Equivariant $K$-theory, affine Grassmannian and perfection

TL;DR

This work develops a detailed framework to compute torus-equivariant algebraic $K$-theory of affine Schubert varieties inside the perfect affine Grassmannian over $\,\mathbb{F}_p$ and relates it to equivariant Hochschild invariants via fixed-point schemes. Central to the approach is the trace map from equivariant $K$-theory to global functions on fixed-point data, shown to be an isomorphism in degree zero after $\,\mathbb{F}_p$-linearization in the GL$_n$ setting and extended to general GL$_n$ via Demazure resolutions and semi-orthogonal decompositions; the theory also asserts that $K^T(X_{\,\le\,\mu})\simeq KH^T(X_{\,\le\,\mu})$ in this perfect context. The paper establishes structural results such as perfect proper excision for $K^T$ and $KH^T$, explains the relation to derived loop spaces, and provides explicit presentations for $K^T_0$ in several small, singular cases (including GL$_2$, GL$_3$). The results yield computable, presentation-level descriptions of equivariant $K$-theory rings in a positive characteristic setting and connect to broader themes in the geometric Langlands program via fixed-point/Hochschild perspectives. These techniques open paths to further computations for higher rank groups and toric varieties, and illuminate how perfection simplifies descent and trace phenomena in equivariant $K$-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.

Theorems & Definitions (126)

• Theorem 1: Theorem \ref{['theorem: trace for GL_n affine grassmannian']}
• Theorem 2: Theorem \ref{['lemma: homotopy fiber square for K_T']}
• Theorem 3: Theorem \ref{['theorem: trace map for perfectly proper toric varieties']}
• Example 1.1
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Remark 1.5
• Lemma 1.6