Seminar Algebra und Topologie
Das Seminar findet normalerweise am Mittwoch in F.13.11, 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.
Marco D'Addezio: Injectivity failure in crystalline comparisons
I will report on a recent work with Daniel Caro, where we study the de Rham-to-crystalline comparison map for affine smooth schemes over the Witt vectors. We answer a question of Esnault-Kisin-Petrov by showing that this comparison map can fail to be injective, even in the presence of good compactifications. We identify an obstruction by looking at the slopes of the Frobenius action on crystalline cohomology. On the positive side, we prove that injectivity holds in cohomological degree 1 and for certain subspaces defined by slope conditions. In this talk, I will explain how the F-gauge structure of crystalline cohomology yields the relevant slope bounds. I will further explain how these techniques can be used to determine algebraic de Rham cohomology modulo torsion in terms of the slopes of rigid cohomology of the special fibre.
Krishna Kumar Madhavan Vijayalakshmi: Relative A^1-Contractibility of Smooth Schemes and Exotic Motivic Spaces
One of the emerging problems in motivic homotopy theory is to uniquely characterize the affine n-space A^n among smooth A^1-contractible (affine) schemes. In this talk, we shall learn the background of this problem and show that A^1-contractibility is strong enough to capture the aforementioned uniqueness in relative dimensions up to 2 over a base scheme. This is a joint work in progress with Paul Arne Østvær and Adrien Dubouloz. Following this, we illustrate the theory in higher dimensions by the family of Koras--Russell prototypes, smooth affine varieties that are A^1-contractible over a field of characteristic zero. We shall further extend their A^1-contractibility over a Noetherian scheme. A foundational consequence of this is the existence of "exotic spheres" in motivic homotopy theory in all dimensions >2. Preprint available at https://krishmv.github.io/research.html (arxiv: 2510:21594)
Christopher Deninger: Rational Witt vectors
We discuss various aspects of rational Witt vectors, in particular their connections with K-theory and algebraic cycles and their sheaf theoretical properties. We also explain how they can be used to construct dynamical systems attached to arithmetic schemes.
Christian Lehn: Tannakian groups of perverse sheaves and E6 geometry
In a joint work with Krämer and Maculan, we prove that the Fano surfaces of lines on smooth cubic threefolds are the only smooth (less than half-dimensional) subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. The proof uses an upgrade of the Krämer-Weissauer formalism from perverse sheaves to Hodge modules, the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi, an intensive computer search, and the geometry of lines on cubic surfaces.
Chirantan Chowdhury: Extending Abstract Six-Functor Formalisms to Ind and Pro Geometric Setups
Abstract six-functor formalisms, as developed by Liu–Zheng and Mann, have played a key role in arithmetic geometry and motivic homotopy theory in recent years. Their extension to algebraic stacks by Khan–Ravi
and Chowdhury, together with the rational formalisms for prestacks constructed by Richarz–Scholbach, illustrates the flexibility of this framework. In this talk, building on Yaylali’s approach to rational motives on pro-algebraic stacks, we construct an extension of the abstract six-functor formalism to a suitable class of Ind- and Pro-geometric setups. Such results yield integral motivic categories for ind-pro algebraic stacks, most notably the Hecke stack, together with a natural convolution product.
Sebastian Bartling: Close local fields and fundamental lemmas
The idea of close local field goes back to work of Krasner, Deligne and Kazhdan and says that as ramification of p-adic fields grows to infinity, their theory approximates the theory of equal characteristic local fields. I want to explain how one can use this philosophy to put many moduli problems appearing in (local) arithmetic geometry (Rapoport-Zink spaces, Wittvector affine Grassmannians, Affine Deligne Lustzig varieties, Affine Springer fibers...) into profinite families. As applications one can deduce the missing function field case of the following statements: Wei Zhang's Arithmetic fundamental lemma, the base change fundamental lemma and remaining small p cases of the standard endoscopic fundamenta lemma as proven by Ngo. This is joint work with Andreas Mihatsch and also with Kazuhiro Ito.
Margherita Pagano: Wild Brauer classes via prismatic cohomology
Let K be a number field and X be a smooth and proper variety over K, I will explain how from the exitance of a global 2-form on X it is possible to deduce information on the set of rational points on X. More precisely, I will show how from the geometry of the reduction modulo p of the variety we can prove the existence of p-torsion Brauer classes with interesting arithmetic properties on the variety X. This is a joint work with Emiliano Ambrosi and Rachel Newton.
Eugen Hellmann: Harish-Chandra Bimodules and stacks of logarithmic derivations
Harish-Chandra bimodules for a semi-simple Lie Algebra g are U(g x g)-modules whose restriction to the diagonal is a direct sum of finite dimensional representations. The monoidal category of such bimodules acts naturally on various categories in representation theory; for example on the category of locally analytic representations to p-adic Lie groups.
As part of the categorical approach to the p-adic local Langlands correspondence one aims to understand a corresponding action of this category on the category of coherent sheaves on the stack of Galois-equivariant vector bundles on the Fargues--Fontaine curve. In this talk I will explain how to (partially conjecturally) realize this picture by embedding Harish-Chandra bimodules into coherent sheave on a Hecke stack for logarithmic derivations. The latter category naturally acts on sheaves on the stack of equivariant vector bundles on the Fargues--Fontaine curve.
Christian Carrick: Higher real K-theories and finite spectra
Producing explicit type n finite spectra, such as Smith-Toda complexes or generalized Moore spectra, is a difficult problem that is closely related to periodicity in the stable homotopy groups of spheres. Finite spectra are in some sense dual to fp spectra in the sense of Mahowald-Rezk, like ko or tmf. Using genuine equivariant homotopy, we produce a new family of fp spectra at the prime 2 and each chromatic height, known as (connective) higher real K-theories. Using an Euler characteristic for fp spectra defined by Ishan Levy, we use these theories to put constraints on the existence of small finite spectra. This results in a lower bound for the number of obstructions to producing a Smith-Toda complex and new constraints on the exponents of the vi that can appear in a generalized Moore spectrum, valid at all heights. This is joint work with Mike Hill.
Logan Hyslop: Constructible and Homological Spectra
In this talk, we will discuss several results around homological spectra for rational rigid 2-rings, as defined by Balmer. Based on forthcoming work with Tobias Barthel and Maxime Ramzi, we will discuss how to realize the underlying set of the homological spectrum as the constructible spectrum in 2-Ring^{rig} when we are rational, together with the interpretation of the image of the comparison map from the constructible spectrum to the Balmer spectrum in general. From there, we will discuss Balmer’s so-called “nerves of steel conjecture,” and our counter-example to said conjecture.
Time permitting, we will discuss how to realize the homological spectrum as an “E_n-constructible spectrum” in general, and how this can be used to show that any tt-category with an E_n refinement (n>=4) has all points of its homological spectrum being geometric in the category of rigid E\_{n-1}-2-Rings.
Shigeo Koshitani: The finite simple Mathieu group M_12 vs. SL(3,3) in characteristic 3
This is joint work with Tetsuro OKUYAMA. We will be discussing relationship between 3-modular representations of the Mathieu group $M_{12}$ which is one of the 26 sporadic finite simple groups and the special linear group SL(3,3). More precisely, we will be looking at the principal 3-blocks $B_0(kG)$ of the group algebras $kG$ of these two finite groups $G$, where $k$ is an algebraically closed field of characteristic 3, and $G$ is one of the above two finite groups. Our result says that the two principal 3-blocks above are derived equivalent. As a byproduct this answers a question by [W. Murphy, J.Group Theory 26 (2023)]
