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

Iarrobino, Anthony

doi:10.1090/S0002-9947-1984-0748843-4

Compressed algebras: Artin algebras having given socle degrees and maximal length
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Trans. Amer. Math. Soc. 285 (1984), 337-378 Request permission

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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18

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