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)

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