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Building Modules From the Singular Locus

Jesse Burke, Lars Winther Christensen, Ryo Takahashi


A finitely generated module over a commutative noetherian ring of finite Krull dimension can be built from the prime ideals in the singular locus by iteration of three procedures: taking extensions, direct summands, and cosyzygies. In 2003 Schoutens gave a bound on the number of iterations required to build any module, and in this note we determine the exact number. This building process yields a stratification of the module category, which we study in detail for local rings that have an isolated singularity.

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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