Seminar Algebra and Topology

The seminar usually takes place Wednesdays in I.13.71 (!), 16:30 - 17:30.
Involved professors: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

 

Henrik Russell: The geometric fundamental group of the affine line over a finite field

The affine line A¹ over a finite field F is taken as a benchmark for the problem of describing geometric étale fundamental groups. Using a reformulation of Tannaka duality we construct a (non-commutative) universal affine group Lu_{A¹_F}, such that any finite and étale Galois covering of A¹ over F is a pull-back of a Galois covering of Lu_{A¹_F}. Then the geometric fundamental group of A¹_F is a completion of the k-points of Lu_{A¹_F}, where k is an algebraic closure of F, and we obtain an explicit description of Lu_{A¹_F}.

Fabian Hebestreit: Basic higher almost ring theory

(with P.Scholze) For a flat and idempotent ideal I in a commutative ring R the localisation of Mod(R) at the maps whose kernel and cokernel is annihilated by I is Falting's category of almost R-modules. Examples of the set-up arise from perfectoid fields, and almost modules prominently feature in the tilting equivalences relating their étale information in characteristic 0 and p. The possibly simplest example has R obtained from K[X] by adjoining all roots of X which together span I. Now, the category of almost R-modules derives many of its good properties from the fact that R/I is derived tensor-idempotent R-algebra on account of the assumptions on I. The flatness assumption is, however, often not satisfied in higher dimensional situations, eg if R and I are similarly obtained from K[X,Y], and effort has been expended into weakening or removing it. In the talk I will explain how this can be achieved rather easily by directly passing to derived categories and replacing the ordinary ring R/I by a certain new animated commutative ring.

Jack Davies: Geometric norms on equivariant elliptic cohomology

A notion of multiplicative structure on equivariant cohomology theories will be called a ``geometric norm structure''. We will then introduce equivariant elliptic cohomology, as defined by Lurie and Gepner-Meier, and discuss how it naturally comes with a geometric norm structure. We will also touch on a few applications of these structures: constructions of C_p-normed algebra structures and integral models for Behrens' Q(N) spectra. This is joint work-in-progress with William Balderrama and Sil Linskens.

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