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Abstract Consider a pair of elementsfandgin a commutative ringQ. Given a matrix factorization offand another ofg, 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 ofd-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.
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Non-existence of Ulrich modules over Cohen-Macaulay local rings
Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal.
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- PAR ID:
- 10600005
- Publisher / Repository:
- AMS
- Date Published:
- Journal Name:
- Communications of the American Mathematical Society
- Volume:
- 5
- Issue:
- 5
- ISSN:
- 2692-3688
- Page Range / eLocation ID:
- 195 to 208
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/cams/44
