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Tensor products of $d$-fold matrix factorizations

Richie Sheng, Tim Tribone

TL;DR

This work develops a comprehensive framework for the tensor product of $d$-fold matrix factorizations, generalizing Knörrer’s original construction and Yoshino’s extension to arbitrary $d$. It analyzes when these tensor products decompose nontrivially, establishing symmetry, associativity, and shift-compatible properties, and extends Knörrer-type direct sum decompositions to the $d$-fold setting. A central focus is the decomposability of tensor products, with results bounding the number of indecomposable summands and providing conditions for indecomposability via reduction and radical techniques. The paper culminates in concrete applications to hypersurface rings: constructing indecomposable maximal Cohen-Macaulay and Ulrich modules through carefully designed $d$-fold tensor products, and giving explicit rank and generator counts that illuminate Ulrich complexity in this setting.

Abstract

Consider a pair of elements $f$ and $g$ in a commutative ring $Q$. Given a matrix factorization of $f$ and another of $g$, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum $f+g$. We will study the tensor product of $d$-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen-Macaulay and Ulrich modules over hypersurface domains of a certain form.

Tensor products of $d$-fold matrix factorizations

TL;DR

This work develops a comprehensive framework for the tensor product of -fold matrix factorizations, generalizing Knörrer’s original construction and Yoshino’s extension to arbitrary . It analyzes when these tensor products decompose nontrivially, establishing symmetry, associativity, and shift-compatible properties, and extends Knörrer-type direct sum decompositions to the -fold setting. A central focus is the decomposability of tensor products, with results bounding the number of indecomposable summands and providing conditions for indecomposability via reduction and radical techniques. The paper culminates in concrete applications to hypersurface rings: constructing indecomposable maximal Cohen-Macaulay and Ulrich modules through carefully designed -fold tensor products, and giving explicit rank and generator counts that illuminate Ulrich complexity in this setting.

Abstract

Consider a pair of elements and in a commutative ring . Given a matrix factorization of and another of , the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum . We will study the tensor product of -fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen-Macaulay and Ulrich modules over hypersurface domains of a certain form.
Paper Structure (12 sections, 29 theorems, 81 equations)

This paper contains 12 sections, 29 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Notation and conventions
  4. Idempotents in the category of matrix factorizations
  5. The tensor product
  6. Properties of the tensor product
  7. Knörrer's direct sum decomposition
  8. Direct sum decompositions of tensor products
  9. Reduced matrix factorizations
  10. The radical of a Krull-Schmidt category
  11. Strongly indecomposable matrix factorizations
  12. Constructing indecomposable Ulrich modules

Key Result

Lemma 2.2

Let $(X,\varphi),(X',\varphi')\in \operatorname{MF}_{Q}^{d}(f)$ and suppose $\alpha=(\alpha_0,\dots,\alpha_{d-1}) \in \textup{Hom}(X,X')$. For any $k \in \mathbb{Z}_d$, replacing $\alpha_k$ with $A\alpha_k B$, for isomorphisms $A$ and $B$, results in a morphism equivalent to $\alpha$.

Theorems & Definitions (67)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • ...and 57 more