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The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

Daniele Faenzi, Joan Pons-Llopis

Abstract

We show that all reduced closed subschemes of projective space that have a Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except for a few cases which we completely classify.

The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

Abstract

We show that all reduced closed subschemes of projective space that have a Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except for a few cases which we completely classify.

Paper Structure

This paper contains 26 sections, 25 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. CM-wild varieties
  3. ACM varieties and modules
  4. ACM subschemes
  5. Terminology on coherent sheaves
  6. Cohen-Macaulay and Ulrich conditions
  7. CM-wildness
  8. CM-wildness from extensions
  9. The functor from the Kronecker quiver to $\mathbf{Coh}_X$
  10. Representation embeddings of the Kronecker quiver
  11. Stable syzygies of Ulrich modules
  12. CM-wildness from syzygies of Ulrich extensions
  13. Wildness of non-integral projective schemes
  14. Varieties of minimal degree
  15. Extension of sheaves over cones
  16. ...and 11 more sections

Key Result

Theorem 1

Let $X\subset \mathbb P^n$ be a reduced non-degenerate closed ACM subscheme of dimension $m \ge 1$. Then $X$ is of wild CM-type unless $X$ is one of the following:

Theorems & Definitions (55)

  • Theorem
  • Conjecture 1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Theorem A
  • ...and 45 more