TITLE: Minimize $\| A x - b \|_{2}^{2}$ Subject To $\| x \|_2 = 1$ and $x \succeq 0$ (Least Squares with Inequality and Non Linear Equality of $ L_2 $ Norm)
QUESTION [3 upvotes]: Given $y \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$, whis is some way for
$$\min_x \| y- Ax\|$$ subject to $\|x\|=1$, and $x \geq 0$ (which means every components of $x$ is nonnegative)?
Is there any book discussing such a problem? Thanks!
Remark:  The objective functions $\left\| A x - y \right\|$ and $\frac{1}{2} {\left\| A x - y \right\|}$ are equivalent, while the latter is differentiable and easier to handle.

REPLY [0 votes]: I would use the Projected Gradient Descend for this case.
Though the problem isn't Convex it will work nicely.
The algorithm is as following:

Calculate the Gradient at the current point.
Update the solution $ x = x - t {A}^{T} A x - {A}^{T} b $ where $ {A}^{T} A x - {A}^{T} b $ is the Gradient of the Objective Function at $ x $ and $ t $ is the step size.
Project the output of previous step into $ {\mathbb{R}}_{+} $ by $ {x}_{i} = \max \left\{ {x}_{i}, 0 \right\} $.
Project the output of previous step onto the Unit Sphere by $ {x}_{i} = \frac{ {x}_{i} }{ \left\| x \right\|_{2} } $.
Go back to (1) (Or check validity of the point, KKT will do even the problem isn't Convex).

In a simple 2D example I created it worked pretty well:

The code is available at my StackExchange Mathematics Q2699867 GitHub Repository.
Remark 001
I'd even considering starting with the solution of the Convex Problem when you replace the equality constraint $ \left\| x \right\|_{2} = 1 $ by $ \left\| x \right\|_{1} = 1 $. You can either use it as a starting point for the above algorithm or approximated solution by itself.
Remark 002
Another approach might be something like I did in - Solution for $ \arg \min_{ {x}^{T} x = 1} { x}^{T} A x - {c}^{T} x $.
Yet after each iteration of updating $ \lambda $ you should also project the output $ x $ into $ \mathbb{R}_{+} $.


text:
\begin{document}

\title[Motivic concentration theorem]{Motivic concentration theorem}
\author{Gon{\c c}alo~Tabuada and Michel Van den Bergh}

\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}
\thanks{G.~Tabuada was partially supported by a NSF CAREER Award.}
\thanks{M.~Van den Bergh is a senior researcher at the Research Foundation -- Flanders}

\address{Michel Van den Bergh, Department of Mathematics, Universiteit Hasselt, 3590 Diepenbeek, Belgium}
\email{michel.vandenbergh@uhasselt.be}
\urladdr{http://hardy.uhasselt.be/personal/vdbergh/Members/~michelid.html}

\subjclass[2010]{14A20, 14A22, 14F30, 14F40, 18D20, 19D25, 19E08}
\date{\today}

\abstract{In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact quasi-separated scheme $X$, we prove that (after inverting some elements of representation ring of $G$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is ``concentrated'' in the subscheme of $G$-fixed points $X^G$. Moreover, in the particular case where $G$ is connected and the action has finite stabilizers, we compute the additive invariants of $[X/G]$ using solely the subgroups of roots of unity of $G$. As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards.}}

\maketitle

\section{Introduction and statement of results}\label{sec:intro}

A {\em dg category $\cA$}, over a base field
$k$ (of characteristic $p\geq 0$), is a category enriched over complexes of $k$-vector spaces; see
\S\ref{sub:dg}. Every (dg) $k$-algebra
$A$ gives naturally rise to a dg category with a single
object. Another source of examples is provided by schemes (or, more generally, by algebraic
stacks) since the
category of perfect complexes $\perf(X)$ of every
quasi-compact quasi-separated $k$-scheme $X$ (or algebraic
stack) admits a canonical dg enhancement $\perf_\dg(X)$; see \S\ref{sub:perfect}. Let us denote by $\dgcat(k)$ the category of (essentially small) dg categories.

An {\em additive invariant} is a functor $E\colon \dgcat(k) \to \mathrm{D}$, with values in an additive category, which inverts Morita equivalences and sends semi-orthogonal decompositions in the sense of Bondal-Orlov \cite{BO} to direct sums; see \S\ref{sec:additive}. Examples of additive invariants include algebraic $K$-theory and its variants, cyclic homology and its variants, topological Hochschild homology and its variants, etc. Given a $k$-scheme $X$ (or algebraic stack) as above, we will often write $E(X)$ instead of $E(\perf_\dg(X))$.

Let $G$ be a smooth diagonalizable group $k$-scheme of finite type and $X$ a smooth quasi-compact quasi-separated $k$-scheme $X$ equipped with a $G$-action. In what follows, we will write $[X/G]$ for the associated (global) quotient stack, $G^\vee$ for the group of characters of $G$, and $R(G)\simeq \bbZ[G^\vee]$ for the representation ring of $G$.

As explained below in \S\ref{sec:action}, given an additive invariant $E\colon \dgcat(k) \to \mathrm{D}$, the Grothendieck ring $K_0([X/G])$, \ie the $G$-equivariant Grothendieck ring of $X$, acts naturally on the object $E([X/G])\in \mathrm{D}$. By pre-composing this action with the ring homomorphism $R(G) \to K_0([X/G])$ (induced by pull-back along the projection map $X\to\bullet:=\mathrm{Spec}(k)$), we hence obtain an action of $R(G)$ on $E([X/G])$. Given a multiplicative set $S \subset R(G)$, consider the following presheaf of abelian groups:
\begin{equation}\label{eq:presheaf}
S^{-1}E([X/G]):= \Hom_{\mathrm{D}}(-,E([X/G]))\otimes_{R(G)} S^{-1}R(G)\,.
\end{equation}
Note that since $S^{-1}R(G)$ can be written as a filtered colimit of free finite $R(G)$-modules, the presheaf \eqref{eq:presheaf} belongs to the category $\mathrm{Ind}(\mathrm{D})$ of ind-objects\footnote{For the general theory of ind-objects, we invite the reader to consult \cite{SGA4,AM}.} in $\mathrm{D}$.

Let $H$ be a closed diagonalizable subgroup of $G$. In what follows, we will write $X^H$ for the smooth subscheme of $H$-fixed points and $S_H$ for the multiplicative set generated by the elements $(1 - \chi) \in R(G)\simeq \bbZ[G^\vee]$, where $\chi \in G^\vee$ is any character of $G$ whose restriction to $H$ is non-trivial.

Under the above notations and assumptions, our first main result is the following:
\begin{theorem}[Motivic concentration]\label{thm:localization}
We have an isomorphism of ind-objects
$$
E(\iota^\ast)\colon S^{-1}_H E([X/G]) \stackrel{\simeq}{\too} S^{-1}_H E([X^H/G])
$$
induced by pull-back along the closed immersion $\iota\colon X^H \hookrightarrow X$.

Moreover, its inverse is given by the following composition
$$
S^{-1}_H E([X^H/G]) \stackrel{(\lambda_{-1}(\cN)\cdot -)^{-1}}{\too} S^{-1}_HE([X^H/G]) \stackrel{E(\iota_\ast)}{\too} S^{-1}_H E([X/G])\,,
$$
where $\cN$ stands for the conormal bundle of the closed immersion $\iota\colon X^H \hookrightarrow X$, $\lambda_{-1}(\cN)$ for the Grothendieck class $\sum_j (-1)^j [\bigwedge^j(\cN)] \in K_0([X^H/G])$, and $-\cdot -$ for the induced action of the ring $K_0([X^H/G])$ on the ind-object $S^{-1}_H E([X^H/G])$.
\end{theorem}
Intuitively speaking, Theorem \ref{thm:localization} shows that (after inverting the multiplicative set $S_H$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is ``concentrated'' in the quotient stack $[X^H/G]$. Since Theorem \ref{thm:localization} holds for {\em every} additive invariant, we named it the ``motivic concentration theorem''.
\begin{remark}[Generalization]
Let $\cH$ be a flat quasi-coherent sheaf of algebras over $[X/G]$, \ie a $G$-equivariant flat quasi-coherent sheaf of algebras over $X$. Given an additive invariant $E\colon \dgcat(k) \to \mathrm{D}$, let us write $E([X/G];\cH)$ for the object $E(\perf_\dg([X/G];\cH)) \in \mathrm{D}$, where $\perf_\dg([X/G];\cH)$ stands for the canonical dg enhancement of the category of $G$-equivariant perfect $\cH$-modules $\perf([X/G];\cH)$. As explained in Remark \ref{rk:generalization}, Theorem \ref{thm:localization} holds more generally with $S^{-1}_HE([X/G])$ and $S^{-1}_HE([X^H/G])$ replaced by $S^{-1}_HE([X/G];\cH)$ and $S^{-1}_HE([X^H/G];\iota^\ast(\cH))$.
\end{remark}
\begin{remark}[Localization at prime ideals]\label{rk:prime}
Let $\rho$ be a prime ideal of the representation ring $R(G)\simeq \bbZ[G^\vee]$. Recall that $G\simeq D(G^\vee)$, where $D(-)$ stands for the classical diagonalizable group scheme construction. On the one hand, similarly to \eqref{eq:presheaf}, we can consider the following presheaf of abelian groups:
$$ E([X/G])_{(\rho)}:=\Hom_{\mathrm{D}}(-,E([X/G]))\otimes_{R(G)}R(G)_{(\rho)}\,.$$
On the other hand, following Segal \cite[Prop.~3.7]{Segal}, we can consider the closed diagonalizable subgroup $H_\rho:=D(G^\vee/K_\rho)$ of $G$ (called the {\em support} of $\rho$), where $K_\rho:=\{\chi \in G^\vee\,|\, 1 - \chi \in \rho \subset \bbZ[G^\vee]\}$. Note that $S_{H_\rho} \cap \rho = \emptyset$ and that $H_\rho$ is maximal with respect to this property. Therefore, by further inverting the elements $R(G)\backslash (S_{H_\rho} \cup \rho)$, we conclude that Theorem \ref{thm:localization} holds similarly with $S^{-1}_HE([X/G])$ and $S^{-1}_H E([X^H/G])$ replaced by $E([X/G])_{(\rho)}$ and $E([X^{H_\rho}/G])_{(\rho)}$, respectively.
\end{remark}
Given an additive category $\mathrm{D}$, let us write $- \otimes_\bbZ -$ for the canonical action of the category of finite free $\bbZ$-modules $\mathrm{free}(\bbZ)$ on $\mathrm{D}$. This action extends naturally to an action of $\mathrm{Ind}(\mathrm{free}(\bbZ))$ on $\mathrm{Ind}(\mathrm{D})$. Our second main result is the following:
\begin{theorem}\label{thm:roots}
Assume that the base field $k$ (of characteristic $p\geq 0$) contains the $l^{\mathrm{th}}$ roots of unity for every~prime~$l \neq p$ such that $(G^\vee)_{l\text{-}\text{torsion}}\neq 0$. Under this assumption, we have an isomorphism of ind-objects:
$$
S^{-1}_{G} E([X/G]) \simeq E(X^G) \otimes_\bbZ S^{-1}_G R(G)\,.
$$
\end{theorem}
Note that when $G$ is moreover connected, \ie a $k$-split torus $T$, the assumption of Theorem \ref{thm:roots} is vacuous. In this case, we have an isomorphism of ind-objects
$$S^{-1}_{T} E([X/T]) \simeq E(X^T) \otimes_\bbZ \bbZ[t_1^\pm, \ldots, t_r^{\pm}][\{(1-t^j_i)^{-1}\}_{i, j}])\,,$$
where $r$ stands for the rank of $T$, $1\leq i \leq r$, and $j\neq 0 \in \bbZ$.

Similarly to Theorem \ref{thm:localization}, Theorem \ref{thm:roots} shows that (after inverting the multiplicative set $S_G$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is ``concentrated'' in the subscheme of $G$-fixed points $X^G$.

\medskip

We now illustrate Theorems \ref{thm:localization} and \ref{thm:roots} in several examples:
\begin{example}[Algebraic $K$-theory]\label{ex:K-theory1}
Algebraic $K$-theory gives rise to an additive invariant $K\colon \dgcat(k) \to \Ho(\Spt)$ with values in the category of spectra; see \cite[\S2.2.1]{book}. Hence, Theorem \ref{thm:localization} applied to $E=K$ yields an isomorphism of ind-objects:
\begin{equation}\label{eq:iso-spectra}
K(\iota^\ast)\colon S^{-1}_HK([X/G]) \stackrel{\simeq}{\too} S^{-1}_H K([X^H/G])\,.
\end{equation}
Consequently, we obtain, in particular, isomorphisms of abelian groups:
\begin{equation}\label{eq:inverse1}
K_\ast(\iota^\ast)\colon S^{-1}_H K_\ast([X/G])  \stackrel{\simeq}{\too} S^{-1}_H K_\ast([X^H/G])\,.
\end{equation}
Several variants of algebraic $K$-theory such as Karoubi-Villamayor $K$-theory, homotopy $K$-theory, and \'etale $K$-theory, are also additive invariants; see \cite[\S2.2.2-\S2.2.6]{book}. Hence, isomorphisms similar to \eqref{eq:iso-spectra}-\eqref{eq:inverse1} also hold for all these variants.

The above isomorphisms \eqref{eq:inverse1} and their explicit inverses, with $S^{-1}_H K_\ast([X/G])$ and $S^{-1}_H K_\ast([X^H/G])$ replaced by $K_\ast([X/G])_{(\rho)}$ and $K_\ast([X^{H_\rho}/G])_{(\rho)}$ (see Remark \ref{rk:prime}), were originally established by Thomason in \cite[Thm.~2.1 and Prop.~3.1]{Thomason} under the weaker assumption that $X$ is a regular algebraic space. Previously, in the particular case of the Grothendieck group, the  isomorphism \eqref{eq:inverse1} and its explicit inverse were established by Nielsen in \cite[Thm.~3.2]{Nielsen} under the stronger assumptions that $X$ is a smooth projective $k$-scheme and that $k$~is~algebraically~closed.
\end{example}
\begin{example}[Mixed complex]\label{ex:mixed1}
Recall from Kassel \cite[\S1]{Kassel} that a {\em mixed complex} is a (right) dg module over the algebra of
  dual numbers $\Lambda:=k[\epsilon]/\epsilon^2$, where
  $\mathrm{deg}(\epsilon)=-1$ and $d(\epsilon)=0$. The mixed complex construction gives rise to an additive invariant
  $\mathrm{C}\colon \dgcat(k) \to \cD(\Lambda)$ with values in the derived category of $\Lambda$; see \cite[\S2.2.7]{book}. Hence, Theorem \ref{thm:localization} applied to $E=\mathrm{C}$ yields an isomorphism of ind-objects:
\begin{equation}\label{eq:iso-mixed}
\mathrm{C}(\iota^\ast)\colon S^{-1}_H\mathrm{C}([X/G]) \stackrel{\simeq}{\too} S^{-1}_H \mathrm{C}([X^H/G])\,.
\end{equation}
Cyclic homology and all its variants such as Hochschild homology, negative cyclic homology, and periodic cyclic homology, factor through $\mathrm{C}$. Consequently, an isomorphism similar to \eqref{eq:iso-mixed} also holds for all these invariants. To the best of the authors' knowledge, all these isomorphisms are new in the literature.
\end{example}
\begin{example}[Periodic cyclic homology]\label{ex:HP}
Assume that $\mathrm{char}(k)=0$. Periodic cyclic homology gives rise to an additive invariant $HP_\pm\colon \dgcat(k) \to \mathrm{Vect}_{\bbZ/2}(k)$ with values in the category of $\bbZ/2$-graded $k$-vector spaces; see \cite[\S2.2.11]{book}. Moreover, thanks to the Hochschild-Kostant-Rosenberg theorem, we have an isomorphism $HP_\pm(Y)\simeq (\bigoplus_{i \,\mathrm{even}}H^i_{\mathrm{dR}}(Y), \bigoplus_{i \, \mathrm{odd}}H^i_{\mathrm{dR}}(Y))$ for every smooth $k$-scheme $Y$, where $H^\ast_{\mathrm{dR}}(-)$ stands for de Rham cohomology. Therefore, Theorem \ref{thm:roots} applied to $E=HP_\pm$ yields, in particular, an isomorphism of $\bbZ/2$-graded $k$-vector spaces:
$$
S^{-1}_G HP_\pm([X/G])\simeq (\bigoplus_{i \, \mathrm{even}}H^i_{\mathrm{dR}}(X^G), \bigoplus_{i \, \mathrm{odd}}H^i_{\mathrm{dR}}(X^G))\otimes_\bbZ S^{-1}_G R(G)\,.
$$
This description of the periodic cyclic homology of the quotient stack $[X/G]$ in terms of the de Rham cohomology of the subscheme of $G$-fixed points $X^G$ is, to the best of the authors' knowledge, new in the literature.
\end{example}
\begin{example}[Topological Hochschild homology]\label{ex:THH}
Topological Hochschild homology gives rise to an additive invariant $THH\colon \dgcat(k) \to \Ho(\Spt)$; see \cite[\S2.2.12]{book}. Hence, Theorem \ref{thm:localization} applied to $E=THH$ yields an isomorphism of ind-objects:
\begin{equation}\label{eq:iso-THH}
THH(\iota^\ast)\colon S^{-1}_HTHH([X/G]) \stackrel{\simeq}{\too} S^{-1}_H THH([X^H/G])\,.
\end{equation}
Topological Hochschild homology and all its variants such as topological cyclic homology, topological negative cyclic homology, and topological periodic cyclic homology, are also additive invariants; consult \cite{Lars,NS}\cite[\S2.2.13]{book}. Consequently, an isomorphism similar to \eqref{eq:iso-THH} also holds for all these variants. To the best of the authors' knowledge, all these isomorphisms are new in the literature.
\end{example}
\begin{example}[Topological periodic cyclic homology]\label{ex:TP}
Assume that $k$ is a perfect field of characteristic $p>0$. Let $W(k)$ be the ring of $p$-typical Witt vectors of $k$ and $K:= W(k)_{1/p}$ the fraction field of $W(k)$. Periodic cyclic homology gives rise to an additive invariant $TP_\pm(-)_{1/p}\colon \dgcat(k) \to \mathrm{Vect}_{\bbZ/2}(K)$ with values in the category of $\bbZ/2$-graded $K$-vector spaces; see \cite[Thm.~2.3]{finite}. Moreover, following Scholze (see \cite[Thm.~5.2]{CD}), we have $TP_\pm(Y)_{1/p} \simeq (\bigoplus_{i \, \mathrm{even}}H^i_{\mathrm{crys}}(Y), \bigoplus_{i \, \mathrm{odd}}H^i_{\mathrm{crys}}(Y))$ for every smooth proper $k$-scheme $Y$, where $H^\ast_{\mathrm{crys}}(-)$ stands for crystalline cohomology. Therefore, Theorem \ref{thm:roots} applied to $E=TP_\pm(-)_{1/p}$ yields, in particular, an isomorphism of $\bbZ/2$-graded $K$-vector spaces:
$$
S^{-1}_G TP_\pm([X/G])_{1/p}\simeq (\bigoplus_{i \, \mathrm{even}}H^i_{\mathrm{crys}}(X^G), \bigoplus_{i \, \mathrm{odd}}H^i_{\mathrm{crys}}(X^G))\otimes_\bbZ S^{-1}_G R(G)\,.
$$
Similarly to the above Example \ref{ex:HP}, this description of the topological periodic cyclic homology of the quotient stack $[X/G]$ in terms of the crystalline cohomology of the subscheme of $G$-fixed points $X^G$ is new in the literature.
\end{example}

\subsection*{Proper push-forwards}

The following result is an immediate application of the above Theorems \ref{thm:localization} and \ref{thm:roots}:
\begin{theorem}[Motivic Lefschetz-Riemann-Roch formula]\label{thm:push-forward}
Given a $G$-equivariant proper map $f\colon X \to Y$, between smooth quasi-compact quasi-separated $k$-schemes, we have the following commutative diagram of ind-objects:
\begin{equation}\label{eq:Euler}
\xymatrix{
S^{-1}_H E([X/G]) \ar[dd]_-{E(f_\ast)} \ar[rr]^{E(\iota^\ast)} && S^{-1}_H E([X^H/G]) \ar[d]^-{(\lambda_{-1}(\cN)\cdot -)^{-1}} \\
&& S^{-1}_H E([X^H/G]) \ar[d]^-{E((f\circ \iota)_\ast)} \\
S^{-1}_H E([Y/G]) \ar@{=}[rr] && S^{-1}_H E([Y/G])\,.
}
\end{equation}
Moreover, in the particular case where $X^G$ consists of a finite set of closed points and $Y=\bullet$, the commutative diagram \eqref{eq:Euler} (with $H=G$) reduces to the following commutative diagram of ind-objects\footnote{In the particular case where $X^H$ consists of a finite set of closed points and $G$ is moreover connected, the $G$-action on $X^H$ is necessarily trivial. Consequently, in this case, the above diagram \eqref{eq:Euler1} holds similarly with $S^{-1}_G E(-)$ and $X^G$ replaced by $S^{-1}_H E(-)$ and $X^H$, respectively.}
\begin{equation}\label{eq:Euler1}
\xymatrix{
S^{-1}_G E([X/G]) \ar[dd]_-{E(f_\ast)} \ar[rr]^{E(\iota^\ast)} && \bigoplus_{x \in X^G} S^{-1}_G E([\bullet/G]) \ar[d]^-{\bigoplus_{x \in X^G}(\lambda_{-1}(T^\vee_x)\cdot -)^{-1}} \\
&& \bigoplus_{x \in X^G} S^{-1}_G E([\bullet/G]) \ar[d]^-{\nabla} \\
S^{-1}_G E([\bullet/G]) \ar@{=}[rr] && S^{-1}_G E([\bullet/G])\,,
}
\end{equation}
where $\nabla$ stands for the co-diagonal map and $T^\vee_x$ for the dual of the tangent bundle of $X$ at the point $x$. Furthermore, whenever $k$ contains the $l^{\mathrm{th}}$ roots of unity for every prime $l\neq p$ such that $(G^\vee)_{l\text{-}\text{torsion}}\neq 0$, the ind-object $\bigoplus_{x \in X^G} S^{-1}_G E([\bullet/G])$ in \eqref{eq:Euler1} can be replaced by the ind-object $E(k) \otimes_\bbZ \bigoplus_{x \in X^G} S^{-1}_G R(G)$.
\end{theorem}
Intuitively speaking, the commutative diagram \eqref{eq:Euler}, resp. \eqref{eq:Euler1}, shows that after inverting the multiplicative set $S_H$, resp. $S_G$, all the information concerning the additive invariants of the push-forward along $f$, resp. along $X \to \bullet$, is ``concentrated'' in the quotient stack $[X^H/G]$, resp. in the finite set~of~closed~points~$X^G$.

To the best of the authors' knowledge, Theorem \ref{thm:push-forward} is new in the literature. In the particular case where $E=K_0(-)$, the diagram \eqref{eq:Euler1} was originally established by Nielsen in \cite[Prop.~4.5]{Nielsen} (under the stronger assumptions that $X$ is a smooth projective $k$-scheme and that $k$ is algebraically closed) and later by Thomason in \cite[Thm.~3.5]{Thomason} (with $S^{-1}_G K_0(-)$ replaced by $K_0(-)_{(\{0\})}$ (see Remark \ref{rk:prime}) under the weaker assumption that $X$ is a regular algebraic space). Note that in this particular case, the diagram \eqref{eq:Euler1} reduces to the classical Lefschetz-Riemann-Roch formula
\begin{equation}\label{eq:LRR}
\sum_i (-1)^i [H^i(X;\cF)] = \sum_{x \in X^G} \frac{[\cF_x]}{\sum_j (-1)^j [\bigwedge^j (T^\vee_x)]}\quad \mathrm{in}\,\, S^{-1}_GR(G)\,,
\end{equation}
which computes the $G$-equivariant Euler characteristic of every $G$-equivariant perfect complex of $\cO_X$-modules $\cF$ in terms of the finite set of closed points $X^G$. It is well-known that the formula \eqref{eq:LRR} implies many other classical formulas such as the Woods Hole fixed-point formula (see \cite{Woods}), the Weyl's character formula (see \cite{Demazure,Weyl}), the Brion's counting formula (see \cite{Brion1}), etc.

\subsection*{Torus actions with finite stabilizers}

In this subsection we assume that $G$ is moreover connected, i.e. a $k$-split torus $T$, and that the $T$-action on $X$ has finite (geometric) stabilizers. Let us denote by $\cC(T)$ the set of all those subgroups $\mu_n \subset T$ such that $X^{\mu_n}\neq \emptyset$. Note that since the $T$-action on $X$ has finite stabilizers, the set $\{n \in \bbN\,|\, \mu_n \in \cC(T)\}$ is finite; in what follows, we will write $r$ for the least common multiple of the elements of this latter set.

Given a subgroup $\mu_n \in \cC(T)$, let $\underline{S}_{\mu_n} \subset R(T)_{1/r}$ be the multiplicative set defined as the pre-image of $1$ under the following $\bbZ[1/r]$-algebra homomorphism
$$R(T)_{1/r} \stackrel{\text{(a)}}{\too} R(\mu_n)_{1/r} \stackrel{\text{(b)}}{\simeq} \frac{\bbZ[1/r][t]}{\langle t^n -1 \rangle} \simeq \prod_{d|n} \frac{\bbZ[1/r][t]}{\langle \Phi_d(t) \rangle} \stackrel{\text{(c)}}{\too} \frac{\bbZ[1/r][t]}{\langle\Phi_n(t)\rangle}\,,$$
where (a) is the restriction homomorphism, (b) is induced by the choice of a(ny) generator $t$ of the character group $\mu_n^\vee$, $\Phi_d(t)$ stands for the $d^{\mathrm{th}}$ cyclotomic polynomial, and (c) is the projection homomorphism. Under these notations and assumptions, our third main result is the following:
\begin{theorem}\label{thm:stabilizers}
For every additive invariant $E\colon \dgcat(k) \to \mathrm{D}$, with values in a $\bbZ[1/r]$-linear category, we have an isomorphism of ind-objects
\begin{equation}\label{eq:stabilizers}
E([X/T]) \stackrel{\simeq}{\too} \bigoplus_{\mu_n \in \cC(T)} \underline{S}^{-1}_{\mu_n} E([X^{\mu_n}/T])
\end{equation}
induced by pull-back along the closed immersions $X^{\mu_n} \hookrightarrow X$. Moreover, the direct sum on the right-hand side is finite.
\end{theorem}
Intuitively speaking, Theorem \ref{thm:stabilizers} shows that all the information concerning the additive invariants of the quotient stack $[X/T]$ (no invertion is needed!) is ``concentrated'' in the quotient stacks $[X^{\mu_n}/T]$.

Thanks to Theorem \ref{thm:stabilizers}, the above isomorphism \eqref{eq:stabilizers} holds for algebraic $K$-theory and all its variants, for cyclic homology and all its variants, for topological Hochschild homology and all its variants, etc. In the particular case of algebraic $K$-theory such an isomorphism was originally established by Vezzosi-Vistoli in \cite[\S3]{VV} under the weaker assumption that $X$ is a regular algebraic space. The remaining isomorphisms are, to the best of the authors' knowledge, new in the literature.

\subsection*{Proofs}

Our proof of Theorem \ref{thm:localization}, resp. Theorem \ref{thm:stabilizers}, is different from the proof of Thomason, resp. of Vezzosi-Vistoli, in algebraic $K$-theory. Nevertheless, we do borrow some ingredients from their proofs. In fact, using a certain category of subschemes of the quotient stack $[X/G]$ (see \S\ref{sec:subschemes}), we are able to ultimately reduce the proof of Theorem \ref{thm:localization}, resp. Theorem \ref{thm:stabilizers}, to the proof of the $K_0$-case of Thomason's result, resp. of Vezzosi-Vistoli's result; consult \S\ref{sec:proof1} for details. Note, however, that
we {\em cannot} mimic Thomason's arguments, resp. Vezzosi-Vistoli's arguments, because they depend in an essential way
on the d\'evissage property of $G$-theory (=$K$-theory for smooth schemes), which does {\em not} hold for many
additive invariants. For example, as explained by Keller in \cite[Example 1.11]{Exact}, Hochschild homology, and consequently the mixed
  complex, do {\em not} satisfy d\'evissage.

\section{Preliminaries}

Throughout the article, $k$ will be a base field of characteristic $p\geq 0$.

\subsection{Dg categories}\label{sub:dg}

Let $(\cC(k),\otimes, k)$ be the category of (cochain) complexes of
$k$-vector spaces. A {\em dg category $\cA$} is a category enriched over $\cC(k)$
and a {\em dg functor} $F\colon\cA\to \cB$ is a functor enriched over
$\cC(k)$; consult Keller's survey \cite{ICM-Keller}. Recall from \S\ref{sec:intro} that $\dgcat(k)$ stands for the category of (essentially small) dg categories.

Let $\cA$ be a dg category. The opposite dg category $\cA^\op$ has the
same objects and $\cA^\op(x,y):=\cA(y,x)$. The category $\dgHo(\cA)$ has the same objects as $\cA$ and morphisms $\dgHo(\cA)(x,y):=H^0 \cA(x,y)$, where $H^0(-)$ stands for the $0^{\mathrm{th}}$-cohomology functor. A {\em right dg
  $\cA$-module} is a dg functor $M\colon \cA^\op \to \cC_\dg(k)$ with values
in the dg category $\cC_\dg(k)$ of complexes of $k$-vector spaces. Let
us write $\cC(\cA)$ for the category of right dg
$\cA$-modules. Following \cite[\S3.2]{ICM-Keller}, the derived
category $\cD(\cA)$ of $\cA$ is defined as the localization of
$\cC(\cA)$ with respect to the objectwise quasi-isomorphisms.

A dg functor $F\colon\cA\to \cB$ is called a {\em Morita equivalence} if it
induces an equivalence on derived categories $\cD(\cA) \stackrel{\simeq}{\to}
\cD(\cB)$; see \cite[\S4.6]{ICM-Keller}. As explained in
\cite[\S1.6]{book}, the category $\dgcat(k)$ admits a Quillen model
structure whose weak equivalences are the Morita equivalences. Let us
denote by $\Hmo(k)$ the associated homotopy category.

The {\em tensor product $\cA\otimes\cB$} of dg categories is defined
as follows: the set of objects is the cartesian product of the sets of objects of $\cA$ and $\cB$ and
$(\cA\otimes\cB)((x,w),(y,z)):= \cA(x,y) \otimes \cB(w,z)$. As
explained in \cite[\S2.3]{ICM-Keller}, this construction gives rise to
a symmetric monoidal structure on $\dgcat(k)$, which descends to $\Hmo(k)$.

A {\em dg $\cA\text{-}\cB$-bimodule} is a dg functor
$\mathrm{B}\colon \cA\otimes \cB^\op \to \cC_\dg(k)$ or, equivalently, a
right dg $(\cA^\op \otimes \cB)$-module. A standard example is the dg
$\cA\text{-}\cB$-bimodule
\begin{eqnarray}\label{eq:bimodule2}
{}_F\mathrm{B}:\cA\otimes \cB^\op \too \cC_\dg(k) && (x,z) \mapsto \cB(z,F(x))
\end{eqnarray}
associated to a dg functor $F:\cA\to \cB$.

\subsection{Additive invariants}\label{sec:additive}

A functor $E\colon \dgcat(k)
\to \mathrm{D}$, with values in an additive category, is called an
{\em additive invariant} if it satisfies the following two conditions:
\begin{itemize}
\item[(i)] It sends the Morita equivalences (see \S\ref{sub:dg}) to isomorphisms.
\item[(ii)] Let $\cA \subseteq \cB$ and $\cC\subseteq \cB$ be dg categories inducing a semi-orthogonal decomposition $\dgHo(\cB)=\langle\dgHo(\cA), \dgHo(\cC) \rangle$ in the sense of Bondal-Orlov \cite{BO}. In this case, the inclusions $\cA \subseteq \cB$ and $\cC\subseteq \cB$ induce an isomorphism $E(\cA) \oplus E(\cC) \stackrel{\simeq}{\to} E(\cB)$.
\end{itemize}
Given small dg categories $\cA$ and $\cB$, let us write $\rep(\cA,\cB)$ for the full triangulated subcategory of $\cD(\cA^\op \otimes \cB)$ consisting of those dg $\cA\text{-}\cB$-modules $\mathrm{B}$ such that for every object $x \in \cA$ the associated right dg $\cB$-module $\mathrm{B}(x,-)\in \cD(\cB)$ belongs to the full triangulated subcategory of compact objects $\cD_c(\cB)$. As explained in \cite[\S1.6.3]{book}, there is a
  natural bijection between $\Hom_{\Hmo(k)}(\cA,\cB)$ and the set of
  isomorphism classes of the category $\rep(\cA,\cB)$. Moreover, under this bijection, the composition law of $\Hmo(k)$ corresponds to the (derived) tensor product of bimodules.

The {\em additivization} $\Hmo_0(k)$ of $\Hmo(k)$ is defined as the category with the same objects as $\Hmo(k)$ and morphisms $\Hom_{\Hmo_0(k)}(\cA,\cB):=K_0\rep(\cA,\cB)$. Since the dg $\cA\text{-}\cB$-bimodules
  \eqref{eq:bimodule2} belong to $\rep(\cA,\cB)$, we have the symmetric monoidal functor:
\begin{eqnarray}\label{eq:universal}
\U\colon \dgcat(k) \too \Hmo_0(k) & \cA \mapsto \cA & (\cA \stackrel{F}{\to} \cB) \mapsto [{}_F\mathrm{B}]\,.
\end{eqnarray}
As explained in
  \cite[\S2.3]{book}, this functor is the {\em universal} additive invariant, \ie given any additive category $\mathrm{D}$, pre-composition with $\U$
  gives rise to an equivalence
\begin{eqnarray}\label{eq:equivalence1}
\Fun_{\mathrm{additive}}(\Hmo_0(k),\mathrm{D}) \stackrel{\simeq}{\too} \Fun_{\mathrm{add}}(\dgcat(k),\mathrm{D})\,,
\end{eqnarray}
where the left-hand side stands for the category of additive functors and the right-hand side for the category of additive invariants.

\subsection{Derived categories of quotient stacks}\label{sub:perfect}

Let $G$ be an affine group $k$-scheme of finite type and $X$ a quasi-compact quasi-separated $k$-scheme equipped with a $G$-action. Let us denote by $\mathrm{Mod}([X/G])$ the Grothendieck category of $G$-equivariant $\cO_X$-modules and by $\mathrm{Qcoh}([X/G])$, resp. by $\mathrm{coh}([X/G])$, the full subcategory of $G$-equivariant quasi-coherent, resp. coherent, $\cO_X$-modules. In what follows, we will write $\cD([X/G]):=\cD(\mathrm{Mod}([X/G]))$ for the derived category of the quotient stack $[X/G]$, $\cD_{\mathrm{Qcoh}}([X/G])\subset \cD([X/G])$ for the full triangulated subcategory of those complexes of $G$-equivariant $\cO_X$-modules whose cohomology belongs to $\mathrm{Qcoh}([X/G])$, $\cD^b(\mathrm{coh}([X/G]))\subset \cD_{\mathrm{Qcoh}}([X/G])$ for the full triangulated subcategory of bounded complexes of $G$-equivariant coherent $\cO_X$-modules, and $\perf([X/G]) \subset \cD^b(\mathrm{coh}([X/G]))$ for the full triangulated subcategory of perfect complexes of $G$-equivariant $\cO_X$-modules.

Let $\cE x$ be an exact category. As explained in \cite[\S4.4]{ICM-Keller}, the {\em derived dg category $\cD_\dg(\cE x)$ of $\cE x$} is defined as the Drinfeld's dg quotient $\cC_\dg(\cE x)/\cA{c}_\dg(\cE x)$ of the dg category of complexes over $\cE x$ by its full dg subcategory of acyclic complexes.

Following the above notations, we will write $\cD_\dg([X/G])$ for the dg category $\cD_\dg(\cE x)$, with $\cE x:=\mathrm{Mod}([X/G])$, and $\cD_{\mathrm{Qcoh}, \dg}([X/G])$, $\cD^b_\dg(\mathrm{coh}([X/G]))$, and $\perf_\dg([X/G])$, for the corresponding full dg subcategories.
\begin{proposition}[Trivial action]\label{prop:trivial}
Assume that the category $\cD_{\mathrm{Qcoh}}([\bullet/G])$ is compactly generated. Under this assumption, whenever the $G$-action on $X$ is trivial, we have the following Morita equivalence:
\begin{eqnarray}\label{eq:MoritaEquivalence}
\perf_{\dg}(X) \otimes \perf_{\dg}([\bullet/G])\too \perf_{\dg}([X/G]) && (\cF, V)\mapsto \cF\boxtimes V\,.
\end{eqnarray}
\end{proposition}
\begin{remark}\label{rk:compactgeneration}
As proved in \cite[Thm.~A]{HR} and \cite[Lem.~4.1]{HNR}, the category $\cD_{\mathrm{Qcoh}}([\bullet/G])$ is compactly generated if and only if $k$ is of characteristic zero or if $k$ is of positive characteristic and $\overline{G}:=G\otimes_k \overline{k}$ does not contains a copy of the additive group $\bbG_a$.
\end{remark}
\begin{proof}
Given any $\cF \in \cD_{\mathrm{Qcoh}}(X)$, any $V \in \cD_{\mathrm{Qcoh}}([\bullet/G])$, and any $\cG \in \cD_{\mathrm{Qcoh}}([X/G])$, we have the following classical tensor-Hom relation:
\begin{equation}\label{eq:relation}
{\bf R}\Hom_{[X/G]}(\cF \boxtimes V, \cG)\simeq {\bf R}\Hom_{[\bullet/G]}(V, {\bf R}\Hom_X(\cF,\cG))\,.
\end{equation}
The relation \eqref{eq:relation} implies that if $\cF$ and $V$ are compact objects, then $\cF\boxtimes V \in \cD_{\mathrm{Qcoh}}([X/G])$ is also a compact object. Moreover, if ${\bf R}\Hom_{[X/G]}(\cF\boxtimes V, \cG)=0$ for every $\cF$ and $V$, then $\cG$ is necessarily equal to zero. Furthermore, if $\cF$ and $V$ are perfect complexes, then $\cF\boxtimes V$ is also a perfect complex. Since the categories $\cD_{\mathrm{Qcoh}}(X)$ and $\cD_{\mathrm{Qcoh}}([\bullet/G])$ are compactly generated by perfect complexes (consult \cite[Thm.~3.1.1]{MRXXX1} and \cite[Thm.~A (b)]{HR}, respectively) the above three facts imply that the category $\cD_{\mathrm{Qcoh}}([X/G])$ is also compactly generated by perfect complexes. Finally, given any two perfect complexes $\cF_1, \cF_2 \in \perf(X)$ and any two $G$-representations $V_1, V_2 \in \perf([\bullet/G])$, note that \eqref{eq:relation} also implies that
$$ {\bf R}\Hom_{[X/G]}(\cF_1 \boxtimes V_1, \cF_2 \boxtimes V_2)\simeq {\bf R}\Hom_{[\bullet/G]}(V_1, V_2) \otimes {\bf R}\Hom_X(\cF_1, \cF_2)\,.$$
This allows us to conclude that the dg functor \eqref{eq:MoritaEquivalence} is a Morita equivalence.
\end{proof}

\subsection{Action of the Grothendieck ring}\label{sec:action}

Let $G$ be an affine group $k$-scheme of finite type and $X$ a quasi-compact quasi-separated $k$-scheme equipped with a $G$-action. Since the tensor product $-\otimes_X-$ makes the dg category $\perf_\dg([X/G])$ into a commutative monoid and the universal additive invariant \eqref{eq:universal} is symmetric monoidal, we obtain a commutative monoid $\U([X/G])$ in the category $\Hmo_0(k)$. Making use of the following natural ring isomorphism
$$ \Hom_{\Hmo_0(k)}(\U(k),\U([X/G])):=K_0\rep(k, \perf_\dg([X/G]))\simeq K_0(\perf([X/G]))\,,$$
we hence conclude that the Grothendieck ring $K_0([X/G])\simeq  K_0(\perf([X/G]))$ acts on the object $\U([X/G])$ (and also that the monoid structure of $\U([X/G])$ is $K_0([X/G])$-linear). Concretely, this action can be explicitly described as follows:
\begin{eqnarray*}
K_0(\perf([X/G]))\too K_0(\rep(\perf_{\dg}([X/G]),\perf_{\dg}([X/G]))) & [\cF]\mapsto [{}_{(\cF\otimes_X -)} \mathrm{B}]\,. &
\end{eqnarray*}
Given any additive invariant $E\colon \dgcat(k) \to \mathrm{D}$, the equivalence of categories \eqref{eq:equivalence1} implies, by functoriality, that $K_0([X/G])$ acts on the object $E([X/G]) \in \mathrm{D}$.

\section{Category of subschemes of a quotient stack}
\label{sec:subschemes}

Let $G$ be a smooth affine group $k$-scheme of finite type and $X$ a smooth quasi-compact quasi-separated $k$-scheme equipped with a $G$-action. In this section, we construct a certain category\footnote{In the case of a constant finite group $k$-scheme $G$, a related category of $G$-equivariant smooth ``covers'' of $X$ was constructed in \cite[\S5]{Additive}.} $\mathbb{S}\mathrm{ub}^G_0(X)$ of $G$-stable smooth closed subschemes of $X$. This category, which is of independent interest, will play a key role in the proof of Theorems \ref{thm:localization} and \ref{thm:stabilizers}; consult \S\ref{sec:proofs} below.

\subsection*{Definition of the category $\mathbb{S}\mathrm{ub}^G_0(X)$} Let $\mathbb{S}\mathrm{ub}^G(X)$ be the category whose objects are the $G$-stable closed immersions $Y \stackrel{\tau}{\hookrightarrow} X$, with $Y$ a smooth quasi-compact quasi-separated $k$-scheme. In what follows, in order to simplify the exposition, we will often write $Y$. Given two objects $Y_1$ and $Y_2$, $\Hom_{\mathbb{S}\mathrm{ub}^G(X)}(Y_1, Y_2)$ is defined as the set of isomorphism classes of the bounded derived category
\begin{equation}\label{eq:coh-support}
\cD^b(\mathrm{coh}_{Y_1 \times_X Y_2}([(Y_1 \times Y_2)/G]))\subset \perf([(Y_1 \times Y_2)/G])
\end{equation}
of those $G$-equivariant coherent $\cO_{Y_1 \times Y_2}$-modules
that are (topologically) supported on the closed subscheme $Y_1 \times_X Y_2$. Note that since the quotient stack $[(Y_1 \times Y_2)/G]$ is smooth, every bounded complex of $G$-equivariant coherent $\cO_{Y_1 \times Y_2}$-modules is perfect. Given three objects $Y_1$, $Y_2$, and $Y_3$, the composition law
$$\Hom_{\mathbb{S}\mathrm{ub}^G(X)}(Y_2,Y_3) \times \Hom_{\mathbb{S}\mathrm{ub}^G(X)}(Y_1,Y_2) \too \Hom_{\mathbb{S}\mathrm{ub}^G(X)}(Y_1,Y_3)$$
is induced by the classical (derived) ``pull-back/push-forward'' formula
\begin{equation}\label{eq:assignment}
(\cE_{23},\cE_{12}) \mapsto (q_{13})_\ast ((q_{23})^\ast(\cE_{23}) \otimes^{\bf L} (q_{12})^\ast(\cE_{12})) \,,
\end{equation}
where $q_{ij}$ stands for the projection from the triple fiber product onto its $ij$-factor. Finally, the identity of an object $Y$ is the (isomorphism class of the) $G$-equivariant structure sheaf $\cO_\Delta$ of the diagonal $\Delta$ in $Y \times Y$.

The {\em additivization} $\mathbb{S}\mathrm{ub}^G_0(X)$ of $\mathbb{S}\mathrm{ub}^G(X)$ is defined by formally adding all finite direct sums to the category which has the same objects as $\mathbb{S}\mathrm{ub}^G(X)$ and morphisms
$$\Hom_{\mathbb{S}\mathrm{ub}^G_0(X)}(Y_1,Y_2):=K_0(\cD^b(\mathrm{coh}_{Y_1 \times_X Y_2}([(Y_1 \times Y_2)/G])))\,.$$
Note that, since the above formula \eqref{eq:assignment} is exact in each one of the variables, the composition law of $\mathbb{S}\mathrm{ub}^G(X)$ extends naturally to $\mathbb{S}\mathrm{ub}^G_0(X)$. Let us denote by $\UU\colon \mathbb{S}\mathrm{ub}^G(X)\to \mathbb{S}\mathrm{ub}^G_0(X)$ the canonical functor. Note also that thanks to Quillen's d\'evissage theorem \cite[\S5]{Quillen} and to the definition of $G$-theory, we have isomorphisms:
$$
\Hom_{\mathbb{S}\mathrm{ub}^G_0(X)}(Y_1, Y_2)\simeq G_0([(Y_1 \times_X Y_2)/G])\,.
$$
In particular, we have ring isomorphisms:
$$\End_{\mathbb{S}\mathrm{ub}^G_0(X)}(Y)\simeq G_0([Y/G]) \simeq K_0([Y/G])\,.$$
\subsection*{Relation(s) between the categories $\mathbb{S}\mathrm{ub}^G_0(X)$ and $\Hmo_0(k)$}
Given two objects $Y_1$ and $Y_2$ of the category $\mathbb{S}\mathrm{ub}^G(X)$, consider the exact functor
$$ \cD^b(\mathrm{coh}_{Y_1 \times_X Y_2}([(Y_1 \times Y_2)/G]))\too \rep(\perf_\dg([Y_1/G]), \perf_\dg([Y_2/G])$$
that sends a bounded complex of $G$-equivariant coherent $\cO_{Y_1 \times Y_2}$-modules $\cE_{12}$ (supported on the closed subscheme $Y_1 \times_X Y_2$) to the following Fourier-Mukai dg-functor:
\begin{eqnarray*}
\label{eq:fm}
\Phi_{\cE_{12}}\colon\perf_\dg([Y_1/G])\too \perf_\dg([Y_2/G]) \qquad \cF \mapsto (q_2)_\ast((q_1)^\ast(\cF ) \otimes^{\bf L} \cE_{12})\,.
\end{eqnarray*}
By definition of the categories $\mathbb{S}\mathrm{ub}^G(X)$ and $\Hmo(k)$, the above constructions lead to a well-defined functor
\begin{eqnarray*}
\mathbb{S}\mathrm{ub}^G(X)\too \Hmo(k) & Y \mapsto \perf_\dg([Y/G]) & \cE_{12} \mapsto {}_{\Phi_{\cE_{12}}}\mathrm{B}\,,
\end{eqnarray*}
which naturally extends to the additive categories:
\begin{eqnarray*}
\Psi\colon \mathbb{S}\mathrm{ub}^G_0(X) \too \Hmo_0(k) && \UU(Y) \mapsto \U([Y/G])\,.
\end{eqnarray*}
\begin{remark}[Sheaves of algebras]\label{rk:sheaves}
Let $\cH$ be a flat quasi-coherent sheaf of algebras over $[X/G]$, \ie a $G$-equivariant flat quasi-coherent sheaf of algebras over $X$. Given two objects $Y_1\stackrel{\tau_1}{\hookrightarrow} X$ and $Y_2 \stackrel{\tau_2}{\hookrightarrow} X$ of $\mathbb{S}\mathrm{ub}^G(X)$, consider the exact functor
$$
\cD^b(\mathrm{coh}_{Y_1 \times_X Y_2}([(Y_1 \times Y_2)/G]))\to  \rep(\perf_\dg([Y_1/G];\tau_1^\ast(\cH)),
\perf_\dg([Y_2/G];\tau_2^\ast(\cH)))
$$
defined, as above, by the assignment $\cE_{12} \mapsto {}_{\Phi_{\cE_{12}}}\mathrm{B}$. This leads to a functor
\begin{eqnarray*}
\mathbb{S}\mathrm{ub}^G(X)\too \Hmo(k) & (Y\stackrel{\tau}{\hookrightarrow} X) \mapsto \perf_\dg([Y/G]; \tau^\ast(\cH)) & \cE_{12} \mapsto {}_{\Phi_{\cE_{12}}} \mathrm{B}\,,
\end{eqnarray*}
which naturally extends to the additive categories:
\begin{eqnarray*}
\Psi_\cH\colon \mathbb{S}\mathrm{ub}^G_0(X) \too \Hmo_0(k) && \UU(Y \stackrel{\tau}{\hookrightarrow} X) \mapsto \U([Y/G]; \tau^\ast(\cH))\,.
\end{eqnarray*}
\end{remark}
\subsection*{Some properties of the category $\mathbb{S}\mathrm{ub}^G_0(X)$ and of the functor $\Psi$} In what follows, we describe three important properties that will be used in the sequel.
\subsubsection{Pull-back and push-forward}\label{rk:pullback}
Let $Y_1\stackrel{\tau_1}{\to} X$ and $Y_2\stackrel{\tau_2}{\to} X$ be two objects of the category $\mathbb{S}\mathrm{ub}^G(X)$. Given a $G$-stable closed immersion $\iota\colon Y_1 \hookrightarrow Y_2$ such that $\tau_2 \circ \iota = \tau_1$, its {\em pull-back} $\UU(\iota^\ast)\colon \UU(Y_2) \to \UU(Y_1)$, resp. {\em push-forward} $\UU(\iota_\ast)\colon \UU(Y_1) \to \UU(Y_2)$, is defined as the Grothendieck class $[(\iota \times_X\id)_\ast(\cO_{Y_1})]$, resp. $[(\id \times_X \iota)_\ast(\cO_{Y_1})]$, of the group $G_0([(Y_2\times_X Y_1)/G])$, resp. $G_0([(Y_1\times_X Y_2)/G])$. Note that $\Psi(\UU(\iota^\ast))=\U(\iota^\ast)$ and $\Psi(\UU(\iota_\ast))=\U(\iota_\ast)$.
\subsubsection{$K_0$-action}\label{rk:K0action}
Let $Y$ be an object of $\mathbb{S}\mathrm{ub}^G(X)$. The push-forward along the diagonal map
$i_\Delta:Y \hookrightarrow Y\times Y$ leads to an exact functor
\begin{equation}\label{eq:diagonal}
(i_\Delta)_\ast\colon \perf([Y/G])\too  \cD^b(\mathrm{coh}_\Delta([(Y\times Y)/G]))
\end{equation}
that sends the tensor product $-\otimes_Y-$ on the left-hand side to the ``pull-back/push-forward'' formula \eqref{eq:assignment} on the right-hand side. Therefore, by applying $K_0(-)$ to \eqref{eq:diagonal}, we obtain an induced ring morphism $K_0([Y/G])\to \End_{\mathbb{S}\mathrm{ub}_0^G(X)}(\UU(Y))$. In other words, we obtain an action of $K_0([Y/G])$ on the object $\UU(Y)$.
\begin{lemma} \label{lem:compat} The functor $\Psi$ interchanges with the $K_0([Y/G])$-action on $\UU(Y)$ (defined above)  with the $K_0([Y/G])$-action on $\U([Y/G])$ (defined in \S\ref{sec:action}).
\end{lemma}
\begin{proof}
Consider the following commutative diagram:
\begin{equation}\label{eq:diagram}
\xymatrix{
\perf([Y/G])\ar@{=}[d]\ar[rrr]^-{(i_\Delta)_\ast}&&& \cD^b(\mathrm{coh}_\Delta([(Y \times Y)/G]))\ar[d]^{\cE\mapsto  {}_{\Phi_\cE}\mathrm{B}}\\
\perf([Y/G])\ar[rrr]_-{\cF\mapsto{}_{(\cF\otimes_Y-)}\mathrm{B}}&&& \rep(\perf_{\dg}([Y/G]),\perf_{\dg}([Y/G]))\,.
}
\end{equation}
By applying $K_0(-)$ to \eqref{eq:diagram}, we obtain the claimed compatibility.
\end{proof}
\subsubsection{$K_0$-linearity}\label{rk:compatibility}
Let $Y\stackrel{\tau}{\hookrightarrow} X$ be an object of $\mathbb{S}\mathrm{ub}^G(X)$. By composing the induced ring homomorphism $\tau^\ast\colon K_0([X/G]) \to K_0([Y/G])$ with the $K_0([Y/G])$-action on $\UU(Y)$ described in \S\ref{rk:K0action}, we obtain a $K_0([X/G])$-action on $\UU(Y)$. A simple verification shows that this $K_0([X/G])$-action is compatible with the morphisms of the category $\mathbb{S}\mathrm{ub}_0^G(X)$. This implies that $\mathbb{S}\mathrm{ub}^G_0(X)$ is a $K_0([X/G])$-linear category. Note that since the projection map $X \to \bullet$ induces a ring homomorphism $R(G) \to K_0([X/G])$, the category $\mathbb{S}\mathrm{ub}^G_0(X)$ is also $R(G)$-linear.

\section{Proofs}\label{sec:proofs}

In this section, making use of the category $\mathbb{S}\mathrm{ub}^G_0(X)$ of $G$-stable smooth closed subschemes of $X$ (consult \S\ref{sec:subschemes}), we prove Theorems \ref{thm:localization}, \ref{thm:roots} and \ref{thm:stabilizers}.

\subsection*{Proof of Theorem \ref{thm:localization}}\label{sec:proof1}

Consider the following morphisms
\begin{eqnarray}\label{eq:morphisms1}
\UU(X) \stackrel{\UU(\iota^\ast)}{\too} \UU(X^H) && \UU(X^H) \stackrel{\UU(\iota_\ast)}{\too} \UU(X)
\end{eqnarray}
in the category $\mathbb{S}\mathrm{ub}^G_0(X)$. Since both these morphisms are $R(G)$-equivariant (see \S\ref{rk:compatibility}), they give rise to well-defined morphisms of ind-objects
\begin{eqnarray}\label{eq:morphisms2}
S^{-1}_H\UU(X) \stackrel{\UU(\iota^\ast)}{\too} S^{-1}_H\UU(X^H) && S^{-1}_H\UU(X^H) \stackrel{\UU(\iota_\ast)}{\too} S^{-1}_H\UU(X)
\end{eqnarray}
in the category $\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))$. Under the ring isomorphisms
$$\End_{\mathbb{S}\mathrm{ub}^G_0(X)}(\UU(X^H))\simeq G_0([X^H/G])\simeq K_0([X^H/G])\,,$$
the composition $\UU(\iota^\ast) \circ \UU(\iota_\ast)$ of the morphisms \eqref{eq:morphisms1} (which by definition is given by $[\cO_{X^H}\otimes_X^{\bf L} \cO_{X^H}]$) corresponds to the Grothendieck class $\sum_j (-1)^j [\bigwedge^j(I/I^2)] \in K_0([X^H/G])$, where $I$ stands for the sheaf of ideals associated to the closed immersion $\iota\colon X^H \hookrightarrow X$. Consequently, since $I/I^2=\cN$, the composition $\UU(\iota^\ast) \circ \UU(\iota_\ast)$ of the morphisms \eqref{eq:morphisms2} corresponds to the following morphism of ind-objects
\begin{equation}\label{eq:morph1}
S^{-1}_H\UU(X^H) \stackrel{\lambda_{-1}(\cN)\cdot -}{\too} S^{-1}_H\UU(X^H)\,,
\end{equation}
where $-\cdot -$ stands for the induced action of the Grothendieck group $K_0([X^H/G])$ on the ind-object $S^{-1}_H\UU(X^H)$ (see \S\ref{rk:K0action}).
\begin{lemma}\label{lem:invertible}
The above morphism of ind-objects \eqref{eq:morph1} is invertible.
\end{lemma}
\begin{proof}
Thanks to the Yoneda lemma, it is enough to show that \eqref{eq:morph1} becomes an isomorphism after application of the functor $\Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))}(S^{-1}_H\UU(X^H),-)$. Recall that $S^{-1}_HR(G)$ can be written as a filtered colimit of free finite $R(G)$-modules. Therefore, it suffices to show that \eqref{eq:morph1} becomes an isomorphism after application of the functor $\Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))}(\UU(X^H),-)$. By definition of the category $\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))$, this latter claim is equivalent to the invertibility of the following homomorphism of abelian groups:
\begin{eqnarray}\label{eq:morph2}
S^{-1}_HK_0([X^H/G]) \too S^{-1}_H K_0([X^H/G]) && \alpha \mapsto \lambda_{-1}(\cN) \cdot \alpha\,.
\end{eqnarray}
As proved by Thomason in \cite[Lem.~3.2]{Thomason}, \eqref{eq:morph2} is indeed invertible.
\end{proof}
Thanks to Lemma \ref{lem:invertible}, we can now consider the following composition:
\begin{equation}\label{eq:composition1}
S^{-1}_H \UU(X^H) \stackrel{(\lambda_{-1}(\cN)\cdot -)^{-1}}{\too} S^{-1}_H\UU(X^H) \stackrel{\UU(\iota_\ast)}{\too} S^{-1}_H\UU(X)\,.
\end{equation}
\begin{proposition}\label{prop:invertible}
The morphism of ind-objects $\UU(\iota^\ast)\colon S^{-1}_H\UU(X) \to S^{-1}_H\UU(X^H)$ is invertible. Moreover, its inverse is given by the above composition \eqref{eq:composition1}.
\end{proposition}
\begin{proof}
Thanks to Lemma \ref{lem:invertible}, $S^{-1}_H\UU(X^H)$ is a direct summand of $S^{-1}_H\UU(X)$. Therefore, using the Yoneda lemma, it is enough to show that $\UU(\iota^\ast)$ becomes an isomorphism after application of the functor $\Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))}(S^{-1}_H\UU(X),-)$. Moreover, similarly to the proof of Lemma \ref{lem:invertible}, it suffices to show that $\UU(\iota^\ast)$ becomes an isomorphism after application of the functor $\Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))}(\UU(X),-)$. By definition of the category $\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))$, this latter claim is equivalent to the invertibility of the following  homomorphism of abelian groups:
\begin{equation}\label{eq:morph3}
K_0(\iota^\ast)\colon S^{-1}_H K_0([X/G]) \too S^{-1}_H K_0([X^H/G])\,.
\end{equation}
As proved by Thomason in \cite[Thm.~2.1 and Lem.~3.3]{Thomason}, \eqref{eq:morph3} is indeed invertible.

Finally, note that the composition \eqref{eq:composition1} is the right-inverse of $\UU(\iota^\ast)$. Since $\UU(\iota^\ast)$ is invertible, \eqref{eq:composition1} is also the left-inverse of $\UU(\iota^\ast)$.
\end{proof}
We now have the ingredients necessary to conclude the proof of Theorem \ref{thm:localization}. As explained in \S\ref{rk:pullback}, resp. \S\ref{rk:K0action}, resp. \S\ref{rk:compatibility}, the functor $\Psi\colon \mathbb{S}\mathrm{ub}^G_0(X) \to \Hmo_0(k)$ is compatible with pull-backs and push-forwards, resp. with $K_0$-actions, resp. with $R(G)$-actions. Moreover, it extends naturally to the categories of ind-objects:
\begin{equation}\label{eq:ind-functor}
\mathrm{Ind}(\Psi)\colon \mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X)) \too \mathrm{Ind}(\Hmo_0(k))\,.
\end{equation}
Therefore, by combining the preceding functor \eqref{eq:ind-functor} with Proposition \ref{prop:invertible}, we conclude that the morphism of ind-objects $\U(\iota^\ast)\colon S^{-1}_H \U([X/G]) \to S^{-1}_H \U([X^H/G])$ is invertible and that its inverse is given by the following composition:
$$S^{-1}_H \U([X^H/G]) \stackrel{(\lambda_{-1}(\cN)\cdot -)^{-1}}{\too} S^{-1}_H \U([X^H/G]) \stackrel{\U(\iota_\ast)}{\too} S^{-1}_H\U([X/G])\,.
$$
This proves Theorem \ref{thm:localization} in the case of the universal additive invariant. The general case follows now from the equivalence of categories \eqref{eq:equivalence1} and from the fact that every additive functor $\Hmo_0(k) \to \mathrm{D}$ extends naturally to the categories of ind-objects.
\begin{remark}[Generalization]\label{rk:generalization}
Let $\cH$ be a flat quasi-coherent sheaf of algebras over $[X/G]$, \ie a $G$-equivariant flat quasi-coherent sheaf of algebras over $X$. A proof similar to the above one, with $\Psi$ replaced by the functor $\Psi_\cH$ (see Remark \ref{rk:sheaves}), allows us to conclude that Theorem \ref{thm:localization} holds more generally with $S^{-1}_HE([X/G])$ and $S^{-1}_HE([X^H/G])$ replaced by $S^{-1}_HE([X/G];\cH)$ and $S^{-1}_HE([X^H/G];\iota^\ast(\cH))$.
\end{remark}

\subsection*{Proof of Theorem \ref{thm:roots}}

Note first that since $R(G)\simeq \bbZ[G^\vee]$ and $G^\vee$ is a finitely generated abelian group, the abelian group $R(G)$ belongs to $\mathrm{Ind}(\mathrm{free}(\bbZ))$.

Since $\overline{G}:=G\otimes_k \overline{k}$ does not contains a copy of the additive group $\bbG_a$ (in any caracteristic) and the $G$-action on $X^G$ is trivial, Proposition \ref{prop:trivial} and Remark \ref{rk:compactgeneration} yield a Morita equivalence $\perf_\dg(X^G)\otimes \perf_\dg([\bullet/G]) \to \perf_\dg([X^G/G])$. Therefore, using the fact that the universal additive invariant \eqref{eq:universal} is symmetric monoidal, we obtain an induced isomorphism
\begin{equation}\label{eq:isomorphism-2}
\U(X^G) \otimes \U([\bullet/G]) \stackrel{\simeq}{\too} \U([X^G/G])\,.
\end{equation}
Recall from \S\ref{sec:additive}-\S\ref{sec:action} that the object $\U([\bullet/G])\in \Hmo_0(k)$ carries a canonical commutative monoid structure and that we have natural ring isomorphisms:
\begin{equation}\label{eq:isos}
\Hom_{\Hmo_0(k)}(\U(k),\U([\bullet/G]))\simeq K_0([\bullet/G]) \simeq R(G) \simeq \bbZ[G^\vee]\,.
\end{equation}
Using the characters of $G$, we hence obtain an induced morphism of ind-objects:
\begin{equation}\label{eq:isomorphism-3}
\U(k)\otimes_\bbZ R(G) \too \U([\bullet/G])\,.
\end{equation}
\begin{proposition}
The above morphism of ind-objects \eqref{eq:isomorphism-3} is invertible.
\end{proposition}
\begin{proof}
Note that, thanks to the ring isomorphisms \eqref{eq:isos}, by applying the functor $\Hom_{\mathrm{Ind}(\Hmo_0(k))}(\U(k),-)$ to \eqref{eq:isomorphism-3} we obtain an isomorphism. Hence, in order to prove that \eqref{eq:isomorphism-3} is invertible, it is enough to show that $\U([\bullet/G])$ is isomorphic to a (possibly infinite) direct sum of copies of $\U(k)$.

Recall that we have an isomorphism $G\simeq \bbG_m^{\times r} \times \mu_{l_1^{\nu_1}} \times \cdots \times \mu_{l_s^{\nu_s}}$ for some prime numbers $l_1, \ldots, l_s$ and non-integers $r, \nu_1, \ldots, \nu_s$. The multiplicative group $k$-scheme $\bbG_m$ is semi-simple. Moreover, the simple $\bbG_m$-representations $V$ are parametrized by the group of characters $\bbG_m^\vee$ and we have $\End_{\bbG_m}(V)\simeq k$. Since, by assumption, $k$ contains the $l^{\mathrm{th}}$ roots of unity, with $l=l_1, \ldots, l_s$, the group $k$-schemes $\mu_{l_1^{\nu_1}}, \ldots,  \mu_{l_v^{\nu_s}}$ are isomorphic to the constant finite group $k$-schemes $C_{l_1^{\nu_1}}, \ldots, C_{l_s^{\nu_s}}$, respectively. In particular, they are semi-simple. Moreover, the simple $\mu_{l^\nu}$-representations $V$ are parametrized by the group of characters $\mu_{l^\nu}^\vee$ and we have $\End_{\mu_{l^\nu}}(V)\simeq k$. These considerations imply that the group $k$-scheme $G$ is also semi-simple and that the dg category $\perf_\dg([\bullet/G])$ is Morita equivalent to the disjoint union $\coprod_{\chi \in G^\vee}k$. Consequently, since $\rep(\coprod_{\chi \in G^\vee}k, \cB)\simeq \prod_{\chi \in G^\vee} \rep(k,\cB)$ for every dg category $\cB$ and the functor $K_0(-)$ preserves arbitrary products, we obtain canonical isomorphisms:
$$ \Hom_{\Hmo_0(k)}(\U([\bullet/G]),\cB)\simeq \prod_{\chi \in G^\vee} \Hom_{\Hmo_0(k)}(\U(k),\cB)\,.$$
This shows not only that the (possibly infinite) direct sum $\bigoplus_{\chi \in G^\vee}\U(k)$ exists in the category $\Hmo_0(k)$, but moreover that $\U([\bullet/G])\simeq \bigoplus_{\chi \in G^\vee}\U(k)$.
\end{proof}
The above isomorphisms \eqref{eq:isomorphism-2} with \eqref{eq:isomorphism-3} yield an isomorphism of ind-objects $\U(X^G) \otimes_\bbZ R(G) \stackrel{\simeq}{\to} \U([X^G/G])$. Under this latter isomorphism, the natural action of $R(G)$ on the right-hand side corresponds to the canonical $R(G)$-action on $R(G)$. Consequently, we obtain an induced isomorphism of ind-objects:
\begin{equation}\label{eq:isomorphism-5}
\U(X^G) \otimes_\bbZ S^{-1}_G R(G) \stackrel{\simeq}{\too} S^{-1}_G \U([X^G/G])\,.
\end{equation}
Finally, by combining \eqref{eq:isomorphism-5} with the (inverse of the) isomorphism of ind-objects $S^{-1}_G \U([X/G]) \stackrel{\simeq}{\to} S^{-1}_G \U([X^G/G])$ provided by Theorem \ref{thm:localization}, we obtain an isomorphism of ind-objects $S^{-1}_G \U([X/G])\simeq \U(X^G) \otimes_\bbZ S^{-1}_G R(G)$. This proves Theorem \ref{thm:roots} in the case of the universal additive invariant. The general case follows now from the equivalence of categories \eqref{eq:equivalence1} and from the fact that the natural extension of every additive functor $\Hmo_0(k) \to \mathrm{D}$ to the categories of ind-objects is compatible with the induced action $-\otimes_\bbZ -$ of the category $\mathrm{Ind}(\mathrm{free}(\bbZ))$.

\subsection*{Proof of Theorem \ref{thm:stabilizers}}

Let us denote by $\Hmo_0(k)_{1/r}$, resp. by $\mathbb{S}\mathrm{ub}^G_0(X)_{1/r}$, the $\bbZ[1/r]$-linear category obtained by tensoring the abelian groups of morphisms of $\Hmo_0(k)$, resp. of $\mathbb{S}\mathrm{ub}^G_0(X)$, with $\bbZ[1/r]$. In the same vein, let us denote by $\Psi_{1/r}\colon \mathbb{S}\mathrm{ub}^G_0(X)_{1/r} \to \Hmo_0(k)_{1/r}$ the induced $\bbZ[1/r]$-linear functor.

\begin{lemma}\label{lem:finite}
The set of ind-objects $\{\underline{S}^{-1}_{\mu_n} \UU(X)_{1/r}\,|\, \mu_n \in \cC(T) \}$ is finite.
\end{lemma}
\begin{proof}
Note first that the ind-object $\underline{S}^{-1}_{\mu_n} \UU(X)_{1/r}$ is trivial if and only if we have $\mathrm{End}_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X)_{1/r})}(\underline{S}^{-1}_{\mu_n} \UU(X)_{1/r})=0$. Recall that $\underline{S}_{\mu_n}^{-1}R(T)_{1/r}$ can be written as a filtered colimit of free finite $R(T)_{1/r}$-modules. Hence, by definition of the category $\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X)_{1/r})$, the ind-object $\underline{S}^{-1}_{\mu_n} \UU(X)_{1/r}$ is trivial if and only if we have:
$$ \Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X)_{1/r})}(\UU(X)_{1/r}, \underline{S}^{-1}_{\mu_n} \UU(X)_{1/r})\simeq \underline{S}^{-1}_{\mu_n} K_0([X/T])_{1/r}=0\,.$$
The proof follows now from the fact that the following set of $\bbZ[1/r]$-modules $\{\underline{S}^{-1}_{\mu_n} K_0([X/T])_{1/r}\,|\, \mu_n \in \cC(T)\}$ is finite; see Vezzosi-Vistoli \cite[Prop.~3.4(ii)]{VV}.
\end{proof}
Consider the following canonical morphism of ind-objects:
\begin{equation}\label{eq:morph-loc}
\UU(X)_{1/r} \too \bigoplus_{\mu_n \in \cC(T)}\underline{S}^{-1}_{\mu_n} \UU(X)_{1/r}\,.
\end{equation}
Note that, thanks to Lemma \ref{lem:finite}, the direct sum on the right-hand side is finite.
\begin{proposition}\label{prop:VV}
The above morphism of ind-objects \eqref{eq:morph-loc} is invertible.
\end{proposition}
\begin{proof}
Thanks to the Yoneda lemma, since $\underline{S}_{\mu_n}^{-1}R(T)_{1/r}$ can be written as a filtered colimit of free finite $R(T)_{1/r}$-modules, it suffices to show that \eqref{eq:morph-loc} becomes an isomorphism after application of the functor $\Hom_{\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X))}(\UU(X)_{1/r},-)$. By definition of the category $\mathrm{Ind}(\mathbb{S}\mathrm{ub}^G_0(X)_{1/r})$, this latter claim is equivalent to the invertibility of the following homomorphism of $\bbZ[1/r]$-modules:
\begin{equation}\label{eq:VV}
K_0([X/T])_{1/r} \too \bigoplus_{\mu_n \in \cC(T)} \underline{S}^{-1}_{\mu_n} K_0([X/T])_{1/r}\,.
\end{equation}
As proved by Vezzosi-Vistoli in \cite[Prop.~3.4(ii)]{VV}, \eqref{eq:VV} is indeed invertible.
\end{proof}
Recall from \S\ref{rk:K0action} and \S\ref{rk:compatibility} that the functor $\Psi$ is compatible with $K_0$-actions and $R(T)$-actions, respectively. The same holds for its $\bbZ[1/r]$-linearization $\Psi_{1/r}$ and for the induced functor $\mathrm{Ind}(\Psi_{1/r})$ between the categories of ind-objects. Therefore, by applying this latter functor to \eqref{eq:morph-loc}, we obtain an isomorphism of ind-objects:
\begin{equation}\label{eq:final1}
\U([X/T])_{1/r} \stackrel{\simeq}{\too} \bigoplus_{\mu_n \in \cC(T)} \underline{S}^{-1}_{\mu_n} \U([X/T])_{1/r}\,.
\end{equation}
\begin{lemma}\label{prop:mult}
For every $\mu_n \in \cC(T)$, we have an isomorphism of ind-objects
\begin{equation}\label{eq:iso-2}
\underline{S}^{-1}_{\mu_n} \U([X/T])_{1/r} \stackrel{\simeq}{\too} \underline{S}^{-1}_{\mu_n} \U([X^{\mu_n}/T])_{1/r}
\end{equation}
induced by pull-back along the closed immersion $X^{\mu_n} \hookrightarrow X$.
\end{lemma}
\begin{proof}
In order to simplify the exposition, let us still denote by $S_{\mu_n}$ the image of the multiplicative set $S_{\mu_n} \subset R(T)$ (see \S\ref{sec:intro}) in the $\bbZ[1/r]$-linearized representation ring $R(T)_{1/r}$. Thanks to Theorem \ref{thm:localization}, we have an isomorphism of ind-objects
\begin{equation}\label{eq:iso-1}
S^{-1}_{\mu_n} \U([X/T])_{1/r} \stackrel{\simeq}{\too} S^{-1}_{\mu_n} \U([X^{\mu_n}/T])_{1/r}
\end{equation}
induced by pull-back along the closed immersion $X^{\mu_n} \hookrightarrow X$. Let $\chi \in T^\vee$ be a character of $T$ whose restriction to $\mu_n$ is non-trivial. As explained by Thomason in the proof of \cite[Lem.~3.6]{VV}, the image of $1- \chi$ under the $\bbZ[1/r]$-algebra homomorphism $R(T)_{1/r} \to \underline{S}^{-1}_{\mu_n} R(T)_{1/r}$ is invertible. Consequently, we obtain an induced $R(T)_{1/r}$-linear homomorphism $S^{-1}_{\mu_n}R(T)_{1/r} \to \underline{S}^{-1}_{\mu_n} R(T)_{1/r}$. Therefore, by applying the functor $-\otimes_{S^{-1}_{\mu_n} R(T)_{1/r}} \underline{S}^{-1}_{\mu_n} R(T)_{1/r}$ to \eqref{eq:iso-1}, we obtain the searched isomorphism of ind-objects \eqref{eq:iso-2}.
\end{proof}
Finally, by combining \eqref{eq:final1} with \eqref{eq:iso-2}, we obtain an isomorphism of ind-objects
$$ \U([X/T])_{1/r} \stackrel{\simeq}{\too} \bigoplus_{\mu_n \in \cC(T)}\underline{S}^{-1}_{\mu_n} \U([X^{\mu_n}/T])_{1/r}$$
induced by pull-back along the closed immersions $X^{\mu_n} \hookrightarrow X$. This proves Theorem \ref{thm:stabilizers} in the case of the universal additive invariant. The general case follows now from the equivalence of categories \eqref{eq:equivalence1} and from the fact that every additive functor $\Hmo_0(k) \to \mathrm{D}$, with values in a $\bbZ[1/r]$-linear category, extends naturally to a $\bbZ[1/r]$-linear functor $\Hmo_0(k)_{1/r} \to \mathrm{D}$ and, consequently, to a $\bbZ[1/r]$-linear functor $\mathrm{Ind}(\Hmo_0(k)_{1/r}) \to \mathrm{Ind}(\mathrm{D})$.

config: arxiv


TITLE: center mass of the solid
QUESTION [1 upvotes]: Find the center mass of the solid bounded by planes $x+y+z=1,x=0,y=0$ and $z=0$, assuming a mass density of $$\rho(x,y,z) = 10 \sqrt{z}.$$
I could not set up the integral!

REPLY [4 votes]: You can get bounds of integration by intersecting your various surfaces. For example: $x+y+z=1$ intersected with $z=0$ gives you $x+y+0=1$ so that $y=1-x$. Then intersect with $y=0$ and get $0=1-x$ so that $x=1$.
In the end, this region can be described as follows: $0 \leq z \leq 1-x-y$, $0 \leq y \leq 1-x$, $0 \leq x \leq 1$.
So to find the mass you'd need to integrate...
$$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} 10\sqrt{z}\,dz\,dy\,dx$$
Then the moment about the $yz$-plane is...
$$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} 10\sqrt{z} \cdot x\,dz\,dy\,dx$$

REPLY [3 votes]: The mass of the volume is
$$M = \int_0^1 dx \, \int_0^{1-x} dy \, \int_0^{1-x-y} dz \, \rho$$
The $x$-coordinate of the COM is
$$\frac1{M} \int_0^1 dx\, x \, \int_0^{1-x} dy \, \int_0^{1-x-y} dz \, \rho$$
etc.


TITLE: How good is the following method for dividing a set of numbers into two subsets with sums that are as close as possible?
QUESTION [5 upvotes]: The problem is part of the NP group, but the following simple method seems to work pretty well.  Sort the list in descending order. Then remove the numbers one at a time and insert into the subset that has the smaller sum at the time of insertion.
The difference between the two sums will definitely be no greater than the maximum of the original set. In executing the algorithm, each time that the sum of one subset overtakes the other, the difference between the sums is guaranteed to be no greater than the last number that was placed.
It is easy to find cases where this algorithm does not give the minimum sum difference, but even in those cases it works pretty well.  I tried it for prime numbers and squares, cubes, Fibonacci numbers and pseudorandom numbers from 1 to 1,000,000, and in each case got a sum difference that were small compared to the total sum of the original set.  The only way I can imagine the algorithm not doing well is if the numbers are far apart and there is just one way to arrange them so that the two sums are close. Is there a way of determining a worst case scenario?

REPLY [6 votes]: This algorithm is known as longest processing time first scheduling and is known to have an approximation ratio of $\frac76$ in the two-partition case, like here: the largest part in the partition obtained by this algorithm will be at most $\frac76$ of the size of the largest part in the optimal partition.
Said approximation ratio is met by the input $\{3,3,2,2,2\}$, where the algorithm outputs $\{3,2,2\},\{3,2\}$ with largest part $7$ but the optimal partition is $\{3,3\},\{2,2,2\}$ with largest part $6$.


TITLE: Compute $\lim_{n \to +\infty} n^{-\frac12 \left(1+\frac{1}{n}\right)} \left(1^1 \cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\frac{1}{n^2}}$
QUESTION [7 upvotes]: How to compute
  $$\displaystyle \lim_{n \to +\infty} n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(1^1\cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\dfrac{1}{n^2}}$$
  I'm interested in more ways of computing limit for this expression

My proof:
Let $u_n$be that sequence we've:
\begin{eqnarray*}
\ln u_n &=& -\frac{n+1}{2n}\ln n + \frac{1}{n^2}\sum_{k=1}^n k\ln k\\
&=& -\frac{n+1}{2n}\ln n + \frac{1}{n^2}\sum_{k=1}^n k\ln \frac{k}{n}+\frac{1}{n^2}\sum_{k=1}^n k\ln n\\
&=&  \frac{1}{n^2}\sum_{k=1}^n k\ln \frac{k}{n}\\
&=& \frac{1}{n}\sum_{k=1}^n \frac{k}{n}\ln \frac{k}{n}\\
&\to&\int_0^1 x\ln x\,dx = -1/4
\end{eqnarray*}
Therefore the limit is $e^{-\frac{1}{4}}$

REPLY [2 votes]: Another approach, considering $$A_n= n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(\prod_{i=1}^n i^i \right)^{\dfrac{1}{n^2}}=n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} H(n)^{\dfrac{1}{n^2}}$$ where appears the hyperfactorial function. Taking logarithms $$\log(A_n)={-\dfrac12 \left(1+\dfrac{1}{n}\right)}\log(n)+\dfrac{1}{n^2}\log(H(n))$$ Now, using Stirling like approximations $$\log(H(n))=n^2 \left(\frac{1}{2} \log(n)-\frac{1}{4}\right)+\frac{1}{2} n
   \log (n))+\left(\log (A)+\frac{1}{12} \log
   \left(n\right)\right)+O\left(\frac{1}{n^2}\right)$$ where appears Glaisher constant.
All of that, once recombined, leads to $$\log(A_n)=-\frac{1}{4}+\frac{\log (A)+\frac{1}{12} \log
   \left(n\right)}{n^2}+O\left(\frac{1}{n^{5/2}}\right)$$ which shows the limit and how it is approached.


TITLE: Closure of a subset of a metric space is closed
QUESTION [2 upvotes]: From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$.
I need to prove that $\overline{E}$ is closed. By a closed set, I mean that all limit points of a set are in the same set. A limit point of a set is a point whereby every neighbourhood of the point contains a $q$ such that $q$ is an element of the set.
The proof provided in Theorem 2.27(a) of Rudin's Principles of Mathematical Analysis is as follows:
If $p \in X$ and $p \notin \overline{E}$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighbourhood which does not intersect $E$. The complement of $\overline{E}$ is therefore open. Hence $\overline{E}$ is closed.
My question is with how "Hence $p$ has a neighbourhood which does not intersect $E$." leads to "The complement of $\overline{E}$ is therefore open.".
To show that the complement of $\overline{E}$ is open, we need to show that $p$ has a neighbouthood which does not intersect $\overline{E}$, not just $E$. Unless it is true that if the neighbourhood (an open set, since neighbourhoods are all open sets) does not intersect $E$, then it also does not intersect $E'$. Is that true?

REPLY [3 votes]: Your definition of limit point is a little off: it should say that $p$ is a limit point of a set $E$ if every nbhd of $p$ contains a point $q\in E$ such that $q\ne p$. Without that last condition $2$ would be a limit point of $[0,1]\cup\{2\}$ in $\Bbb R$, which is not what we want the term to mean.
You’ve started with a point $p\in X\setminus\overline E$, meaning that $p\notin E$ and $p\notin E'$. Since $p\notin E'$, $p$ has a nbhd $U$ that contains no point of $E$ different from $p$. Since $p\notin E$, this means that $U$ contains no point of $E$ at all: $U\cap E=\varnothing$. Suppose that $x\in U\cap E'$; then $U$ is a nbhd of $x$, and $x\in E'$, so $U\cap E\ne\varnothing$, which we just saw is not the case. Thus, $U\cap E'=\varnothing$, and therefore $U\cap\overline E=\varnothing$, as desired.


TITLE: Is my definition of division by zero problematic?
QUESTION [1 upvotes]: I am a complete amateur, but I've attempted to define division by zero. This it's usually classified as undefined. After all, if 1/0 = b, where b is some non-zero number, then b times 0 would have to equal 1, and there exists no real number that, when multiplied by zero, equals 1.
Therefore, it was evident that my definition would necessitate the creation of a new kind of number. This number, which I call Z, is defined as Z=1/0.  Z is thus the reciprocal of 0.
Playing with this new number for awhile less me to discover that 2/0=2Z, and so on where n/0=nZ.  This leads to a whole set of numbers, Z numbers, similar to imaginary numbers. After all, imaginary numbers were created to define something that was formerly undefined, namely, a number whose square was -1.  This leads to the whole set of imaginary / complex numbers.
Already I've discovered interesting facts about this number Z, such as the square of Z is equal to Z, since 1/0 times 1/0 equals 1/0.
My question is, is there some way in which my new set of numbers to define division by zero falls apart under closer scrutiny? I have yet to discover one, but I'm very new to mathematical exploration.
I apologize for not being familiar with the Math font used to write equations.

REPLY [4 votes]: With your definition of Z, we have $1/Z= 1/(1/0) =0$
Now we get $$2/Z = 2(0)=0 =1/Z$$ Multiply by $Z$ and we get $1=2$
Similarly you can prove $m=n$ for any two integers which we do not approve of.


TITLE: Matrix Representation of Lorentz Group Generators
QUESTION [1 upvotes]: Let $\Lambda^{\alpha}{}_{\beta}$ denote a generic Lorentz transformation.
Then, an infinitesimal transformation can be written like
$$\Lambda^{\mu}{}_{\nu}  = \delta^{\mu}{}_{\nu} + \omega^{\mu}{}_{\nu} $$
where
$$\omega^{ij} = \epsilon^{ijk}\theta_k$$
$$\omega^{i0} = - \omega^{0i} = \delta^i$$
where $i,j,k$ run from 1 to 3 and $\delta^i$ is a parametre related with boosts. Then, an infinitesimal transformation has a matrix representation
\begin{pmatrix}
1 & -\delta_1 & -\delta_2 & -\delta_3\\
-\delta_1 & 1 & \theta_3 & -\theta_2\\
-\delta_2 & -\theta_3 & 1 & \theta_1\\
-\delta_3 & \theta_2 & -\theta_1 & 1
\end{pmatrix}
However, we can also write
$$\Lambda^{\mu}{}_{\nu}  = \delta^{\mu}{}_{\nu} + i\frac{\omega^{\alpha \beta}}{2}\left(J_{\alpha \beta} \right)^{\mu}{}_{\nu} $$
where $J_{\alpha \beta}$ are the generators of the group. I want to prove that $J_{01}$ is of the form
\begin{pmatrix}
0 & -i & 0 & 0 \\
-i  & 0  & 0  & 0 \\
0  & 0  & 0  & 0 \\
0  & 0  & 0  & 0
\end{pmatrix}
My problem is in understanding the notation in
$$\Lambda^{\mu}{}_{\nu}  = \delta^{\mu}{}_{\nu} + i\frac{\omega^{\alpha \beta}}{2}\left(J_{\alpha \beta} \right)^{\mu}{}_{\nu} $$
For example, I tried to compute $\left(J_{01} \right)^{0}{}_{1}$ by doing
$$\Lambda^{0}{}_{1}  = \delta^{0}{}_{1} + i\frac{\omega^{01}}{2}\left(J_{01} \right)^{0}{}_{1} $$
$$\Leftrightarrow - \delta_1 = 0 -i \frac{\delta_1}{2}\left(J_{01} \right)^{0}{}_{1}$$
which yields $\left(J_{01} \right)^{0}{}_{1} = -2i$, which is not correct. What am I doing wrong?

REPLY [3 votes]: In the following text, we use $\eta_{\mu\nu} = \text{diag}(-1,+1,+1,+1)$.
For an infinitesimal homogeneous Lorentz transformation, we have
$$ {\omega^\mu}_\nu = \begin{pmatrix} 0 & \zeta_1 & \zeta_2 & \zeta_3 \\ \zeta_1 & 0 & -\theta_3 & \theta_2 \\ \zeta_2 & \theta_3 & 0 & -\theta_1 \\ \zeta_3 & -\theta_2 & \theta_1 & 0  \end{pmatrix}, $$
and
$$\begin{aligned}
{\Lambda^\mu}_\nu &= {\delta^\mu}_\nu + {\omega^\mu}_\nu \\
&= {\delta^\mu}_\nu + \eta^{\rho\mu}\omega_{\rho\nu} \\
&= {\delta^\mu}_\nu + \eta^{\rho\mu}\omega_{\rho\sigma}{\delta^\sigma}_\nu \\
&= {\delta^\mu}_\nu + \frac{1}{2}\omega_{\rho\sigma}(\eta^{\rho\mu}{\delta^\sigma}_\nu - \eta^{\sigma\mu}{\delta^\rho}_\nu) \\
&= {\delta^\mu}_\nu + \frac{i}{2}\omega_{\rho\sigma}{(S_V^{\rho\sigma})^\mu}_\nu \\
\end{aligned}$$
where the vector representation of the generators are defined as
$$ {(S_V^{\rho\sigma})^\mu}_\nu \equiv -i({\eta}^{\rho\mu}{\delta^\sigma}_\nu - {\eta}^{\sigma\mu}{\delta^\rho}_\nu). $$
Note that $\omega_{\rho\sigma}$ and $S_V^{\rho\sigma}$ are antisymmetric in the indices $(\rho\sigma)$, and
$$ {(S_V^{0i})}^\dagger = - S_V^{0i}, \quad {(S_V^{ij})}^\dagger = S_V^{ij}. $$

If we define
$$ \boldsymbol\theta \equiv (\theta_1, \theta_2, \theta_3), \quad \boldsymbol\zeta \equiv (\zeta_1, \zeta_2, \zeta_3), $$
$$ \mathbf{J} \equiv (S_V^{23},S_V^{31},S_V^{12}),\quad \mathbf{K} \equiv (S_V^{01},S_V^{02},S_V^{03}), $$
then
$$ {\Lambda^\mu}_\nu = {\delta^\mu}_\nu - i{{(\boldsymbol\theta \cdot \mathbf{J} + \boldsymbol\zeta \cdot \mathbf{K})}^\mu}_\nu. $$
The explicit expressions of the matrices $J_i$ and $K_i$ are
$$
J_1 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{pmatrix}, \quad
J_2 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix}, \quad
J_3 = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}
$$
$$
K_1 = \begin{pmatrix} 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad
K_2 = \begin{pmatrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad
K_3 = \begin{pmatrix} 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}
$$


TITLE: Theory of Computation - Intersection of 2 Languages
QUESTION [0 upvotes]: Let's say that L1 is regular and that L1∩L2 is regular as well.
What L2 can be?Is the only option regular?
I have ruled out CF due to a theorem that says CF intersection Regular = Regular.
Furthermore,is it even worth trying to find a solution via the Venn diagram?Because I find odd that the union of 2 CF can be regular.

REPLY [1 votes]: If L1 is defined on even length strings, and L2 is defined to include all even-length strings, and an arbitrary subset of the odd-length strings, then if L1 is regular, L1 $\cap$ L2 $=$ L1 has to be regular, but L2 can be arbitrary.