Journals / Turkish Journal of Mathematics / 2014 / Cilt: 38 - Sayı: 1
On Betti series of the universal modules of second order derivations of k[x1,x2,...,xs] (f)
- Pages
- 25–28
- DOI
- —
Abstract
Let R be a coordinate ring of an affine irreducible curve represented by \frac{k[x1,x2,...,xs]}{(f)} and m be a maximal ideal of R. In this article, the Betti series of W2(Rm) is studied. We proved that the Betti series of W2(Rm), where W2(Rm) denotes the universal module of second order derivations of Rm, is a rational function under some conditions.Derivations and their universal modules have been studied by many mathematicians. Erdo˘gan [2] has studied when the Betti series of a universal module of second order derivations is a rational function. In this work, theanalogue of this question for the Betti series of Ω2(Rm), where R = k[x1,x2,...,xs] (f) and m is a maximal ideal of R, has been studied. At the end, we give an example to illustrate our result. All rings we will study in this work will be commutative with identity. Now, we recall some important properties.
Özet
Let R be a coordinate ring of an affine irreducible curve represented by \frac{k[x1,x2,...,xs]}{(f)} and m be a maximal ideal of R. In this article, the Betti series of W2(Rm) is studied. We proved that the Betti series of W2(Rm), where W2(Rm) denotes the universal module of second order derivations of Rm, is a rational function under some conditions.Derivations and their universal modules have been studied by many mathematicians. Erdo˘gan [2] has studied when the Betti series of a universal module of second order derivations is a rational function. In this work, theanalogue of this question for the Betti series of Ω2(Rm), where R = k[x1,x2,...,xs] (f) and m is a maximal ideal of R, has been studied. At the end, we give an example to illustrate our result. All rings we will study in this work will be commutative with identity. Now, we recall some important properties.