Hermitian K-theory of stable $\infty$-categories
Speaker:
Fabian Hebestreit
Fabian Hebestreit
Abstract:
Associated to a commutative ring $R$ is a hermitian K-, or Grothendieck-Witt-spectrum $GW^s(R)$, that relates to unimodular symmetric forms over $R$ in the same way that its plain K-theory K(R) relates to projective modules (and there are versions for quadratic, skew-symmetric, symplectic... forms as well).
Associated to a commutative ring $R$ is a hermitian K-, or Grothendieck-Witt-spectrum $GW^s(R)$, that relates to unimodular symmetric forms over $R$ in the same way that its plain K-theory K(R) relates to projective modules (and there are versions for quadratic, skew-symmetric, symplectic... forms as well).
By work of Karoubi and Schlichting, these are fairly well-understood relative to K-theory if $2 \in R$ is a unit in $R$ by considering the natural map $hyp \colon K(R) \rightarrow GW^s(R)$ arising from the association of the evaluation form on $P \oplus P^*$ to a projective module. This map factors through the orbits of the $C_2$-action on the source by duality and one of the main results is that the cofibre of this factorisation is $4$-periodic and closely related to the Witt groups of $R$.
I will explain an generalisation of this result to rings in which $2$ is not assumed invertible, replacing Witt-groups by certain (non 4-periodic) L-groups arising in surgery theory. In fact, I will try to indicate that one obtains a much more general statement along with a simple proof by extending the domain of the Grothendieck-Witt- and L-theory functors to stable $\infty$-categories equipped with a so-called Poincar\‘e structure and establishing a universal property for these extensions.
All of this is joint with Calmès, Dotto, Harpaz, Land, Moi, Nardin, Nikolaus, and Steimle.