## Rocky Mountain Journal of Mathematics

### On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

#### Abstract

Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM$, $P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p$ its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4$ and $\dim _k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $m$ for $n\in [2,c]$ and equal to $1$ for $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 413-433.

Dates
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.rmjm/1469537470

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-413

Mathematical Reviews number (MathSciNet)
MR3529076

Zentralblatt MATH identifier
06624467

#### Citation

Casnati, Gianfranco; Jelisiejew, Joachim; Notari, Roberto. On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence. Rocky Mountain J. Math. 46 (2016), no. 2, 413--433. doi:10.1216/RMJ-2016-46-2-413. https://projecteuclid.org/euclid.rmjm/1469537470

#### References

• L. Avramov, A. Kustin and M. Miller, Poincaré series of modules over local rings of small embedding codepth or small linking number, J. Algebra 118 (1988), 162–204.
• L. Avramov and G. Levin, Factoring out the socle of a Gorenstein ring, J. Algebra 55 (1978), 74–83.
• W. Bruns and J. Herzog, Cohen-Macaulay rings, II, Cambridge University Press, Cambridge, 1998.
• G. Casnati, J. Elias, R. Notari and M.E. Rossi, Poincaré series and deformations of Gorenstein local algebras with low socle degree, Comm. Algebra 41 (2013), 1049–1059.
• G. Casnati and R. Notari, The Poincaré series of a local Gorenstein ring of multiplicity up to $10$ is rational, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), 459–468.
• ––––, A structure theorem for $2$-stretched Gorenstein algebras, preprint arXiv: 1312.2191 [math.AC], J. Commutative Algebra, to appear.
• J. Elias and M.E. Rossi, Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system, Trans. Amer. Math. Soc. 364 (2012), 4589–4604.
• J. Elias and G. Valla, A family of local rings with rational Poincaré series, Proc. Amer. Math. Soc. 137 (2009), 1175–1178.
• J. Emsalem, Géométrie des points épais, Bull. Soc. Math. France 106 (1978), 399–416.
• R. Hartshorne, Algebraic geometry, Grad. Texts Math. 52, Springer, New York, 1977.
• T.H. Gulliksen and G. Levin, Homology of local rings, Queen's Papers Pure Appl. Math. 20 (1969), x+192.
• A.V. Iarrobino, Associated graded algebra of a Gorenstein Artin algebra, Mem. Amer. Math. Soc. 107 (1994), viii+115.
• C. Jacobsson, A. Kustin and M. Miller, The Poincaré series of a codimension four Gorenstein ring is rational, J. Pure Appl. Algebra 38 (1985), 255–275.
• J.D. Sally, The Poincaré series of stretched Cohen-Macaulay rings, Canad. J. Math. 32 (1980), 1261–1265.
• R.P. Stanley, Combinatorics and commutative algebra, Birkhäuser, Berlin, 1983.
• J. Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.
• H. Wiebe, Über homologische Invarianten lokaler Ringe, Math. Ann. 179 (1969), 257–274.