RESULTS ON BETTI SERIES OF THE UNIVERSAL MODULES OF SECOND ORDER DERIVATIONS

Year 2011, Volume: 40 Issue: 3, 449 - 452, 01.03.2011

Ali Erdoğan

Abstract

RESULTS ON BETTI SERIES OF THE UNIVERSAL MODULES OF SECOND ORDER DERIVATIONS

Abstract

Let R be the coordinate ring of an affine irreducible curve presented by
k[x,y]
(f)
and m a maximal ideal of R. Assume that Rm, the localization
of R at m, is not a regular ring. Let Ω2(Rm) be the universal module
of second order derivations of Rm. We show that, under certain conditions, B(Ω2(Rm), t), the Betti series of Ω2(Rm), is a rational function.
To conclude, we give examples related to B(Ω2(Rm), t) for various rings
R.

Keywords

Universal module , Universal differential operators , Betti series , Minimal resolution

References

• Erdo˘gan, A. Homological dimension of the universal modules for hypersurfaces, Comm. Algebra 24 (5), 1565–1573, 1996.
• Nakai, Y. High order derivations 1, Osaka J. Math. 7, 1–27, 1970.
• Sweedler, M. E. and Heynemann, R. G. Affine Hopf algebras, J. Algebra 13, 192–241, 1969.