A bound for the rate of connections

Document Type : Research Paper

Authors

Mathematics Research Institute,IPM Institute For Research In Fundamental Sciences, Tehran

Abstract

Let  $K$  be a field, $R$ a standard  $K$-algebra and ‎$M$ a finitely generated  $R$-module. The growth rate of ‎$M$ is denoted by ‎$rate_R(M)$, which is a measure of the growth rate of connections in the minimal graded free resolution of the $R$-module  $M$. In this article, the behavior of this invariant under change of rings is investigated. Additionally, the growth rate for Artinian algebras is precisely determined. Moreover, an upper bound for the growth rate of the tensor product of two modules is determined based on the growth rates of each module.

Keywords

Main Subjects


[1] A. Aramova, S. Bărcănescu and J. Herzog, On the rate of relative Veronese submodules, Rev. Roumaine Math. Pures Appl., 40 (1995) 243–251.
[2] L. L. Avramov, Infinite free resolutions, in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math., 166, Birkhäuser, Basel, (1998) 1–118.
[3] L. L. Avramov and D. Eisenbud, Regularity of modules over a Koszul algebra, J. Algebra, 153 (1992) 85–90.
[4] L. L. Avramov and I. Peeva, Finite regularity and Koszul algebras, Amer. J. Math., 123 (2001) 275–281.
[5] J. Backelin, On the rates of growth of the homologies of Veronese subrings, in Algebra, algebraic topology and their interac-tions, Lecture Notes in Math., 1183, Springer, New York, (1986) 79–100.
[6] J. Backelin and R. Fröberg, Veronese subrings, Koszul algebras and rings with linear resolutions, Rev. Roumaine Math. Pures Appl., 30 (1985) 85– 97.
[7] W. Bruns, A. Conca and T. Römer, Koszul homology and syzygies of veronese subalgebras, Math. Ann., 351 (2011) 761–779.
[8] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
[9] A. Conca, M. E. Rossi, and G.Valla, Gröbner flags and Gorenstein algebras, Compositio Math. 129(1) (2001) 95–121.
[10] F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc., 26 (1927) 531–555.
[11] J. Sally, Stretched Gorenstein rings, J. London Math. Soc. 20 (1979) 19–26.