Categorical Milnor squares and K-theory of algebraic stacks

Bachmann, Tom; Khan, Adeel Ahmed; Ravi, Charanya; Sosnilo, Vladimir

doi:10.48550/arxiv.2011.04355

Cited by 2 publications

(8 citation statements)

References 20 publications

(34 reference statements)

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“…is cartesian (see e.g. [BKRS,Thm. 2.2.3]), in view of the definition of compact objects, the fact that filtered colimits of spectra are exact (i.e., commute with finite limits), and that formation of mapping spectra in a stable ∞-category commutes with limits.…”

Section: Now Consider the Compositementioning

confidence: 99%

“…B], there exists an affine Nisnevich cover Y ′ 0 ↠ Y cl and a finite locally free E 0 on Y ′ 0 lifting N i 0 . By derived invariance of the étale site and [BKRS,Lem. A.2.6], we can lift this data to an affine Nisnevich cover Y ′ ↠ Y and a finite locally free E on Y ′ .…”

Section: Now Consider the Compositementioning

confidence: 99%

“…A.2.6], we can lift this data to an affine Nisnevich cover Y ′ ↠ Y and a finite locally free E on Y ′ . As the statement is Nisnevich-local on Y, we may replace Y by Y ′ and thereby assume that there exists a finite locally free [BKRS,Prop. A.3.4], φ lifts along the π 0 -surjection…”

Section: Now Consider the Compositementioning

confidence: 99%

“…where I is the ideal sheaf of the closed immersion Ho3,Lem. 2.17] or [BKRS,Prop. A.3.4]), this admits a splitting and the resulting morphism…”

Section: Now Consider the Compositementioning

confidence: 99%

“…This was previously known in the cases of tame quasi-Deligne-Mumford classical stacks (see [HR2,Thm. A]) and quasi-compact derived stacks with affine diagonal and nice stabilizers (see [BKRS,Thm. A.3.2]).…”

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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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Section: Now Consider the Compositementioning

confidence: 99%

Section: Now Consider the Compositementioning

confidence: 99%

Section: Now Consider the Compositementioning

confidence: 99%

“…where I is the ideal sheaf of the closed immersion Ho3,Lem. 2.17] or [BKRS,Prop. A.3.4]), this admits a splitting and the resulting morphism…”

Section: Now Consider the Compositementioning

confidence: 99%

mentioning

confidence: 99%