Cohomology of the moduli stack of algebraic vector bundles

Annala, Toni; Iwasa, Ryomei

doi:10.48550/arxiv.2104.09559

Cited by 1 publication

(6 citation statements)

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“…for every stack X (Corollary 4.2.7). This is a refinement of the main theorem in [AI22], which assumes the existence of transfers. The proof is mostly parallel to that of [AI22].…”

Section: Cohomology Of the Moduli Stack Of Vector Bundlesmentioning

confidence: 78%

“…Let S be a qcqs derived scheme. Let Sh tr pbf (Sch S ) be the ∞-category of pbf-local sheaves with transfers in the sense of [AI22]. Then there is a canonical symmetric monoidal left adjoint St S * → Sh tr pbf (Sch S ), which carries 1 to an invertible object.…”

Section: Lemma An Orientable Motivic Spectrum In Is Fundamentalmentioning

confidence: 99%

“…We remark that if a motivic spectrum is orientable then it is fundamental. Then we formulate projective bundle formula for oriented motivic spectra, develop a theory of Chern classes, and calculate the cohomology of the moduli stack ect n of rank n vector bundles by adopting the argument in [AI22]. 0.2.2.…”

Section: Introductionmentioning

confidence: 99%

“…Remark. All cohomology theories treated in [AI22] were assumed to have finite quasi-smooth transfers. We have succeeded in removing this assumption by introducing the notion of oriented motivic spectra.…”

Section: Introductionmentioning

confidence: 99%

“…The formulation of Theorem 0.1.1 is due to Dustin Clausen. We would like to thank him for the essential remark that our previous work [AI22] may be useful in proving Theorem 0.1.1, and for many helpful discussions. We also thank Lars Hesselholt, Markus Land, and Shuji Saito for helpful discussions.…”

Section: Introductionmentioning

confidence: 99%

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Motivic spectra and universality of $K$-theory

Annala¹,

Iwasa²

2022

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We develop a theory of motivic spectra in a broad generality; in particular 1 -homotopy invariance is not assumed. As an application, we prove that K-theory of schemes is a universal Zariski sheaf of spectra which is equipped with an action of the Picard stack and satisfies projective bundle formula. CONTENTS TONI ANNALA AND RYOMEI IWASA A.3. Constructions of -monoidal structures 41 A.4. Day convolution 42 A.5. Smashing localizations 43 References 43 1.3.4. Definition (Lax c-spectrum). Let S c be the free commutative algebra in Σ generated by F 1 (c). For an ∞-categoroy presentably tensored over , we define Sp lax c ( ) := Mod S c ( Σ ) and call it the ∞-category of lax c-spectra in . Then Sp lax c ( ) admits a presentably symmetric monoidal structure in a canonical way and Sp lax c ( ) is a presentably tensored over Sp lax c ( ). 1.3.5 (Adjunction). We consider the following adjunctions: -The adjunction (F, U) together with S c ⊗ − induces an adjunction F c := S c ⊗ F : ⇄ Sp lax c ( ): U. Then F c : → Sp lax c ( ) is symmetric monoidal and F c : → Sp lax c ( ) is -linear. -The adjunction (s + , s − ) induces an adjunction s + : Sp lax c ( ) ⇄ Sp lax c ( ): s − . Then s + : Sp lax c ( ) → Sp lax c ( ) is Sp lax c ( )-linear. 1.3.6. Lemma.

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“…for every stack X (Corollary 4.2.7). This is a refinement of the main theorem in [AI22], which assumes the existence of transfers. The proof is mostly parallel to that of [AI22].…”

Section: Cohomology Of the Moduli Stack Of Vector Bundlesmentioning

confidence: 78%

Section: Lemma An Orientable Motivic Spectrum In Is Fundamentalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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