Michael Groechenig

The index map in algebraic K-theory

In this paper we provide a detailed description of the K-theory torsor constructed by S. Saito for a Tate R-module, and its analogue for general idempotent complete exact categories. We study the classifying map of this torsor in detail, construct an explicit simplicial model, and link it to the index theory of Fredholm operators. The torsor is also related to canonical central extensions of loop groups. More precisely, we compare the K-theory torsor to previously studied dimension and determinant torsors.

Cohomologically rigid local systems and integrality

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety X is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld’s theorem on the existence of

Tate Objects in Exact Categories (with appendix by Jan \vS\vtov\'\i\vcek and Jan Trlifaj)

\ell

Moduli stacks of maps for supermanifolds

ℓ-adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity.

A Generalized Contou-Carr\`ere Symbol and its Reciprocity Laws in Higher Dimensions

The adèles of a scheme have local components—these are topological higher local fields. The topology plays a large role since Yekutieli showed in 1992 that there can be an abundance of inequivalent topologies on a higher local field and no canonical way to pick one. Using the datum of a topology, one can isolate a special class of continuous endomorphisms. Quite differently, one can bypass topology entirely and single out special endomorphisms (global Beilinson–Tate operators) from the geometry of the scheme. Yekutieli’s “Conjecture 0.12” proposes that these two notions agree. We prove this.