Derived Algebraic Geometry Student Seminar Fall 2023

Our current plan is to read Toen’s survey. We are currently meeting in Evans 762 on Tuesdays at 2 p.m. All are welcome; please e-mail me to be added to the mailing list.

If you’d like to join remotely, send me an e-mail so that I open Zoom. Here’s the Zoom link.

8/3: Formulated the plan and discussed how to compute homotopy (co)fibers in the category of chain complexes (with the projective model structure).
8/10: Discussed Illusie’s cotangent complex and did some simple calculations with it. Figured out the relation between Grothendieck-Riemann-Roch and virtual fundamental classes. This still corresponds to Section 1 of the survey.
8/17: Formulation of deformation theory as a functor out of a category of local Artin rings following Schlessinger. Also Section 1 of the survey.
8/24: Discussed model categories, focusing on the main examples of topological spaces and chain complexes. This is Section 2 of the survey. Resources: Mazel-Gee’s Notes and Sophie Morel’s Notes.
8/29: Discussed construction of quasi-categoies via the horn-filling property, i.e. via weak Kan complexes. Also discussed Kan complexes. Supplemental to section 2 of the survey.
9/5: Gave some examples of infinity categories via \(\mathsf{Top}\) and \(\mathsf{Ch}\). Discussed homotopy pullbacks and pushouts in these categories, and how to pass from \(\mathsf{Top}\) to \(\mathsf{Ch}\) via the chains functor.
9/12: Discussed stacks, descent, \(BG\) as a (truncated) stack, Cech nerves, prestacks, and derived schemes. Section 2 of the survey.
9/19: Discussed quasi-coherent shaves on prestacks, the cotangent complex of a prestack, tangent spaces/complexes (on locally almost of finite type prestacks), and geometric \(n\)-stacks, also known as Artin \(n\)-stacks. Section 3.1 of the survey.
9/26: Yuji led us through a calculation of the self-intersection of a point in the affine line, a calculation of the blowup of the affine plane at the origin along with a calculation of the derived fiber, a general construction of the cotangent complex for categories with abelian group objects, and a discussion of non-flat base change in classical algebraic geometry and how it relates to the derived setting. References: Cotangent complex, Base change, Blowups.
10/3: Justin covered monoidal Dold Kan, comparison between simplicial commutative algebras, dgas, and \(\mathbb E_\infty\) algebras, and a computation of cotangent complex of a quotient stack \(\mathbb L_U/G = \text{Fib}(\mathbb L_U \to f^* g^*)\). References: nlab monoidal Dold Kan, Lurie’s thesis, MO post on cdgas in char p, Adeel Khan notes on Stacks & DAG
10/10: Justin led us through a calculation of the cotangent complex of the stack of perfect complexes \(X \mapsto \mathsf{Perf}(X)\), but we weren’t able to figure out some details. He also led us through some intersection theory. We discussed quasi-smooth maps between stacks, locally complete intersections, and \(\mathbb A^1\)-homotopy theory. In this context, Justin walked us through some arguments about how Chow homology classes and the Gysin homomorphism behave nicely in the setting of derived algebraic geometry, namely because the derived intersection of subvarieties provides a more natural intersection theory than classical intersections.
10/17: Skipped because of Tony’s talk at the Representation Theory seminar.
10/24: Vivasvat talked about \(n\)-Artin stacks, groupoid objects, \(K(G,n)\)’s as n-Artin stacks, and the stack of perfect complexes. We also took a long tangent on the \(B\) functor which maps a group to its classifying space; we succeeded in understanding \(BX\) when \(X=BG\) is the classifying space of an abelian group. We tried to work through some examples of gerbes but were unable to. References: Perfect complexes, Section 3 of the survey.
10/31: Ansuman told us about gerbes, especially how we can think of them as 2-cocycles. There were a few examples, and we worked out the example of how a \(B\mathbb Z\)-bundle is a \(\mathbb Z\)-gerbe.
11/7: John gave a way to think about the cotangent complex as classifying lifts of square-zero extensions. For \(X\) a classical, smooth, proper scheme, he gave a calculation of the cotangent complex of the mapping stack \(\text{Map}(X,Y)\) for \(Y\) a finitely-presented \(n\)-Artin stack. He also proved a theorem: Obstructions to lifts live in \(\pi_0(\Omega^\infty \text{Hom}(\mathbb L_Y,M))\), and these form a torsor under \(\Omega^\infty \text{Hom}(\mathbb L_Y,M)\), which mirrors the classical “main” theorem about cotangent complexes. References: Halpern-Leistner-Preygel, Toen-Vezzosi, HAG II section 1.4, Toen’s survey Section 4.1 and parts of Section 3, John’s notes.
11/14: Daigo told us about the HKR theorem following Loop Spaces and Connections. We went through a full proof of HKR for derived schemes, and discussed some examples by studying \(B\mathbb Z=S^1 \to B\mathbb G_a\). We also got confused as to why there was a requirement that the compatibility maps for (pro-)cotangent spaces be isomorphisms for the (pro-)cotangent complex. Initial thoughts were it may have something to do with the procompletion of QCoh not forcing these maps to be isomorphisms.
11/21: No meeting this week.
11/28: I talked about the derived de Rham complex and its relation to algebraic de Rham cohomology following Bhatt. This was motivated by a discussion of “formal descent,” which is a descent property for closed immersions of classical schemes for a certain “formal completion” in the category of derived schemes. In turn, this leads to an algebro-geometric analogue of the fact that we can recover the connected component of a point in a topological space by taking the delooping of the based loop space. References: My notes following Bhatt, and Section 4.2 of Toen’s survey.
12/5: Justin covered the content of a paper of Toen which is a new proof that proper lci maps preserve perfect complexes using DAG. The proof is a reduction to the noetherian case, but in the process we are forced to work with derived schemes.