{
  "id": "2406.17139",
  "version": "v1",
  "published": "2024-06-24T21:15:22.000Z",
  "updated": "2024-06-24T21:15:22.000Z",
  "title": "Algebras Associated to Inverse Systems of Projective Schemes",
  "authors": [
    "Andrew Conner",
    "Peter Goetz"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AG",
    "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-24T21:15:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S38",
      "16W50"
    ],
    "keywords": [
      "inverse system",
      "projective schemes",
      "van den bergh",
      "local cohomology modules",
      "noncommutative projective algebraic geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}