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Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Takehiko Yasuda

Abstract

The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen-Macaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular.

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

Abstract

The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen-Macaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular.

Paper Structure

This paper contains 20 sections, 29 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Pure subrings of regular local rings and NCCRs (Section \ref{['sec-NCRs-pure']})
  3. A result on wild quotient singularities (Section \ref{['sec-byproduct']})
  4. NCCRs via Frobeniuses (Section \ref{['sec-NCCRs-Frob']})
  5. Noncommutative Frobeniuses and global dimensions (Sections \ref{['sec-NC-Koh-Lee']} and \ref{['sec-NC-Frob-gldim']})
  6. Convention
  7. Acknowledgments
  8. NCRs of pure subrings
  9. Preliminaries
  10. NCRs of pure subrings
  11. A result on wild quotient singularities
  12. NCCRs via Frobenius
  13. Noncommutative Herzog-Koh-Lee and exact order-raising endofunctors
  14. Noncommutative Herzog-Koh-Lee
  15. Exact order-raising endofunctors and the finiteness of global dimension
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $S$ be a commutative Noetherian complete local CM ring and $M$ a finitely generated MCM $S$-module which includes every indecomposable MCM $S$-module as a direct summand. Then $\mathop{\mathrm{End}}\nolimits_{S}(M)$ has finite global dimension.

Theorems & Definitions (71)

  • Example 1.1
  • Theorem 1.3: Leuschke:2007va
  • Theorem 1.4: Theorem \ref{['thm-pure-finite-gldim-main']}
  • Corollary 1.5: Corollary \ref{['cor-pure-subring-regular-CMEnd->NCCR']}
  • Corollary 1.7: Corollary \ref{['cor-byproduct']}
  • Remark 1.8
  • Theorem 1.10: Corollary \ref{['cor-generator-by-Frobenius']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 61 more