Algebraic K-theory of quasi-smooth blow-ups and cdh descent

Khan, Adeel A.

doi:10.5802/ahl.55

Cited by 11 publications

(8 citation statements)

References 26 publications

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“…In fact, although the result is an immediate consequence of Corollary B(iv), we also give a more direct argument, independent of the formalism of six operations, by using the cdh descent criterion in [Kh2] (see Remark 10.6).…”

mentioning

confidence: 82%

“…E,Rem. 5.11(c)], we can give a direct proof of Corollary G by following [Kh2,5.3.4]. The new input in our setting is Remark 10.4 and the localization theorem for H * (Theorem 3.19), which together imply closed descent (cf.…”

Section: Smentioning

confidence: 99%

“…The new input in our setting is Remark 10.4 and the localization theorem for H * (Theorem 3.19), which together imply closed descent (cf. [Kh2,Ex. 5.9]).…”

Section: Smentioning

confidence: 99%

“…This already yields unstable representability of KH (see Construction 10.3), which is enough to deduce Corollaries F and G (see Remarks 10.5 and 10.6). The stable representability (Theorem 10.7) is not much more difficult than in the case of schemes or algebraic spaces [Ci,Kh2] and quotient stacks [Ho4].…”

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confidence: 99%

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Generalized cohomology theories for algebraic stacks

Khan¹,

Ravi²

2021

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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

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confidence: 82%

Section: Smentioning

confidence: 99%

“…The new input in our setting is Remark 10.4 and the localization theorem for H * (Theorem 3.19), which together imply closed descent (cf. [Kh2,Ex. 5.9]).…”

Section: Smentioning

confidence: 99%

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confidence: 99%

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Generalized cohomology theories for algebraic stacks

Khan¹,

Ravi²

2021

Preprint

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“…(See also [BS,Theorem 6.7], [Kh20,Theorem 3.3], [J21,Theorem B.3]. ) (2) (The projectivization formula) In general, if F is not locally free, the above semiorthogonal sequence no longer spans the whole category D qc (P(F )).…”

Section: Semiorthogonal Decompositionsmentioning

confidence: 99%

Derived projectivizations of complexes

Jiang¹

2022

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In this paper, we study the counterpart of Grothendieck's projectivization construction in the context of derived algebraic geometry. Our main results are as follows: First, we define the derived projectivization of a connective complex, study its fundamental properties such as finiteness properties and functorial behaviors, and provide explicit descriptions of their relative cotangent complexes. We then focus on the derived projectivizations of complexes of perfect-amplitude contained in [0, 1]. In this case, we prove a generalized Serre's theorem, a derived version of Beilinson's relations, and establish semiorthogonal decompositions for their derived categories. Finally, we show that many moduli problems fit into the framework of derived projectivizations, such as moduli spaces that arise in Hecke correspondences. We apply our results to these situations. 7.1. The Fourier-Mukai functors and semiorthogonal decompositions 69 7.2. The local situation 72 7.3. Applications to classical examples 75 8. Hecke correspondence moduli as derived projectivizations 84 8.1. Hecke correspondences: the case of one-step flags 85 8.2. Hecke correspondences: the case of general flags 88 8.3. Hecke correspondences for surfaces 91 Appendix A. Simplicial Models for Symmetric and Koszul Algebras 95 A.1. Dold-Puppe and Quillen's construction 95 A.2. Simplicial symmetric algebras 95 A.3. Simplicial Koszul algebras 96 References 98

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