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

J. Emsalem and the author showed in [18] that a general polynomialf of degree j in the ring SP= k[yl,...,yr] has (jr+-rl 1) linearly independent partial derivates of order i, for i = 0,1, . . ., 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[[xl sXr]] having given socle type E. These algebras are Artin algebras having maximal length dimk A possible, given the embedding degree r and given the socle-type sequence E = ( els . . . s eS), where ei is the number of generators of the dual module 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 AN 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 + + k)-for dimension reasons. For some choices of the sequence E, D. Buchsbaum, 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 = dimk (socle A) and are defined by an ideal I with certain given numbers of generators in R =

[1]  G. Gotzmann,et al.  Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes , 1978 .

[2]  A. Iarrobino,et al.  Some zero-dimensional generic singularities ; finite algebras having small tangent space , 1978 .

[3]  G. Mazzola.  Generic finite schemes and Hochschild cocycles , 1980 .

[4]  David A. Buchsbaum,et al.  Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3 , 1977 .

[5]  A. Iarrobino.  The Number of Generic Singularities , 1973 .

[6]  R. Stanley.  Hilbert functions of graded algebras , 1978 .

[7]  Judith D. Sally,et al.  Numbers of generators of ideals in local rings , 1978 .

[8]  J. Emsalem.  Géométrie des points épais , 1978 .

[9]  D. Bayer.  The division algorithm and the hilbert scheme , 1982 .

[10]  F. S. Macaulay.  On a method of dealing with the intersections of plane curves , 1904 .

[11]  E. Green.  Complete intersections and Gorenstein ideals , 1978 .

[12]  A. Iarrobino,et al.  Reducibility of the families of 0-dimensional schemes on a variety , 1972 .

[13]  Peter Schenzel.  Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe , 1980 .

[14]  Joël Briançon,et al.  Description de Hilbn C{x, y} , 1977 .

[15]  David A. Buchsbaum,et al.  Schur Functors and Schur Complexes , 1982 .

[16]  E. Zatini,et al.  Relations between the type of a point on an algebraic variety and the type of its tangent cone , 1980 .

[17]  E. Kunz,et al.  On the deviation and the type of a Cohen-Macaulay ring , 1973 .

[18]  Anthony Iarrobino,et al.  Punctual Hilbert Schemes , 2005 .

[19]  David R. Berman.  The number of generators of a colength N ideal in a power series ring , 1981 .

[20]  J. Sally.  Stretched Gorenstein Rings , 1979 .

[21]  Simplicity of a vector space of forms: Finiteness of the number of complete Hilbert functions , 1977 .

[22]  H. Kleppe.  DEFORMATION OF SCHEMES DEFINED BY VANISHING OF PFAFFIANS , 1978 .