ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 13, 11 Cipu, Mihai, Institute of Mathematics, Bucharest, Romania Multiplicities and pure resolutions for Gorenstein codimension four algebras A homogeneous algebra S over a field K is isomorphic to R/I, where R := K[x_{1}, x_{2}, …, x_{n}] is a commutative polynomial ring and I is a homogeneous ideal contained in (x_{1}, x_{2}, …, x_{n})R. Such an algebra is said to have pure resolution if there exists a graded exact sequence of the form

0 ⟶ R(−d_{c})^{β_{c}} ⟶ … ⟶ R(−d_{1})^{β_{1}} ⟶ R ⟶ S ⟶ 0 ,

where the shifts d_{i} and Betti numbers β_{i} are positive integers. We prove that the shifts in the pure resolution of a Gorenstein codimension four homogeneous algebra are uniquely determined by the multiplicity e(S). This extends a result of Huneke and Miller who used the additional hypothesis e(S) < 10^{12}. Reference: C. Huneke and M. Miller, A note on the multiplicity of Cohen–Macaulay algebras with pure resolutions, Canad. J. Math. , 37(1985), 1149–1162