Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks
In the classical setting of a compact group acting on a topological space one has basically two approaches to equivariant cohomology. One is due to [\textit{A. Borel}, Seminar on transformation groups. With contributions by G. Bredon, E. E. Floyd, D. Montgomery and R. Palais, Ann. Math. Stud. 46 (1960; Zbl 0091.37202)] and the other is due to \textit{G. Bredon} [``Equivariant cohomology theories''. (1967; Zbl 0162.27202)]. The Bredon-style cohomology theory can be constucted in the following way. Let \({\mathcal R}^G: (G\)-topology on \(X)\rightarrow \)abelian groups be the presheaf defined by \({\Gamma}(U,{\mathcal R}^G)=K^{0}_{G}(U),\) where \(K^{0}_{G}(U)\) denotes the Atiyah-Segal \(K\)-theory of \(U\). Given an abelian presheaf \(P\) one then may define \(H^{*}_{G,\text{Br}}=R{\Gamma}(X,(P\otimes {\mathcal R}^G){\widetilde{\phantom I}})\). Here \({\tilde{}}\) denotes the sheafification of a presheaf and \(R{\Gamma}(X,*)\) denotes the derived functor of the global sections computed on \(G\)-topology of \(X.\) In the paper under review, the author introduces the Bredon-style equivariant homology and cohomology theories for algebraic stacks and proves variants of Riemann-Roch theorems in this situation. The equivariant cohomology and homology theories introduced in the paper generalize both Bredon-style theories for group actions and Bloch-Ogus-style theories for schemes and algebraic spaces [\textit{D. Knutson}, ``Algebraic spaces''. Lect. Notes Math. 203. (1971; Zbl 0221.14001)].
scientific article; zbMATH DE number 5123967
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