Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited (Q2796771)

For a topological space \(X\) with an action of a compact Lie group \(G\) and a given prime \(\ell\), the study of the \(\mathrm{mod} \, \ell \) equivariant cohomology algebras \(H^*_G(X, \mathbb{F}_{\ell } )\) goes back to \textit{D. Quillen}'s paper [Ann. Math. (2) 94, 549--572, 573--602 (1971; Zbl 0247.57013)].NEWLINENEWLINEQuillen has shown that for any noetherian commutative ring \(\Lambda\) the algebra \(H^*_G(X, \Lambda )\) is finitely generated over \(\Lambda\). Furthermore, he has given structure theorems relating the ring structure of \(H^*_G(X, \mathbb{F}_{\ell } )\) to the group structure of \(G\), namely to the elementary abelian \(\ell\)-subgroups \(A\) of \(G\) and the components of the fixed points set \(X^A\).NEWLINENEWLINEThe authors of this present article study the algebraic situation and give an analogue to Quillen's results: \(k\) denotes an algebraically closed field with \(\mathrm{char} \, k \neq {\ell}\), \(\Lambda\) a noetherian commutative ring annihilated by an integer invertible in \(k\). Suppose \(G\) is an algebraic group over \(k\) and \(X\) a separated algebraic space of finite type over \(k\) with an action of \(G\). Under these assumptions it is shown that the étale cohomology ring \(H^*([X/G], \Lambda)\) of the quotient stack \([X/G]\) is a finitely generated algebra over \(\Lambda\).NEWLINENEWLINEAnother main result of this paper states that the ring homomorphism \(H^*([X/G],\mathbb{F}_{\ell } ) \to {\displaystyle \lim_{\longleftarrow} }\, H^*(BA,\mathbb{F}_{\ell })\) given by restrictions is a uniform \(F\)-isomorphism (the inverse limit taken over the category of pairs \((A,C)\) for elementary abelian \(\ell\)-subgroups \(A\) of of \(G\) and connected components \(C\) of \(X^A\)). Here the notion of a \textit{uniform \(F\)-isomorphism} originates from Quillen's work and means there exists some \(n\) such that the \(\ell^n\)-th power of any Element of \(\mathrm{ker} \, (F)\) and \(\mathrm{coker} \, (F)\) vanishes.NEWLINENEWLINEThis theorem in the spirit of Quillens result is further generalized to a structure theorem for \(H^*([X/G], K)\), where \(K\) is a constructible complex of sheaves on \([X/G]\) with a commutative ring structure, \(K \in D^+_c([X/G],\mathbb{F}_{\ell }) \).NEWLINENEWLINEThe paper starts with part 1 summarizing the necessary material on quotient stacks, étale cohomology of Artin stacks and multiplicative structures in derived categories.NEWLINENEWLINEPart 2 is devoted to formulate and prove the main results: Finiteness theorems for equivariant cohomology rings and the structure theorem.NEWLINENEWLINEThe continuity property -- key ingredient of Quillens original proof -- has to be replaced in the algebraic case: For this purpose, the notions of geometric points and of \(\ell\)-elementary points of Artin stacks are introduced (the latter notion adapted to formulate the general structure theorem). Doing so, Quillen's property can be replaced by an analysis of the specialization of points of the quotient stack \([X/G]\).NEWLINENEWLINEThe structure theorems for equivariant cohomology algebras follow from the main theorem 8.3 of this paper, formulated in a condensed form this way (authors' introduction):NEWLINENEWLINE``If \({\mathcal X} = [X/G]\) or \({\mathcal X}\) is a Deligne-Mumford stack of finite presentation and finite inertia over \(k\), and if \(K\in D^+_c({\mathcal X} ,\mathbb{F}_{\ell }) \) is endowed with a commutative ring structure, then the ring homomorphism \(H^*({\mathcal X} , K) \to \underset{{\overset{\longleftarrow} {x:{\mathcal S}\to {\mathcal X}}}}\lim H^*({\mathcal S},x^*K)\) given by restriction maps is a uniform \(F\)-isomorphism. Here the limit is taken over the category of \(\ell\)-elementary points of \({\mathcal X}\).''NEWLINENEWLINEThe authors point out the contributions communicated by P. Deligne (proof of the finiteness theorem, generalizing their original version) and J.-P. Serre (finiteness of orbit types).NEWLINENEWLINEAs a motivation of their work, they mention an analog of Quillen's localization theorem [loc. cit., Theorem 4.2] which they had obtained in [\textit{L. Illusie} and \textit{W. Zheng}, Int. Math. Res. Not. 2013, No. 1, 1--62 (2013; Zbl 1360.14120)]. Further, the authors refer to [\textit{L. Illusie}, in: De la géométrie algébrique aux formes automorphes (II). Une collection d'articles en l'honneur du soixantième anniversaire de Gérard Laumon. Paris: Société Mathématique de France (SMF). 177--195 (2015; Zbl 1327.14094)] for a report on that paper and the present article.