Math @ Duke

Publications [#199124] of Ezra N Miller
Papers Published
 with Isabella Novik and Ed Swartz, Face rings of simplicial complexes with singularities,
Mathematische Annalen, vol. 351
(2011),
pp. 857875 [math.AC/1001.2812], [DOI:10.1007/s0020801006205]
(last updated on 2012/12/14)
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.


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