Extensions of the Multiplicity conjecture

Migliore, Juan; Nagel, Uwe; Römer, Tim

doi:10.1090/S0002-9947-07-04360-7

Extensions of the Multiplicity conjecture
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by Juan Migliore, Uwe Nagel and Tim Römer PDF

Trans. Amer. Math. Soc. 360 (2008), 2965-2985 Request permission

Abstract:

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-space, and propose an upper bound for finitely generated graded torsion modules. We establish this bound for torsion modules whose codimension is at most two.

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