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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compressed algebras: Artin algebras having given socle degrees and maximal length

Author: Anthony Iarrobino
Journal: Trans. Amer. Math. Soc. 285 (1984), 337-378
MSC: Primary 13E10; Secondary 13H10, 14B07, 35E99
MathSciNet review: 748843
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Abstract: J. Emsalem and the author showed in [18] that a general polynomial $ f$ of degree $ j$ in the ring $ \mathcal{R} = k[ {{y_1},\ldots,{y_r}} ]$ has $ \left( {\begin{array}{*{20}{c}} {j + r - 1} \\ {r - 1} \\ \end{array} } \right)$ linearly independent partial derivates of order $ i$, for $ i = 0,1,\ldots,t = [ {j/2} ]$. Here we generalize the proof to show that the various partial derivates of $ s$ polynomials of specified degrees are as independent as possible, given the room available.

Using this result, we construct and describe the varieties $ G(E)$ and $ Z(E)$ parametrizing the graded and nongraded compressed algebra quotients $ A = R/I$ of the power series ring $ R = k[[{x_1},\ldots,{x_r}]]$, having given socle type $ E$. These algebras are Artin algebras having maximal length $ \dim {_{k}}A$ possible, given the embedding degree $ r$ and given the socle-type sequence $ E = ({e_1},\ldots,{e_s})$, where $ {e_i}$ is the number of generators of the dual module $ \hat A$ of $ A$, having degree $ i$. The variety $ Z(E)$ is locally closed, irreducible, and is a bundle over $ G(E)$, fibred by affine spaces $ {{\mathbf{A}}^N}$ whose dimension is known.

We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable--have no deformation to $ (k + \cdots + k)$--for dimension reasons. For some choices of the sequence $ E,{\text{D}}$. Buchsbaum, $ {\text{D}}$. Eisenbud and the author have shown that the graded compressed algebras of socle-type $ E$ have almost linear minimal resolutions over $ R$, with ranks and degrees determined by $ E$. Other examples have given type $ e = {\dim_k}\;({\text{socle}}\;A)$ and are defined by an ideal $ I$ with certain given numbers of generators in $ R = k[[{x_1},\ldots\;,{x_r}]]$.

An analogous construction of thin algebras $ A = R/({f_1},\ldots,{f_s})$ of minimal length given the initial degrees of $ {f_1},\ldots,{f_s}$ is compared to the compressed algebras. When $ r = 2$, the thin algebras are characterized and parametrized, but in general when $ r > 3$, even their length is unknown. Although $ k = {\mathbf{C}}$ through most of the paper, the results extend to characteristic $ p$.

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Keywords: Artin algebra, Gorenstein algebra, Hilbert function, socle, type, generators, dualizing module, nonsmoothable algebras, minimal resolutions, almost linear resolutions, derivatives of polynomials, linearly independent dérivates, maximal rank, zero-dimensional schemes, parametrization, Hilbert scheme, deformation, irreducible, Hankel matrix, cactalecticant, invariants, power sum decomposition, forms, general polynomials, unking, compressed algebra
Article copyright: © Copyright 1984 American Mathematical Society