Derived equivalences via Tate resolutions

TL;DR

This paper establishes a connection between Koszul complexes and Tate resolutions to derive equivalences between certain categories of modules over a commutative noetherian ring, with implications in prime characteristic settings.

Contribution

It introduces a method to factor Koszul complexes through Tate resolutions, leading to new derived equivalences for modules supported on specific ideals.

Findings

Derived equivalence between modules supported on $V(I)$ with finite $ ext{A}$-dimension and $ ext{A}$-modules with support on $V(I)$

Factorization of Koszul complexes via Tate resolutions for large powers of elements

Special case in prime characteristic relating finite projective dimension modules to projective modules

Abstract

For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$ factors through the Tate resolution on $s_1^n, \ldots , s_d^n$. Using this, for any resolving subcategory $\mathcal A$ of mod($R$) and any ideal $I$ such that it has a filtration $\{ I_n \}$ which is equivalent to the $I$-adic filtration and $\textrm{dim}_{\mathcal A}(R/I_n) < \infty$, we show a derived equivalence between the bounded derived category of finitely generated modules supported on $V(I)$ having finite $\mathcal A$-dimension and the bounded derived category of $\mathcal A$ with homologies supported on $V(I)$. As a special case, when $R$ is of prime characteristic and $I$ is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on $V(I)$ and the bounded derived category of projective modules with homologies supported on $V(I)$.