# Associated Graded Algebra of a Gorenstein Artin Algebra

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*Anthony A. Iarrobino*

In 1904, Macaulay described the Hilbert function of the intersection of two plane curve branches: It is the sum of a sequence of functions of simple form. This monograph describes the structure of the tangent cone of the intersection underlying this symmetry. Iarrobino generalizes Macaulay's result beyond complete intersections in two variables to Gorenstein Artin algebras in an arbitrary number of variables. He shows that the tangent cone of a Gorenstein singularity contains a sequence of ideals whose successive quotients are reflexive modules. Applications are given to determining the multiplicity and orders of generators of Gorenstein ideals and to problems of deforming singular mapping germs. Also included are a survey of results concerning the Hilbert function of Gorenstein Artin algebras and an extensive bibliography.

#### Table of Contents

# Table of Contents

## Associated Graded Algebra of a Gorenstein Artin Algebra

- TABLE OF CONTENTS vii8 free
- 1. GORENSTEIN ARTIN ALGEBRAS AND DUALITY 110 free
- 2. THE INTERSECTION OF TWO PLANE CURVES 2332
- 3. EXTREMAL DECOMPOSITIONS 3140
- 4. COMPONENTS OF THE HILBERT SCHEME STRATA 4 4655
- 4A. Hilbert functions of m[sup(u) and 0 :m[sup(v)] and semicontinuity 4756
- 4B. Components of Gor[sub(T)]R when T = (1,3,3,2,1,1) 4958
- 4B.i. The variety Gor[sub(T)]R 4958
- 4B.ii. Fibration Gor[sub(T)]R to G[sub(T)]: Parametrization as fibred varieties 5564
- 4C. Finding Q(a) from a generator of the dual module 5867

- 5. WHAT DECOMPOSITIONS D AND SUBQUOTIENTS Q(a) CAN OCCUR? 6271
- 5A. Hilbert function of a graded Gorenstein Artin algebra 6473
- 5B. Numerical conditions on the decomposition D 6978
- 5C. Gorenstein Artin algebras with given decomposition D 7382
- 5D. Applications 8190
- 5E. Problems 8897
- 5F. Appendix: Hilbert function decompositions for lengths n ≤ 21 when e ≤ 3, and for n ≤ 16 9099

- 6. RELATIVELY COMPRESSED ARTIN ALGEBRAS 103112
- BIBLIOGRAPHY 105114
- LIST OF THEOREMS, DEFINITIONS, AND EXAMPLES 109118
- INDEX 112121