Generalized cohomology theories for algebraic stacks

Abstract

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer-Vietoris. We also prove a fixed point localization formula for torus actions. Finally, the construction is contrasted with a "limit-extended" stable motivic homotopy category: we show for example that limit-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin-Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism. A. A. KHAN AND C. RAVI 3.4. Exactness of i * 3.5. Derived invariance 3.6. Localization 4. The stable homotopy category 4.1. Thom anima 4.2. Characterization 4.3. Functoriality 4.4. The basic case 4.5. Proof of Theorem 4.4 4.6. Proof of Theorem 4.9(i) 4.7. Proof of Theorem 4.9(ii) 4.8. Proof of Theorem 4.9(iii) 5. Axiomatization 5.1. ( * , ♯, ⊗)-formalisms 5.2. The Voevodsky conditions 5.3. Constructible separation 5.4. Nisnevich descent 5.5. The example of SH * 6. Proper base change 6.1. Statement 6.2. Cdh descent 6.3. Relative purity 6.4. Reductions 6.5. Proof of Theorem 6.1, projective case 6.6. Proof of Theorem 6.1, general case 7. The !-operations 7.1. Statement 7.2. Compactifications 7.3. Proof of Theorem 7.1 7.4. Constructible separation 7.5. Purity 7.6. Descent 8. The Euler and Gysin transformations 8.1. Euler transformation 8.2. Gysin transformation 8.3. Proof of Theorem 8.4 8.4. Self-intersection formula 9. Cohomology and Borel-Moore homology theories 9.1. Definitions 9.2. Operations 9.3. Properties 9.4. Fundamental classes and Poincaré duality 10. Examples 10.1. Homotopy invariant K-theory 10.2. Algebraic cobordism 10.3. Motivic cohomology 11. Fixed point localization 12. Limit extensions 12.1. Limit-extended categories 67 12.2. Cohomology 68 12.3. The Borel construction 69 12.4. Proof of Theorem 12.9 for motivic cohomology 70 12.5. Proof of Theorem 12.9 in general 71 12.6. Equivariant Chow groups, cobordism and K-theory 72 References 74

Six-Functor-Formalisms on Higher Stacks

“…Every morphism in M (0) which is in C (cf. Definition 7.2) satisfies (in the appropriate situation) (H1(f )-H3(f )), (C1(f )-C3(f )), and (CH(f )) for the restrictions 28 of D !,h , D * ,h , and D ′ , to M (0) , M (0),op , and H cor (M (0) ), respectively.…”

Absolute Poincaré duality in étale cohomology

“…Finally, the construction can be refined from étale cohomology to motivic cohomology. For this one can use the limit-extended motivic cohomology of algebraic stacks defined in [KRa,§12] as a substitute for [LZ]. Note that the trace formalism for flat maps (as developed in [SGA4, Exp.…”

Equivariant generalized cohomology via stacks

“…We begin in Sect. 1 by recalling the lisse extension construction from [KR,§12]. The key technical result (Proposition 2.2) is proven in Sect.…”

“…The most relevant distinction between the above setup and the motivic one is the failure of étale descent in the latter setting; for example, algebraic Ktheory only satisfies Nisnevich descent. 1 Surprisingly, there are nevertheless well-behaved extensions of these theories to stacks 2 via the mechanism of lisse extension, see [KR,Kha4] 3 . We may thus use the above approach with quotient stacks to define equivariant versions of generalized cohomology theories:…”