Equivariant $K$-theory, affine Grassmannian and perfection

TL;DR

This paper investigates the torus-equivariant algebraic $K$-theory of affine Schubert varieties in perfect affine Grassmannians over $F_p$, establishing isomorphisms with Hochschild homology and providing new computational methods.

Contribution

It introduces a constructive approach to compare equivariant $K$-theory with Hochschild homology, yielding new computations and structural insights for perfect algebraic $K$-theory.

Findings

Established an $F_p$-linear isomorphism between $K$-theory and Hochschild homology.

Developed new computational techniques for equivariant $K$-theory rings.

Proved structural results of independent interest for equivariant perfect algebraic $K$-theory.

Abstract

We study torus-equivariant algebraic $K$-theory of affine Schubert varieties in the perfect affine Grassmannians over $\mathbb{F}_p$. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that $\mathbb{F}_p$-linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these $K$-theory rings. We establish various structural results for equivariant perfect algebraic $K$-theory on the way; we believe these are of independent interest.