Math @ Duke

Publications [#243904] of Ezra Miller
Papers Published
 with Miller, E; Novik, I; Swartz, E, Face rings of simplicial complexes with singularities,
Mathematische Annalen, vol. 351 no. 4
(2011),
pp. 857875, ISSN 00255831 [math.AC/1001.2812], [DOI:10.1007/s0020801006205], [doi]
(last updated on 2018/07/19)
Abstract: The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of CohenMacaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an rdimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension rc; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules. © 2010 SpringerVerlag.


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