Georg Tamme

Algebraic K -theory and descent for blow-ups

We prove that algebraic K-theory satisfies ‘pro-descent’ for abstract blow-up squares of noetherian schemes. As an application we derive Weibel’s conjecture on the vanishing of negative K-groups.

Regulators and cycle maps in higher-dimensional differential algebraic K-theory

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lotts relation between short exact sequences of geometric bundles with a higher analytic torsion form.

On an analytic version of Lazard’s isomorphism

We prove a comparison theorem between locally analytic group cohomology and Lie algebra cohomology for locally analytic representations of a Lie group over a nonarchimedean field of characteristic 0. The proof is similar to that of van-Ests isomorphism and uses only a minimum of functional analysis.

The Beilinson regulator is a map of ring spectra

Abstract We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E ∞ -ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R . The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E ∞ -refinement and prove a uniqueness result for these refinements.

Multiplicative differential algebraic K-theory and applications

We construct a version of Beilinsons regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Blochs construction of K_3-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.

Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

We construct a variant of Karoubi’s relative Chern character for smooth separated schemes over the ring of integers in a p-adic field, and prove a comparison with the rigid syntomic regulator. For smooth projective schemes, we further relate the relative Chern character to the etale p-adic regulator via the Bloch–Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers, and generalizes it to all smooth projective schemes as above.