Geodesic
On the Degree of Hilbert Polynomials of Derived Functors
Matematičeskie zametki, Tome 106 (2019) no. 3, pp. 450-456.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a $d$-dimensional Cohen–Macaulay local ring $(R,\mathfrak m)$, let $I$ be an $\mathfrak{m}$-primary ideal, and let $J$ be a minimal reduction ideal of $I$. If $M$ is a maximal Cohen–Macaulay $R$-module, then, for $n$ large enough and $1\le i\le d$, the lengths of the modules $\operatorname{Ext}^i_R(R/J,M/I^nM)$ and $\operatorname{Tor}_i^R(R/J,M/I^nM)$ are polynomials of degree $d-1$. It is also shown that $$ \operatorname{deg}\beta_i^R(M/I^nM) =\operatorname{deg}\mu^i_R(M/I^nM)=d-1, $$ where $\beta_i^R(\,\cdot\,)$ and $\mu^i_R(\,\cdot\,)$ are the $i$th Betti number and the $i$th Bass number, respectively.
Mots-clés : Hilbert–Samuel polynomial
Keywords: derived functors. 
@article{MZM_2019_106_3_a10,
     author = {H. Saremi and A. Mafi},
     title = {On the {Degree} of {Hilbert} {Polynomials} of {Derived} {Functors}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {450--456},
     publisher = {mathdoc},
     volume = {106},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a10/}
}
TY  - JOUR
AU  - H. Saremi
AU  - A. Mafi
TI  - On the Degree of Hilbert Polynomials of Derived Functors
JO  - Matematičeskie zametki
PY  - 2019
SP  - 450
EP  - 456
VL  - 106
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a10/
LA  - ru
ID  - MZM_2019_106_3_a10
ER  - 
%0 Journal Article
%A H. Saremi
%A A. Mafi
%T On the Degree of Hilbert Polynomials of Derived Functors
%J Matematičeskie zametki
%D 2019
%P 450-456
%V 106
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a10/
%G ru
%F MZM_2019_106_3_a10
H. Saremi; A. Mafi. On the Degree of Hilbert Polynomials of Derived Functors. Matematičeskie zametki, Tome 106 (2019) no. 3, pp. 450-456. http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a10/

[1] V. Kodiyalam, “Homological invariants of powers of an ideal”, Proc. Amer. Math. Soc., 118:3 (1993), 757–764 | DOI | MR

[2] E. Theodorescu, “Derived functions and Hilbert polynomials”, Math. Proc. Cambridge Philos. Soc., 132:1 (2002), 75–88 | DOI | MR

[3] E. Theodorescu, “Bivariate Hilbert functions for the torsion functor”, J. Algebra, 265:1 (2003), 136–147 | DOI | MR

[4] S. Iyengar, T. J. Puthenpurakal, “Hilbert–Samuel functions of modules over Cohen–Macaulay rings”, Proc. Amer. Math. Soc., 135:3 (2007), 637–648 | DOI | MR

[5] D. Katz, E. Theodorescu, “On the degree of Hilbert polynomials associated to the torsion functor”, Proc. Amer. Math. Soc., 135:10 (2007), 3073–3082 | DOI | MR

[6] G. Failla, M. La Barbiera, P. L. Staglianó, “Betti numbers of powers of ideals”, Matematiche (Catania), 63:2 (2008), 191–195 | MR

[7] T. Marley, Hilbert Functions in Cohen–Macaulay Rings, Ph.D. Thesis, Purdue Univ., 1989

[8] W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge Univ. Press, Cambridge, 1998 | MR

[9] E. Matlis, “The Koszul complex and duality”, Comm. Algebra, 1 (1974), 87–144 | DOI | MR

[10] E. Enochs, “A modification of Grothendieck's spectral sequence”, Nagoya Math. J., 112 (1988), 53–61 | DOI | MR

[11] T. Cortadellas, S. Zarzuela, “On the depth of the fiber cone of filtrations”, J. Algebra, 198:2 (1997), 428–445 | DOI | MR

[12] T. Cortadellas, “Fiber cones with almost maximal depth”, Comm. Algebra, 33:3 (2005), 953–963 | DOI | MR

[13] S. Huckaba, T. Marley, “Hilbert coefficients and the depths of associated grade rings”, J. London Math. Soc. (2), 56:1 (1997), 64–76 | DOI | MR

[14] T. J. Puthenpurakal, “Ratliff-Rush filtration, regularity and depth of higher associated graded modules. I”, J. Pure Appl. Algebra, 208:1 (2007), 159–176 | DOI | MR