Algebras Associated to Inverse Systems of Projective Schemes

Abstract

Artin, Tate and Van den Bergh initiated the field of noncommutative projective algebraic geometry by fruitfully studying geometric data associated to noncommutative graded algebras. More specifically, given a field $\mathbb K$ and a graded $\mathbb K$-algebra $A$, they defined an inverse system of projective schemes $\Upsilon_A = \{{\Upsilon_d(A)}\}$. This system affords an algebra, $\mathbf B(\Upsilon_A)$, built out of global sections, and a $\mathbb K$-algebra morphism $\tau: A \to \mathbf B(\Upsilon_A)$. We study and extend this construction. We define, for any natural number $n$, a category ${\tt PSys}^n$ of projective systems of schemes and a contravariant functor $\mathbf B$ from ${\tt PSys}^n$ to the category of associative $\mathbb K$-algebras. We realize the schemes ${\Upsilon_d(A)}$ as ${\rm Proj \ } {\mathbf U}_d(A)$, where ${\mathbf U}_d$ is a functor from associative algebras to commutative algebras. We characterize when the morphism $\tau: A \to \mathbf B(\Upsilon_A)$ is injective or surjective in terms of local cohomology modules of the ${\mathbf U}_d(A)$. Motivated by work of Walton, when $\Upsilon_A$ consists of well-behaved schemes, we prove a geometric result that computes the Hilbert series of $\mathbf B(\Upsilon_A)$. We provide many detailed examples that illustrate our results. For example, we prove that for some non-AS-regular algebras constructed as twisted tensor products of polynomial rings, $\tau$ is surjective or an isomorphism.