designers and work-horses, Mac and Niall.
Of the remaining 12 targets, 8 formed the second Elimination round course, with four removed to be used in the Individual Semi-Finals which were held on the same day.
The second course was finalised while the first was being shot and the Semi-Final course, which took place on a section of sloping field, was built which the second course was being shot. The Semi-Final course was shot by all disciplines freeing up personnel to build the courses for the Team competitions.
My duties consisted of putting out faces in the morning, manning a Water Point (which included assisting archers send their scores by WiFi to HQ as they passed through) during the competition and doing whatever was necessary after the shooting was over to prepare for the next day.
Despite the strong sunshine there was a strong, gusty, cold wind on day two. It caused difficulties especially on the high, exposed shots.
This varied from removing faces and pegs, transporting dignitaries or supplies. Or even Port-a-loos. One evening was quite windy and we were there well after dark filling and placing sand bags to stop tents blowing away overnight. Starting times varied from 6.50 to 8.00 to 9.00 on the last day. You finished when the work was done, often after dark.
At the end of the event all courses had to be dismantled, tents taken down and equipment transported away for storage. Other people were involved in ferrying archers to and from hotels.
The Day of Official Practice was the first day the volunteer field crew were present. After an introductory overview of how the next few days would go we were walked around the course to get to know it as we had to lead out shooting groups the following morning and later in the day replace faces, as needed, while shooting was still going on. The walk out routes were organised in such a fashion that the archers wouldn't get previews of targets before they had to shoot them. This was particularly important on the first (unmarked) day of shooting.
The archers muster before lead out in their groups at a numbered peg. A big group for each course.
We also needed to know the course for repositioning butts or, later in the event, dismantling the course.
Lunches were provided by the company employed to organise and run the event but it was no harm to bring your own to supplement those as the day was long. We also had access to a caravan which was our base and source of hot water for drinks.
At times it was hectic as target faces had to be out early in the morning so the judges could walk the course before shooting started. Then it was quiet as you manned your Water Point before it got busy again in the evening after the shooting was over, repositioning butts and pegs as necessary, so again, the judges could check all was well for the following morning.
The judges were brilliant. On our course they came from Australia, Croatia, Germany and one of the Nordic countries for our course. They caused us no hassle and often came up with useful suggestions as to how to do things. We worked with them and they with us. They put in some mileage with the number of times they walked the course. They guided us rather than ruled us, the way it should be.
Practice field is empty after the Official Practice is over.
A sunny morning in Ireland in late September. What's wrong with this picture?
Despite being busy at times my role allowed me access to the archers that you couldn't have gotten without qualifying as an archer for the competition. I got to meet several people I knew through the internet and get autographs from a lot of my favourite competitors. I was able to watch them practising and shooting for real.
And the prize for the highest eyebrow goes to….
They all impressed with their friendliness. Chantal Porte was a true lady. After asking for her autograph she sought me out and brought me a few nick-knacks as mementos. Needless to say I was very glad when she took away the gold.
The White and Orange courses were a mixture of open field and sloping forestry. The Green course had less terrain for the designers to play with. But they did. Mathias was counting the misses after the first day of shooting and was very pleased with the results. The courses were well received by the judges and archers and we received several comments, by mouth and in the media, that delighted the designers. The target pins for the 60 cm target faces were different to the 80 cm…. scaled to give the same relative size to add to the confusion.
60 cm targets butts were also available in two sizes. One defending champion admitted that most of the time she wasn't sure which face was which. I think there were very few 'up' shots until the finals but while none of the targets were extreme, none were easy. At least as far as I saw because being stuck on one course I never saw the others.
I was watching the forecast carefully in the two weeks leading up to the event. At first it was forecast as a dry week then the remnants of a dying hurricane came into the equation. Luckily that passed by without problem but it may have been responsible for the gusty conditions of the second day of shooting.
The rainbow was nearly right… we got a couple of more days out of it….
The night before we were filling sand bags as the site is in a wind alley and there was a risk tents in the Athletes Village may have taken flight.
Luckily the only rain we had was on the Saturday for the Team competitions. It was heavy and unpleasant for the morning but largely absent for the afternoon.
Standing in the Team Semi-Finals course with the Team Quarter Finals taking place in the distance.
If we had to have rain one day, that would have been the day for it. I felt sorry for the archers, I was standing out in it doing marshalling duties with a full rain suit on and I was still quite wet by the time it cleared.
The choice of venue proved to be superb. Kilruddery House and its gardens provided a majestic background to the finals of the event and the vast open spaces with varied terrain for the earlier rounds.
Killruddery House is one of the most successful EIizabethan-Revival mansions in Ireland.
Looking down from the top of the rock out-crop.
There have been some great locations for the finals over the years but the rocky out crop of Kilruddery is going to rank with the best for a long time. What a location!!
World Archery photographer, Dean Alberga, is more usually found on the other end of a camera shot.
Digital clock gives the time remaining in seconds.
When the final arrow was shot, it would be fair to say that there was a sense of huge relief. There had been very few problems and those had been minor…. The ladies Port-a-loos ran out of paper at one stage…the organisers hadn't arranged for enough tents initially but the army came to the rescue.
In terms of volunteers and field crew, I think we had about 40 in total and some of those even came over from the UK to help out. I could be wrong on that but I do remember a comment that other Championships would have had about 100 more and not have run as smoothly. The Competition Manager was able to stay at HQ and have others do the running, something that doesn't happen all the time.
The pressure the main men had been under was intense and they were visibly wilted by the time it came to taking everything down while the archers prepared for the Closing banquet. For those heavily involved it had been a long road. Most had donated their annual holidays to the event and would be back at work very soon. The budget for the championship was small, very small, for such an event of such scale and importance. It had to be organised to run on a shoe-string. And it was delivered on that budget. Months and months of meetings and preparation had paid off though.
It's all over and the field Crew let rip.
In essence the Championships were mainly run by members associated to one club so they were a cohesive and experienced team. The biggest field competition that was held in Ireland prior to this, was a national level event that was held over three days with two courses. And it was the only one. So the organisers went from that to the World Championships in one leap without major mishap. I asked what was next. Rohan just shrugged. Where do you go once you've reached the top?


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Jim spent more than 20 years in prison. Tough, and feared as "the meanest guy in prison," Jim didn't care if he lived or died. That changed when a friend introduced him to Jesus. "God brought about an internal change," says Jim. When Jim was released, he joined Prison Fellowship in Canada and returned to prison, a free man, to share his testimony with inmates. His story is now heard by thousands of inmates through The Prisoner's Journey in-prison Bible study program. Watch Jim's story below.
Singer/songwriter Glen Soderholm led PFI's Convocation participants in inspiring worship music throughout the five-day conference.


Despite conservative efforts to get rid of the bill, the Senate passed it to ensure people, no matter what gender, feel included.
"I'm very, very happy. There's been 30 years plus of activity trying to make our national anthem, this important thing about our country, inclusive of all of us," Independent Ontario Sen. Frances Lankin noted.


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If there's one thing I love as much as hygge, it's fragrance. I have a collection of around 40 perfumes, and I'm always looking to add more. But it's not just perfume – I've always had a thing for scents in general. I have a very sharp sense of smell, and there are lots of scents I associate with hygge. In this post I'm going to talk you through some of my favourites – maybe we share some of the same hyggeligt scents?
For some reason, I always appreciate the way the air smells more after dark. In summer, it may be because the heat is released from the ground, coaxing all the scents out in the warmth, but I have absolutely no idea why winter air smells so good. In summer, I love the smell of honeysuckle and jasmine, and the smell of wet grass left to dry. It makes me think about when my grandma used to water the garden when I was little, and how I'd help out (ish) by watering all my favourite flowers the most – the teeny tiny London Pride and the pink roses. The scent of a clear winter night always reminds me of the first time I went to Norway back in 2008. That pure, clean air always makes me smile, even though I don't know how to describe it – darkness? Frost? Stars?
It makes me sad that 'vanilla' is often used to mean 'bland' or 'boring', because it smells and tastes so delicious! I find it a very comforting smell, and it reminds me of baking fairy cakes as a child. I love the smell of an expensive vanilla extract or a vanilla pod, and I'm just as happy with 'fake vanilla' too. My favourite Bath & Body Works scent is Warm Vanilla Sugar (reminds me of my trip to New York where I first discovered it in 2014), and one of my favourite fragrances is Aerin's Tangier Vanille (which is insanely light for a gourmand perfume), which reminds me of a birthday visit to Sketch for afternoon tea where I received it as a gift!
How can anyone not love the smell of a freshly blown-out match? When I light candles now, I'm transported back to childhood birthday cakes every single time. Mine and my brother's birthdays are only five days apart (Halloween and Bonfire Night!), so we'd both blow out candles on both our own and each other's birthdays to avoid any squabbles. On a related note, I LOVE the smell of bonfires. Autumn is my absolute favourite time of year, and there is nothing I love more than getting off the bus on a November evening to be greeted with those plumes of heady smoke seeping out of someone's back garden. The smell of a real fire too – you can't beat it can you?
When my brother and I were little, we had a couple of elderly neighbours called Len and Mrs Len (we never actually knew Mrs Len's real name, but she didn't mind our highly imaginative nickname). Len showed us the little baby greenfly that lived in the foam on our lavender plants, and always danced over to the ice cream van to buy a couple of cones. Every Christmas and Easter, our mum would send us down to Len and Mrs Len's house to give them a present, which was absolutely always a hyacinth in a bucket (no, nothing to do with the character). I LOVED the smell of hyacinths then and I still do now, and it's a bonus that those little blue and pink bells are so pretty.
Strawberries aren't my favourite berry (I'm sure you were dying to know that), but the smell makes me so happy. My ultimate favourite summer activity is going to a pick-your-own farm and filling a basket with berries, plums and apples. I've had a little strawberry plant for a couple of years that flowers at the weirdest times (I was pretty surprised to see delicate white petals peeking out just before Christmas), and the smell of freshly chopped strawberries served with a generous slump of whipped cream is divine.
As if I wasn't going to talk about perfume! My ultimate hygge fragrance is my own signature scent, Alien. I've been wearing this for about 10 years, and I'm not even remotely sick of it. Everyone knows me by this scent, and even though I flirt with other fragrances, this is always the one I come back to. I sometimes spritz it on before bed just because it makes me feel like myself. Chanel features heavily in my sensory hygge – Coco, because it's my mum's 'special occasion' scent, Coco Mademoiselle because virtually every friend of mine has worn this at some point (as have I) and Chance Eau Tendre because it is a beautifully blossoming spring scent. Jo Malone's Mimosa & Cardamom and Blackberry & Bay colognes are staples of my autumn fragrance wardrobe, and for men I love Penhaligon's Juniper Sling.
Now over to you! Which scents would I find in your hygge moments?
Love love this. Thank you!
Vanilla, yes. I love orange as well, and cinnamon and ginger. I'm very Christmas happy.
Perfume-wise, there's a scent called Norman Hartnell's In Love which is the scent of my childhood. My Mum loved it and wore it whenever she wore perfume.
I am loving Green Tea from the Body Shop this year. It's quite sharp, but that's good. And always, always Lily of the Valley and Lavender. Love them both.
I find scent the most powerful way to tap into old memories. Sometimes I smell perfume on someone and have to shuffle through my mental fragrance catalogue to figure out where I recognise it from.
Great Article about Hygge. Keep it up!
The smell of a cats' fur! I know it sounds strange, but imagine snuggling with your kitty cat and burying your face in their fur- that's the smell I love!
I also love the smell of freshly blown-out matches after lighting a candle!
Yes! The scent of night air is one of my favorites in the world! And the smell of bonfires! The two together! However, my favorite scent is that of a small child, just come in from playing outdoors. It reminds me of freedom and undisturbed happiness. Or the scent of an old, yellowing book. I'm that person at the library.
Nothing wrong with being that person at the library 😉 The smell of bonfires always makes me happy – in the UK it always means we're in the throes of autumn – divine!
Morning air always smells great! This morning my town (Brighton, UK) was very humid, and I could smell the incoming rain in the air – I love that!
Great blog from fellow scent lover and appreciator!
Hi Kayleigh, I love the concept of cosyness and I'm glad I found your blog, instant hygge!:) And perfume… it's the magic potion that can change our mood in a second..
Thanks Angela! So glad you like my blog – and yes, perfume absolutely has the power to change our state of mind, it's amazing!


text:
Norquist urges GOP leaders to seek one-year delay of ObamaCare
By Elise Viebeck - 08/07/13 02:31 PM EDT
A group of conservatives led by anti-tax activist Grover Norquist is pushing for a one-year delay of ObamaCare in the government funding bills that must be passed this fall.
In a letter to Speaker John Boehner John Andrew BoehnerA time for war, a time for peace — and always a time to defend America Esper's chief of staff to depart at end of January Soleimani killing deepens distrust between Trump, Democrats MORE (R-Ohio) and Senate Minority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellDemocrats worry a speedy impeachment trial will shut out public George Conway group drops ad seeking to remind GOP senators of their 'sworn oaths' ahead of impeachment trial GOP senator 'open' to impeachment witnesses 'within the scope' of articles MORE (R-Ky.), Norquist and 18 other conservatives urged support for a one-year delay as part of any spending package.
"It is wrong to force people to participate in a system that is simply not ready," the activists wrote.
The letter presents the GOP with an alternative to the government shutdown threat backed by powerful conservative and Tea Party interests.
Lawmakers on that side said they will oppose any government funding bill that includes money for ObamaCare, a strategy likely to prompt a stalemate that could cause the government to shutter.
Other Republicans, including Norquist, the president of Americans for Tax Reform, are skeptical of the plan.
The alternative strategy outlined in Wednesday's letter calls for a delay in the individual mandate to buy health insurance, the tax subsides to make coverage more affordable and the range of taxes designed to pay for the law.
Deferring these provisions would throw a big wrench in the law's implementation.
Without the individual mandate, for example, premiums on the new insurance exchanges would rise sharply as young, healthy people fail to enter the market.
The current government funding bill expires on Sept. 30, setting the stage for negotiations after Congress returns from August recess.
Wednesday's letter was signed by Norquist, former chief policy adviser to Mitt Romney Lanhee Chen, James Capretta of the Ethics and Public Policy Center, and others.
Tags Mitch McConnell John Boehner Boehner

redpajama_set_name: RedPajamaCommonCrawl


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text:
Home NHL Atlantic Division Montreal Canadiens
Montreal Canadiens vs Toronto Maple Leafs Game 1
by Chris Wassel
in Montreal Canadiens, NHL, Opinion, Stanley Cup Playoffs, Toronto Maple Leafs
Montreal Canadiens heading into tonight…
This best of seven series between the Montreal Canadiens and Toronto Maple Leafs will be different in every way. Fans early on will not be allowed into either arena. There will be about a week of wait for that.
While the raging over who is and isn't in the lineup goes on, there is one clear thing.
Marc Bergevin has a vision and Dominique Ducharme is in sync with it. For better or for worse, the Montreal Canadiens are opting for experience tonight for Game 1.
Carey Price will start in goal and Brendan Gallagher will return.
One good infographic thanks to Jason to start us off…
pessimist ▶ OMG! the #habs fell ass-backwards into the playoffs, the worst team to make the playoffs, on a bad run.
optimist ▶ what doesn't kill you makes you stronger; the regular season ain't the playoffs.
Realist ▶ _________________ .#MTLvsTOR pic.twitter.com/jhfioKdJbR
— Jason Paul 🥥 (@WaveIntel) May 18, 2021
Again, this team rises or falls depending on Brendan Gallagher and how he works in the Montreal lineup. So much went right in that first ten-game window. With the huge period of rest, Montreal might come out that way again. No one truly knows.
Every pundit talks about how this series is 42 years in the making. Honestly, funk that! This series may just determine the fate of the vision of Marc Bergevin.
It may not happen right away but this may be a crossroads.
All it takes is a little bit of patience. For once, the logic of not putting Cole Caufield and others into the "fire", makes sense. Now, what other decisions this duo makes are of greater uncertainty.
If anyone believes they know exactly what will happen in Game 1 then I want the lottery numbers for Friday's Mega Millions. Kindly, also take a listen to this — a great RapidHabs podcast.
Latest Canadiens News
NHL Rumors: Montreal Canadiens, Columbus Blue Jackets, Detroit Red Wings
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NHL Rumors: Toronto Maple Leafs, Vancouver Canucks, Montreal Canadiens
Some other Montreal Canadiens news and notes…
It has been a fun challenge listening to zoom meetings in two languages the past couple of months. This will be something much cherished. There are certain nuances to the way different media ask different questions about the same team.
Also, the Laval Rocket finished their AHL season 23-9-4. That is some mark over 36 games. It is all the more impressive considering the circumstances. The fact that Carey Price and Brendan Gallagher got to play in that last game made it even more fun and fascinating. It was like a bridge to the future. Give full credit to a guy like Joel Bouchard.
The words "program" and "academy" are tossed around more and more when it comes to hockey teams and organizations these days. To Bouchard, he just does not casually throw the word program out there.
"Why is it a program? Because it's not a team. The (Canadiens) are a team. We have too much movement in personnel, too much stuff we don't control that you have to call it a program," Bouchard said. "Because you can be playing really well, and you can lose seven or eight guys just like that. You're hit from everywhere with your own injuries, the NHL club's, and you don't make all the decisions. As a coach, I'm not making all the decisions, I get handcuffed by the reality of the AHL."
The quote speaks volumes upon volumes. We heard this on a previous Zoom interview as well. Coaching in the AHL, especially this year, is exceedingly fluid. One day everything is going well and the next day nothing is. That is part of the reason why Bouchard has been so successful in his tenure.
Scoring Regular Season Table
Tyler Toffoli 28 C 52 28 16 44
Jeff Petry 33 D 55 12 30 42
Nick Suzuki 21 C 56 15 26 41
Tomas Tatar 30 LW 50 10 20 30
Josh Anderson 26 RW 52 17 7 24
Phillip Danault 27 C 53 5 19 24
Brendan Gallagher 28 RW 35 14 9 23
Jonathan Drouin 25 LW 44 2 21 23
Corey Perry 35 RW 49 9 12 21
Jesperi Kotkaniemi 20 C 56 5 15 20
Team Total 56 158 257 415
Generated 5/20/2021.
Some final words…
Everything in Round 1 between these two teams is fluid. Realize that first and foremost. Montreal and Toronto are going to have some battle. This is both teams' best chance to get to a Stanley Cup "Final Four" in many years.
Enjoy the wave err ride and don't sweat the small stuff. Literally.
Tags: AHLDominique DucharmeJoel BouchardLaval RocketMarc BergevinMontreal CanadiensNHL PlayoffsToronto Maple Leafs
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