Horrocks correspondence on arithmetically Cohen–Macaulay varieties

Abstract

We describe a vector bundle $\mathcal{ℰ}$ on a smooth $n$-dimensional arithmetically Cohen–Macaulay variety in terms of its cohomological invariants $H_{\ast }^i\left(\mathcal{ℰ}\right)$, $1\le i\le n-1$, and certain graded modules of “socle elements” built from $\mathcal{ℰ}$. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems, where we construct vector bundles from these invariants, and uniqueness theorems, where we show that these data determine a bundle up to isomorphism. The cases of the quadric hypersurface in ${\mathbb{P}}^{n+1}$ and the Veronese surface in ${\mathbb{P}}^5$ are considered in more detail.