Guest seminar Algebra and Topology

The seminar usually takes place Wednesdays in F.13.11, 16:30 - 17:30.
Involved professors: Jens Hornbostel, Sascha Orlik, Tobias Schmidt, Britta Späth, Kay Rülling, Matthias Wendt.

 

Andrea Lachmann: Chow-Witt groups of classifying spaces

The Chow-Witt groups of a scheme are a quadratic refinement of the more well-known Chow groups and play an important role in intersection theory. For any linear algebraic group G, one can define the Chow-Witt ring of its motivic classifying space, which is the ring of characteristic classes for principal G-bundles over smooth varieties with values in Chow-Witt groups. In this talk we will focus on a geometric model for motivic classifying spaces called an admissible gadget. This is a class of algebraic varieties approximating BG in the sense that their Chow-Witt groups coincide in a suitable range of dimensions. We will then discuss some hands-on computations for some classical algebraic groups and finite groups using this machinery.

 

Otto Overkamp: A proof of Chai's conjecture

The base change conductor is an invariant which measures the failure of a semiabelian variety to have semiabelian reduction. It was conjectured by Chai that this invariant is additive in certain exact sequences. I shall report on recent joint work with Takashi Suzuki which implies this conjecture. Time permitting, I shall also discuss counterexamples to a generalisation of Chai’s conjecture.

Andreas Langer: Overconvergent prismatic cohomology

In this note I define an overconvergent version of prisms and prismatic cohomology as introduced by Bhatt and Scholze and show that overconvergent prismatic cohomology specialises to p-adic cohomologies, like Monsky-Washnitzer resp. rigid cohomology for smooth varieties over a perfect field, the de Rham cohomology of smooth weak formal schemes over a perfectoid ring and the étale cohomology of its generic fibre. Besides, I give an overconvergent version of the complex AΩ of Bhatt-Morrow-Scholze and relate it to overconvergent prismatic cohomology.

Shuji Saito: A pro-cdh topology and motivic cohomology of schemes

Recently, Elmanto and Morrow gave a construction of motivic complexes for qcqs schemes. A basic idea coming from the trace method for algebraic K-theory, is to modify the cdh-local motivic complex (defined as the cdh sheafication of the left Kan extension of the motivic complex for smooth schemes over a field) by using other complexes such as Hodge completed derived de Rham complexes and syntomic complexes. In this talk, we propose a different construction of a motivic complex using a new Grothendieck topology, pro-cdh topology, and sketch a proof of the comparison theorem of two motivic complexes for Noetherian schemes over a field. This is a joint work with Shane Kelly.

 

Laurent Berger: Bounded functions on the character variety

The character variety X is a rigid analytic curve defined by Schneider and Teitelbaum, in their work on p-adic Fourier theory. Here is a simple question about it: what is the ring of bounded functions on X? This question seems to be rather difficult. I will discuss it, as well as some related results.

Ahina Nandy: An interpolation between the algebraic cobordism spectrum MGL and the special linear algebraic cobordism spectrum MSL

Just like the universal complex oriented cohomology theory, complex cobordism (MU), we have a similar notion of universal oriented cohomology theory in the realm of stable A¹-homotopy theory. Algebraic cobordism (MGL) would be our analog of MU. Similar to SU-cobordism MSU, there is a notion of “universal” special linear oriented theory in the motivic world. MSL or special linear algebraic cobordism plays the role of MSU here. I would like to talk a bit more about MSL. Conner-Floyd determined the torsion in SU-bordism already back in the late 60’s. The main ingredient of their work was an interpolation between MSU and MU. It results in a filtration of MU in terms of MSU and facilitates further computation. I will talk about an exactly similar interpolation between MSL and MGL we have shown. Now, I am trying to use this filtration and the resulting spectral sequence to get some information about stable homotopy groups of MSL. I would also like to talk a bit about that.

Guido Bosco: Rational p-adic Hodge theory for rigid-analytic varieties

The goal of this talk will be to discuss the rational p-adic Hodge theory of general rigid-analytic varieties (without properness or smoothness assumptions). The study of this subject for varieties that are not necessarily proper (e.g. Stein) is motivated in part by the desire of finding a geometric incarnation of the p-adic Langlands correspondence in the cohomology of local Shimura varieties. In this context, one difficulty is that the relevant cohomology groups (such as the p-adic (pro-)étale, and de Rham ones) are usually infinite-dimensional, and, to study them, it becomes important to exploit the topological structure that they carry. But, in doing so, one quickly runs into several topological issues: for example, the category of topological abelian groups is not abelian, and the cohomology groups of a complex of topological vector spaces can be pathological in the case the differentials do not have closed image. I will explain how to overcome these issues, using the condensed and solid formalisms developed by Clausen and Scholze, and I will report on a general comparison theorem describing the rational p-adic (pro-)étale cohomology in terms of de Rham data. If time permits, I will also discuss ongoing work toward a generalization to coefficients of the latter comparison, via a stacky approach.