Hochster's theta invariant and the Hodge–Riemann bilinear relations

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor i R ( M , N ) ≅ Tor i + 2 R ( M , N ) for i ≫ 0 ). Since R has only an isolated singularity, the...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2011-01, Vol.226 (2), p.1692-1714
Main Authors: Moore, W. Frank, Piepmeyer, Greg, Spiroff, Sandra, Walker, Mark E.
Format: Article
Language:English
Subjects:
Algebraic K-theory
Chern characters
Graded rings
Resolutions
Riemann–Roch theorems
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Summary:Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor i R ( M , N ) ≅ Tor i + 2 R ( M , N ) for i ≫ 0 ). Since R has only an isolated singularity, these torsion modules are of finite length for i ≫ 0 . The theta invariant of the pair ( M , N ) is defined by Hochster to be length ( Tor 2 i R ( M , N ) ) − length ( Tor 2 i + 1 R ( M , N ) ) for i ≫ 0 . H. Dao has conjectured that the theta invariant is zero for all pairs ( M , N ) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj ( R ) .
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2010.09.005