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Algebras Associated to Inverse Systems of Projective Schemes (2406.17139v1)

Published 24 Jun 2024 in math.AG and math.RA

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.

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