Coherent $I$-indexed algebras and noncommutative projective schemes

TL;DR

This paper characterizes noncommutative projective schemes arising from coherent $I$-indexed algebras, extending Polishchuk's results from $bZ$-indexed to more general locally finite directed poset indices.

Contribution

It generalizes the characterization of noncommutative projective schemes to coherent $I$-indexed algebras with a broader class of index sets.

Findings

Provides a new characterization of noncommutative projective schemes

Extends Polishchuk's results to more general index sets

Establishes properties of sequences of objects analogous to tensor powers

Abstract

We study coherent $I$-indexed algebras and associated noncommutative projective schemes, where the index set $I$ is a locally finite directed poset. Our main result is a characterisation of such noncommutative projective schemes in terms of sequences of objects with properties analogous to the sequence of tensor powers of an ample line bundle, extending a similar characterisation given by Polishchuk for coherent $\mathbb{Z}$-indexed algebras.