@article{f5ea5a0fe2ea422d84834e08bc0965f3,

title = "Some extensions of theorems of Kn{\"o}rrer and Herzog–Popescu",

abstract = "A construction due to Kn{\"o}rrer shows that if N is a maximal Cohen–Macaulay module over a hypersurface defined by f+y2, then the first syzygy of N/yN decomposes as the direct sum of N and its own first syzygy. This was extended by Herzog–Popescu to hypersurfaces f+yn, replacing N/yN by N/yn−1N. We show, in the same setting as Herzog–Popescu, that the first syzygy of N/ykN is always an extension of N by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring Λ over which every finitely generated module has an eventually 2-periodic projective resolution, prompting us to call it a “non-commutative hypersurface ring”. Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form x1a1+…+xdad.",

keywords = "Huneke issue, Hypersurface rings, Maximal Cohen–Macaulay modules, Stable module category, Syzygies",

author = "Dugas, {Alex S.} and Leuschke, {Graham J.}",

note = "Funding Information: This work was begun during the authors' participation in the research program “IRTATCA: Interactions between Representation Theory, Algebraic Topology and Commutative Algebra”, at the Centre de Recerca Matem{\`a}tica in Barcelona in 2015. The authors thank the CRM for a pleasant and productive experience. GJL was supported by NSF grant DMS-1502107 , and ASD was supported by NSF conference grant DMS-58502086 . Funding Information: This work was begun during the authors{\textquoteright} participation in the research program “IRTATCA: Interactions between Representation Theory, Algebraic Topology and Commutative Algebra” at the Centre de Recerca Matem{\`a}tica in Barcelona in 2015. The authors thank the CRM for a pleasant and productive experience. GJL was supported by NSF grant DMS-1502107, and ASD was supported by NSF conference grant DMS-58502086. Funding Information: This work was begun during the authors' participation in the research program ?IRTATCA: Interactions between Representation Theory, Algebraic Topology and Commutative Algebra?, at the Centre de Recerca Matem?tica in Barcelona in 2015. The authors thank the CRM for a pleasant and productive experience. GJL was supported by NSF grant DMS-1502107, and ASD was supported by NSF conference grant DMS-58502086. The authors are grateful to Tokuji Araya for pointing out a gap in an earlier version of this paper. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2021",

month = apr,

day = "1",

doi = "10.1016/j.jalgebra.2018.11.021",

language = "English (US)",

volume = "571",

pages = "94--120",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

}