Associated graded algebra of a Gorenstein Artin algebra.

*(English)*Zbl 0793.13010
Mem. Am. Math. Soc. 514, 115 p. (1994).

Let \(A=\bigoplus_{i=0}^ sA_ i\), \(A_ s \neq 0\), denote a graded Artin \(k\)-algebra, \(k\) a field. Then \(A\) is a Gorenstein ring if and only if there is an exact pairing \(A \times A \to k\). In particular, this implies the symmetry \(h_ i=h_{s-i}\) of the Hilbert function \(H(A)=(h_ 0,\dots,h_ s)\), where \(h_ i=\dim _ kA_ i\), \(i=0,\dots,s\). The main theme of this memoir is the question: To what extends this behaviour in the situation of a non-graded Gorenstein \(k\)- algebra \(A\), in particular, what is the behaviour of the Hilbert function?

In order to pursue this question it is natural to consider the associated graded ring \(A^*=Gr_ m(A)\), where \(m\) denotes a maximal ideal of \(A\). In general \(A^*\) is no longer a Gorenstein \(k\)-algebra. The author’s fundamental investigation is the construction of a finite descending sequence of ideals of \(A^*\), \(A^*=C(0) \supseteq C(1) \supseteq \dots\), whose subsequent quotients \(C(a)/C(a+1)\) are reflexive \(A^*\)- modules, called the reflexive factors of \(A^*\). This is done by intersecting the \(m\)-adic filtration with the Loewy filtration \(\{0:_ A m^ n\}_{n \geq 0}\) on \(A\). This ingenious idea yields a decomposition of the Hilbert function into a finite sum of symmetric summands. In particular, the first subquotient \(C(0)/C(1)\) is always a graded Gorenstein algebra and \(A^*\) itself is Gorenstein if and only if \(H(A)\) is symmetric in which case \(A=A^*=C(0)/C(1)\).

The construction, extending F. S. Macaulay’s work [Trans. Am. Math. Soc. 5, 385-410 (1904)] for determinating the possible Hilbert functions of the intersections of two plane curves, has several applications, among them:

(1) Determination of the multiplicity and of the orders of generators of Gorenstein ideals;

(2) Lefschetz conditions in order to characterize the difference function of the Hilbert function as introduced by R. P. Stanley [SIAM J. Algebraic Discrete Methods 1, 168-184 (1980; Zbl 0502.05004)] and J. Watanabe [Adv. Math. 76, No. 2, 194-199 (1989; Zbl 0703.13019)];

(3) study of compressed Gorenstein Artin algebras in the sense of the author [Trans. Am. Math. Soc. 285, 337-378 (1984; Zbl 0548.13009)] and R. Fröberg and D. Laksov [in Complete intersections, Lect. 1st. Sess. CIME, Acireale 1983, Lect. Notes Math. 1092, 121-151 (1984; Zbl 0558.13007)];

(4) study of finite dimensional map germs.

Each chapter is motivated by an introduction exploring the main ideas. It makes also difficult parts of the text well readable. An extensive bibliography completes the picture about recent research on this subject.

Upon request the author has available scripts (written by D. Eisenbud, the author and J. Yameogo) for the computer algebras system MACAULAY. Among others these scripts allow the decomposition of \(A^*\) into its reflexive factors.

In order to pursue this question it is natural to consider the associated graded ring \(A^*=Gr_ m(A)\), where \(m\) denotes a maximal ideal of \(A\). In general \(A^*\) is no longer a Gorenstein \(k\)-algebra. The author’s fundamental investigation is the construction of a finite descending sequence of ideals of \(A^*\), \(A^*=C(0) \supseteq C(1) \supseteq \dots\), whose subsequent quotients \(C(a)/C(a+1)\) are reflexive \(A^*\)- modules, called the reflexive factors of \(A^*\). This is done by intersecting the \(m\)-adic filtration with the Loewy filtration \(\{0:_ A m^ n\}_{n \geq 0}\) on \(A\). This ingenious idea yields a decomposition of the Hilbert function into a finite sum of symmetric summands. In particular, the first subquotient \(C(0)/C(1)\) is always a graded Gorenstein algebra and \(A^*\) itself is Gorenstein if and only if \(H(A)\) is symmetric in which case \(A=A^*=C(0)/C(1)\).

The construction, extending F. S. Macaulay’s work [Trans. Am. Math. Soc. 5, 385-410 (1904)] for determinating the possible Hilbert functions of the intersections of two plane curves, has several applications, among them:

(1) Determination of the multiplicity and of the orders of generators of Gorenstein ideals;

(2) Lefschetz conditions in order to characterize the difference function of the Hilbert function as introduced by R. P. Stanley [SIAM J. Algebraic Discrete Methods 1, 168-184 (1980; Zbl 0502.05004)] and J. Watanabe [Adv. Math. 76, No. 2, 194-199 (1989; Zbl 0703.13019)];

(3) study of compressed Gorenstein Artin algebras in the sense of the author [Trans. Am. Math. Soc. 285, 337-378 (1984; Zbl 0548.13009)] and R. Fröberg and D. Laksov [in Complete intersections, Lect. 1st. Sess. CIME, Acireale 1983, Lect. Notes Math. 1092, 121-151 (1984; Zbl 0558.13007)];

(4) study of finite dimensional map germs.

Each chapter is motivated by an introduction exploring the main ideas. It makes also difficult parts of the text well readable. An extensive bibliography completes the picture about recent research on this subject.

Upon request the author has available scripts (written by D. Eisenbud, the author and J. Yameogo) for the computer algebras system MACAULAY. Among others these scripts allow the decomposition of \(A^*\) into its reflexive factors.

Reviewer: P.Schenzel (Berlin)

##### MSC:

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |

13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |

13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |