# Motives Seminar-WS 2021/22

## Motivic homotopy theory of algebraic stacks

This semester the motives seminar will look at extensions of the stable homotopy category to a theory on suitable algebraic stacks, together with a six-functor formalism. Stacks as generalisations of schemes occur in many natural settings; one that is quite important is their role in the theory of moduli. Many invariants arising in modern enumerative geometry rely on constructing virtual fundamental classes in the Chow ring of suitable moduli stacks, if one wishes to refine these invariants to theories such as the Chow-Witt groups, hermitian K-theory, or the cohomology of the Witt sheaves, one needs to extend the foundations of these theories from schemes to stacks. Here we will look at a way of defining motivic stable homotopy categories for stacks using the Nisnevich topology, which should allow one to obtain these finer invariants.

The program consists of a quick overview of some of the necessary material about infinity categories before starting the main block on the category of stacks with good properties for the Nisnevich topology, and Chirantan Chowdhury's construction of the motivic stable homotopy category for such stacks. This will occupy most of the semester. We will also look at some related theories, such as Hoyois' construction of equivariant versions of the motivic stable homotopy category, as well as work of Khan-Ravi on other approaches to the theory. .

We will hold the seminar "in person'' with an online link via Zoom, so all those who we were happy to have take part from a distance will still be able to do so. We meet on Tuesdays, 16-18 Uhr in WSC-S-U-3.01. If you are interested in attending the seminar or giving one of the lectures, please contact me (Marc Levine) at marc.levine@uni-due.de.

### Program: Motivic homotopy theory of algebraic stacks

Lecture 0 (12.10-Chirantan Chowdhury) Introduction and overview
Lecture 0 Video

Lecture 1 (19.10-Herman Rohrbach) Introduction to $\infty$-categories
Lecture 1 Video

We recall the basic notion of $\infty$-categories. We shall begin with recalling the definition and introduce basic terminologies like objects, morphisms, compositions, homotopy category, initial/final objects, limits and mapping spaces. The two main examples of $\infty$-categories that we are interested in arise from simplicial categories and $2$-categories. We shall end the talk by recalling the notion of fibrations of simplicial sets, categorical equivalence and Cartesian fibrations.

References: [Lur09, Chapter 1-2], [Lur18a, Chapter 1, 2.2,2.3] and [Rez, Part 4].

Lecture 2 (26.10-Marc Levine) Presentable stable $\infty$-categories
Lecture 2 Notes
Lecture 2 Video

We define the notion of presentability and stability in the setting of $\infty$-categories. The talk shall start by recalling the notion of presheaves, Yoneda lemma and adjoint functors in the setting of $\infty$-categories. We then move on defining presentable $\infty$-categories by filtered categories and ind-objects. We briefly recall the notion of $\infty$-sheaves. The talk ends by defining the notion of stable $\infty$-categories and the $\infty$-category of presentable stable $\infty$-categories $\text{Pr}^L_{stb}$.

References: [Lur09], [Lur17] and [Lur18b].

Lecture 3 (02.11-Manuel Hoff) Symmetric monoidal $\infty$-categories and module objects.
Lecture 3 Notes
Lecture 3 Video

We define the algebra and module objects in the setting of $\infty$-categories. In particular, we are interested in defining symmetric monoidal $\infty$-categories. In order to make sense of these notions, we start by defining $\infty$-operads which are generalized notions of colored operads. Symmetric monoidal $\infty$-categories are special kinds of $\infty$-operads. The talk ends by defining module objects over $\infty$-operads and explaning a higher categorical generalization of the fact that a morphism between algebra objects $A \to B$ gives $B$ an $A$-module structure.

References: [Lur17, Chapter 2,3] and [Rob14, Section 9.4.1.2].

Lecture 4 (09.11-Dhyan Aranha) Motivic homotopy theory of schemes
Lecture 4 Video

We define the motivic homotopy theory of schemes in the language of $\infty$-categories. We shall briefly recall the definition of unstable, pointed and stable homotopy theory. Then we recall the functoriality and six operations. We shall state the existence of the exceptional pushforward functors without proving them. This shall be later explained while discussing the enhanced operation map in Talk 10. The talk ends by stating properties like localization, homotopy purity, homotopy invariance and explaining the construction of $\alpha_f$ and purity transformation $\rho_f$.

References: [Rob14], [CD19] and [Hoy17].

Lecture 5 (16.11-Anneloes Viergever) Algebraic stacks
Lecture 5 Notes
Lecture 5 Video

We discuss the notion of algebraic spaces and algebraic stacks. We also discuss some examples of algebraic stacks like local quotient stacks, moduli stack of vector bundles. We also define the deformation to the normal cone in the setting of algebraic stacks. The talk ends by discussing the notion of resolution property of algebraic stacks.

References:[Sta21], [Knu71], [Hei10], [LMB00] [Tot04] and [Gro17].

Lecture 6 (23.11-Pietro Gigli) Descent along sections
Lecture 6 Notes
Lecture 6 Video

In this talk, we recall the $\infty$-categorical setup generalizing the classical statement that descent along morphisms admitting sections is automatic. The talks by recalling the classical statement in ordinary category theory using the notion of split forks. Then we move in explaining the skeletal descriptions of simplicial category $\Delta$, augmented simplicial category $\Delta_+$ and the split-simplicial category $\Delta_{-\infty}$. The talks end by stating the main result which is due to [Lur09, Lemma 6.1.3.16].

References:[Lur09, Section 6.1] and [Cho, Chapter 2]

Lecture 7 (30.11-Jan Hennig) Kan extensions, descent theory and localization of $\infty$-categories.
Lecture 7 Notes
Lecture 7 Video

In this lecture, we recall some important notions in higher category theory that we need for proving [Cho, Theorem 3.4.1], which allows us to extend sheaves from schemes to a large class of algebraic stacks. We start the talk by introducing the notion of Kan extensions in the setting of $\infty$-categories. Kan extensions allow us to associate limits of diagrams in a functorial manner. The second part of the lecture introduces the notion of descent theory and states specific conditions when $\infty$-sheaves can be realised by descent along Čech nerves of coverings. The last part of the lecture reviews localization of $\infty$-categories and explains the existence of localization along any class of morphisms.

References:[Lur09, Section 4.2,5.5], [Lur18b, Appendix A 3.1-3.3],[Cho, Appendix A] and [Lan21, Section 2.4].

Lecture 8 (07.12-Gabriela Guzman) Enhancement of sheaves along coverings with local sections.
Lecture 8 Video

We prove [Cho, Theorem 3.4.1] and thus extend the stable homotopy functor $\mathcal{SH}^{\otimes}(-)$ from schemes to the $(2,1)$-category $\text{Nis-locSt}$. The talk starts by introducing $\mathcal{T}$-local sections in a site $(\mathcal{C},\mathcal{T})$ and stating some properties. We then move to defining the $(2,1)$-category of stacks admitting $\mathcal{T}$-local sections of which the category of qcqs algebraic spaces and the $(2,1)$-category $\text{Nis-locSt}$ are examples. We state[Cho, Theorem 3.4.1] and explain the proof using the theory of Kan extensions and localizations explained in the previous lecture. The lecture ends defining the functor $\mathcal{SH}^{\otimes}_{\text{ext}}(-)$ ([Cho, Corollary 3.5.3]) and constructing the four functors $f^*,f_*,-\otimes-$ and $\text{Hom}(-,-)$ on $\text{Nis-locSt}$.

References:[Cho, Chapter 3].

Lecture 9 (14.12-Alessandro D'Angelo) Compactification in the setting of $\infty$-categories.
Lecture 9 Video

We briefly describe the idea of Deligne's compactification in $\infty$-categorical setting due to Liu and Zheng [LZ12]. The talks starts with briefly recalling Deligne's argument of constructing the exceptional pushforward $f_!$ in étale cohomology. The rest of talk introduces the terminology of multi-marked and multi-tiled simplicial sets which is needed to state the theorem of $\infty$-categorical compactification ([LZ12, Theorem 0.1]) and to construct the enhanced operation map (which shall be done in the next talk). The talk ends with a brief sketch of the idea of [LZ12, Theorem 0.1].

References:[Del73, Section 3][LZ12] and [Cho, Section 4.2, Appendix D].

Lecture 10 (21.12-Viktor Kleen) Enhanced operations for stable homotopy theory of algebraic stacks.
Lecture 10 Video

We extend the exceptional pushforward and pullback functors $f_!$ and $f^!$ from schemes to $\text{Nis-locSt}$. This is extended by the so called enhanced operation map due to Liu and Zheng [LZ17, Section 2.2]. The talk starts with recalling the setting of six operations with smooth (and proper) base change and projection formula on the level of schemes and constructs the enhanced operation map. Then we explain how the enhanced operation map encodes the exceptional pushforward functors, base change and projection formula. The rest of talk deals with explaining extending the enhanced operation map from schemes to $\text{Nis-locSt}$ ([Cho, Proposition 4.4.2]) with a brief sketch of the proof.

References:[Cho, Chapter 4, Appendix D], [LZ17, Section 1, Section 3] and [Rob14, Section 9.4].

Lecture 11 (11.01.22-Postponed to 25.01.22-Chirantan Chowdhury) Six operations for $\mathcal{SH}_{\text{ext}}(\mathcal{X})$.
Lecture 11 Video

In this talk, we prove relations between the six operations that we enocuntered in the previous talks in particular; localization, homotopy invariance and homotopy purity. The talk starts with proving smooth and proper base change and then move on to prove localization and homotopy invariance. We construct the natural transformations $\alpha_f$ and $\rho_f$. The talks ends with proving the homotopy purity theorem via the deformation to the normal cone.

References: [Cho, Chapter 5].

(18.01) Special lecture by Vijaylaxmi Trivedi, TIFR, Mumbai.
Special Lecture Video
Title: The quadric hypersurfaces and some Hilbert-Kunz multiplicity conjectures related to them.
Abstract: After the class of polynomial rings, the quadric hypersurfaces could be considered the next best class of rings. However, a well studied characteristic p invariant introduced by P. Monsky, namely the Hilbert-Kunz (HK) multiplicity, is still unknown for these hypersurfaces.
Here we discuss a long standing conjecture of Watanabe-Yoshida on the HK multiplicities of quadric hypersurfaces. Some basic facts and subtleties about the HK multiplicity and the progress made by several people on this conjecture will be surveyed.
Then, using the classification of ACM bundles on the smooth quadric via matrix factorizations, we describe the HK density functions of the quadrics and prove a part of the WY conjecture for all dimensions. Moreover, for large p, we give a closed formula for HK multiplicities of quadric hypersurfaces.

Lecture 12 (01.02-Charanya Ravi) Generalized cohomology theories of algebraic stacks

We recall the construction of generalized cohomology theories on scalloped algebraic stacks and also define limit-extended cohomology theories.

References: [KR21].

[Ayo07a] Joseph Ayoub. Les six opérations de Grothendieck et le formalisme des cycles évanes- cents dans le monde motivique. I. Astérisque, (314):x+466 pp. (2008), 2007.

[Ayo07b] Joseph Ayoub. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Astérisque, (315):vi+364 pp. (2008), 2007.

[CD19] Denis-Charles Cisinski and Frédéric Déglise. Triangulated categories of mixed motives. Springer Monographs in Mathematics, 2019.

[Cho] Chirantan Chowdhury. Motivic homotopy theory of algebraic stacks. C. Chowdhury Thesis

[Del73] P. Deligne. Cohomologie a supports propres. In Théorie des Topos et Cohomologie Etale des Schémas, pages 250–480, Berlin, Heidelberg, 1973. Springer Berlin Heidel- berg.

[FHT11] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman. Loop groups and twisted k -theory i. Journal of Topology, 4(4):737–798, Dec 2011.

[Gro17] Philipp Gross. Tensor generators on schemes and stacks. Algebr. Geom., 4(4):501– 522, 2017.

[Hei10] Jochen Heinloth. Lectures on the moduli stack of vector bundles on a curve. In Affine flag manifolds and principal bundles, Trends Math., pages 123–153. Birkhäuser/Springer Basel AG, Basel, 2010.

[Hoy17] Marc Hoyois. The six operations in equivariant motivic homotopy theory. Adv. Math., 305:197–279, 2017.

[Knu71] Donald Knutson. Algebraic spaces, volume 203. Springer, Cham, 1971.

[KR21] Adeel A. Khan and Charanya Ravi. Generalized cohomology theories for algebraic stacks, 2021.

[Lan21] Markus Land. Introduction of infinity-categories. Birkhäuser, Cham, 2021.

[LMB00] Gérard Laumon and Laurent Moret-Bailly. Champs algébriques. Springer Berlin Heidelberg, Berlin, Heidelberg, 2000.

[LO08a] Yves Laszlo and Martin Olsson. The six operations for sheaves on Artin stacks. I. Finite coefficients. Publ. Math. Inst. Hautes Études Sci., (107):109–168, 2008.

[LO08b] Yves Laszlo and Martin Olsson. The six operations for sheaves on Artin stacks. II. Adic coefficients. Publ. Math. Inst. Hautes Études Sci., (107):169–210, 2008.

[Lur09] Jacob Lurie. Higher topos theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009.

[Lur17] Jacob Lurie. Higher algebra., 2017.

[Lur18a] Jacob Lurie. Kerodon. 2018.

[Lur18b] Jacob Lurie. Spectral algebraic geometry. 2018.

[LZ12] Yifeng Liu and Weizhe Zheng. Gluing restricted nerves of $\infty$-categories. arXiv e-prints, arXiv:1211.5294, November 2012.

[LZ17] Yifeng Liu and Weizhe Zheng. Enhanced six operations and base change theorem for higher artin stacks, arxiv:1211.5948 2017.

[MV99] Fabien Morel and Vladimir Voevodsky. $\mathbb{A}^1$-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math., (90):45–143 (2001), 1999.

[Rez] Charles Rezk. Introduction to quasicategories. quasicats.pdf.

[Rob14] Marco Robalo. Théorie homotopique motivique des espaces noncommutatifs. These. 2014.

[Rob15] Marco Robalo. K-theory and the bridge from motives to noncommutative motives. Adv. Math., 269:399–550, 2015.

[Sta21] The Stacks project authors. The stacks project. 2021.

[Tot99] Burt Totaro. The Chow ring of a classifying space. In Algebraic K-theory (Seattle, WA, 1997), volume 67 of Proc. Sympos. Pure Math., pages 249–281. Amer. Math. Soc., Providence, RI, 1999.

[Tot04] Burt Totaro. The resolution property for schemes and stacks. J. Reine Angew. Math., 577:1–22, 2004.