Motives seminar-SS 2020: Derived algebraic geometry and algebraic cobordism

This semester the Motives Seminar will look at some new approaches to (geometric) algebraic cobordism that rely on methods of derived algebraic geometry. The original construction of algebraic cobordism by Levine and Morel is limited to the setting of quasi-projective schemes over a field of characteristic zero and the construction of pull-back maps is quite delicate. Derived algebraic geometry allows one to give a more direct construction of these pullback maps, with the cost being a reliance on much of the theory of derived algebraic geometry. The most recent construction, due to Annala and Yokura, gives a theory that has cohomological properties for "singular" schemes, giving for the first time a geometric presentation of algebraic $K$-theory (not its $\mathbb{A}^1$-homotopy invariant quotient).

After giving an overview of the "classical" theory, the plan is to present a brief introduction to aspects of derived algebraic geometry needed in the constructions, then look at three derived versions: the first by Lowrey-Schürg, then a version building on this by Annala and finally a sketch of a new version by Annala (relying on work with Yokura) that gives a theory with good properties over a general base-scheme.

Due to the corona virus, we will be running the seminar on an online basis. The talks will be given via Zoom Meetings, and will be recorded for posting on the motives seminar webpage. We will "meet" at the usual time for our seminar: Tuesdays, 14:00-16:00. Iif you would like to receive an invitation to the Zoom Meetings, please send me an email mnl@ude.de.

Program

Lecture 1 (April 21-Marc Levine). An overview of geometric algebraic cobordism.
Lecture 1 slides

Lecture 2 (April 28-Viktor Kleen). The comparison theorem $\Omega_*(X)=\text{MGL}^\text{BM}_{2*,*}(X)$.
Lecture 2 Notes
Lecture 2 Video

Lecture 3 (May 5-Fangzhou Jin) Introduction to derived algebraic geometry 1: Derived schemes and the cotangent complex
Lecture 3 Notes
Lecture 3 Video

Lecture 4 (May 12-Pavel Sechin), An introduction to derived algebraic geometry 2: $Mod_R$, $QCoh(X)$ and $Perf(X)$. Vector bundles and projective space bundles.
Lecture 4 Slides
Lecture 4 Video
Lecture 4 Chats

Lecture 5 (May 19-Ran Azouri). An introduction to derived algebraic geometry.3: Derived blow ups.
Lecture 5 Notes
Lecture 5 Video
Lecture 5 Chats

Lecture 6 (May 26-Gabriela Guzman). Lowrey-Schürg derived algebraic cobordism: the basic construction [LS] § 1-3 .
Lecture 6 Notes
Lecture 6 Video
Lecture 6 Chats

Lecture 7 (June 2-Fangzhou Jin). Lowrey-Schürg derived algebraic cobordism: quasi-smooth pull-backs [LS] § 4
Lecture 7 Notes
Lecture 7 Video
Lecture 7 Chats

Lecture 8 (June 9-Pavel Sechin). Lowrey-Schürg derived algebraic cobordism: the algebraic Spivak theorem and consequences [LS] § 5, 6. .
Lecture 8 Slides
Sorry, I forgot to record the talk!

Lecture 9 (June 16-Heng Xie). Annala's bivariant derived algebraic cobordism: Summary of results and introduction to bivariant theories [A1] § 1,2, also [Ann-Y] as needed.
Lecture 9 Slides
Lecture 9 Video
Lecture 9 Chats

Lecture 10 (June 23-Heng Xie). Annala's bivariant derived algebraic cobordism: the basic construction [A1], § 3
Lecture 10 Slides
Lecture 10 Video
Lecture 10 Chats

Lecture 11 (June 30-Alessandro D'Angelo). Annala's bivariant derived algebraic cobordism: The Conner-Floyd theorem and an intersection ring for singular varieties [A1], § 4.
Lecture 11 Notes
Lecture 11 Video
Lecture 11 Chats

Lecture 12 (July 7-Tariq Syed). Annala's derived algebraic cobordism over a general base: sketch of the construction, the projective bundle formula and summary of main results. [A2]

Lecture 13 (July 14) ???

Some details:

Lecture 1: Give an overview of Levine-Morel algebraic cobordism $\Omega_*$ in the setting of oriented cohomology theories/oriented Borel-Moore homology theories, from [LM]. Give a sketch of the construction and a summary of the main results: explain the functoriality wrt smooth pullback and proper pushforward, the 1st Chern class operators, projective bundle formula, Chern classes for vector bundles. State the main properties for the lci pullbacks but do not give the construction. State the unversality of $\Omega_*$, $\Omega_*(F)\cong Laz$ for $F$ a field of char. 0, the Conner-Floyd Theorem $\Omega^*\otimes_{Laz}\mathbb{Z}[\beta, \beta^{-1}]\cong K_0[\beta, \beta^{-1}]$ and the relation with the Chow ring $\Omega^*\otimes_{Laz}\mathbb{Z}\cong \text{CH}^*$.

 

Lecture 2. Give report on the paper [L1], with a sketch of the proof of the main result. [L1] relies on [L2] for the construction of Borel-Moore homology of an oriented theory, but this can be done using the six-functor formalism in $SH(k)$ by setting \[ \mathcal{E}^{BM}_{a,b}(X):=\mathcal{E}^{-a, -b}(\pi_{X!}(1_X)) \] for $\pi_X:X\to \text{Spec }k$ a finite type $k$-scheme. One can give this definition and list its properties giving as much detail here as time permits. Also if time permits, you can give a sketch of the proof of the isomorphism $\Omega_*(F)\cong Laz$ for $F$ a field of char. 0.

 

Lecture 3: Definition of the ($\infty$)-category of simplicial rings (SCR for short): see § 1,2 in [Khan 1] or [SAG 25.1.1.5]; equivalently state the existence of a model category structure on simplicial rings (as ordinary category) and use [HH 5.5.9.3], more precisely this is Prop.1.11 in [Khan 1]. Mention the three common different models for SCR: (connective) cdga, (connective) $E_{\infty}$ rings and simplicial rings. They are all equivalent in char=0, see [SAG 25.1.2.2, 25.1.2.4]. This is still in [Khan 1, § 1.10]

If there's time for it, mention as an example that any ring $R$ could be embedded as a constant simplicial ring or as the spectrum $HR$. Another less trivial example is given by an affine regular embedding $\text{Spec }A/I\to \text{Spec }A$ seen as the Koszul complex associated to the surjection $A\to A/I$ (see [Khan 6, Exc. 4.3, Ex. 4.4])

Remark: as just said in the example before, the category of Commutative Rings can be embedded in $SCR$ sending $R$ to the constant simplicial ring $R$. On the other way round there's a functor called the truncation: see [Toen, p.31].

State the definition of derived scheme as a functors of points [Khan 1, Def. 2.6]. To be more concise, the naive definition could be "a sheaf of spaces that is locally a $SCR$". Then give the more workable definition of a topological space with a sheaf of $SCR$ as in [Ann-Y, 2.7] State the fact that $dSch$ admits fiber products: locally they are given by the derived tensor product of $SCR$.

Give the definition of strong morphism [HAG II, 2.2.2.3], then we can take as a definition of resp. Zariski open immersion, flat, smooth and \'etale morphism to be: $f$ is a strong morphism of $SCR$ + the morphism induced on the truncation $\pi_0(f)$ is resp. Zariski open, flat, smooth, \'etale. Define the cotangent complex in the affine case [Khan 5]. For a derived scheme a cotangent complex exists and is given by gluing locally what we know on affines. State the properties of the cotangent complex [Ann-Y, Thm 2.59]. Then define a quasi-smooth morphism as in [Ann-Y, 2.61] and state [Ann-Y, 2.66]

 

Lecture 4: $QCoh(X)$: take Prop.3.5 in [Khan 1] as a definition. Locally $Mod_R$ could be seen as the derived category of chain complexes ([HA 1.3.2.7, 1.3.4.6]) by a thm of Schwede - ``Stable Categories are Categories of Modules'' that generalize Dold-Kan. Define $Perf(X)$ and give some properties of $QCoh$ and $Perf$ as in [Ann-Y, § 2.4]. For example: define vector bundles, define duals of perfect complexes, state Prop 2.36 [Ann-Y], define tor-amplitude, state push-pull formula. etc. (time permitting).

Define $\mathbb{P}^n$ as in [Binda 3] that is taken from [SAG 19.2.6.1]
Define the zero locus of a vector bundle and state [Ann-Y, Prop. 2.51]
Define projective bundles and state some of their properties like [Ann-Y, 2.57]

 

Lecture 5: The goal is to define derived blow ups and to state and explain Thm 2.72 and 2.74 of [Ann-Y]. Good references for giving a talk, apart from the original paper itself [Kh-Ry], could be [Khan 7] and [Binda 6].

 

References:

[HH] Lurie-Higher Topos Theory
[HA] Lurie - Higher Algebra
[SAG] Lurie - Spectral Algebraic Geometry.
[HAG II] Toen, Vezzosi - Homotopical Algebraic Geometry II, Geometric stacks and applications. Mem. Amer. Math. Soc. 193 (2008), no. 902
[Khan 1] Khan- https://www.preschema.com/lecture-notes/kdescent/lect1.pdf
[Khan 5] Khan-https://www.preschema.com/lecture-notes/kdescent/lect5.pdf.
[Khan 6] Khan- https://www.preschema.com/lecture-notes/kdescent/lect6.pdf.
[Khan 7] Khan-https://www.preschema.com/lecture-notes/kdescent/lect7.pdf
[Binda 3] Binda-https://www.preschema.com/lecture-notes/grr/lect3.pdf
[Binda 6] Binda-https://www.preschema.com/lecture-notes/grr/lect6.pdf
[Kh-Ry] Khan, Rydh - Virtual Cartier Divisors and Blow-Ups, arXiv:1802.05702.
[Ann-Y] Annala, Yokura -Bivariant Algebraic Cobordism with Bundles, arXiv:1911.12484.
[Toen] Toen - Derived Algebraic Geometry, EMS Surv. Math. Sci. 1 (2014), no. 2, 153-240.
[LM] Levine, Morel-Algebraic cobordism. Springer Monographs in Mathematics. Springer, Berlin, 2007.
[L1] Levine-Comparison of cobordism theories. J. Algebra 322 (2009), no. 9, 3291-3317.
[L2] Levine-Oriented cohomology, Borel-Moore homology, and algebraic cobordism. Special volume in honor of Melvin Hochster. Michigan Math. J. 57 (2008), 523-572
[LS] Lowrey, Schürg-Derived algebraic cobordism. J. Inst. Math. Jussieu 15 (2016), no. 2, 407-443.
[A1] Annala-Bivariant derived algebraic cobordism, arXiv:1807.04989.
[A2] Annala-Chern classes in precobordism theories, arXiv:1911.12493.