Derived equivalences via Tate resolutions

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 $V(I)$.