Artinian Gorenstein algebras with linear resolutions

作者：El Khoury Sabine; Kustin Andrew R^*

来源：Journal of Algebra, 2014, 420: 402-474.

DOI：10.1016/j.jalgebra.2014.07.026

• Artinian rings
• Buchsbaum-Eisenbud ideals
• Build resolution directly from inverse system
• Compressed algebras
• Flat family of Gorenstein algebras
• Gorenstein rings
• Linear presentation
• Linear resolution
• Macaulay inverse system
• Parameterization of Gorenstein ideals with linear resolutions
• Parameterization of linearly presented Gorenstein algebras
• Pfaffians
• Resolutions

For each pair of positive integers n, d, we construct a complex (G') over tilde (n) of modules over the bi-graded polynomial ring (R) over tilde = Z[x(1),...{t(M)}], where M roams over all monomials of degree 2n - 2 in {x(1),..., x(d)}. The complex (G') over tilde (n) has the following universal property. Let P be the polynomial ring k[x(1),..., x(d)], where k is a field, and let I-n([d]) (k) be the set of homogeneous ideals I in P, which are generated by forms of degree n, and for which P/I is an Artinian Gorenstein algebra with a linear resolution. If I is an ideal from I-n([d]) (k), then there exists a homomorphism (R) over tilde -> P, so that P circle times ((R) over tilde) (G') over tilde (n) is a minimal homogeneous resolution of PII by free P-modules. The construction of (G') over tilde (n) is equivariant and explicit. We give the differentials of (G') over tilde (n) as well as the modules. On the other hand, the homology of (G') over tilde (n) is unknown as are the properties of the modules that comprise (G') over tilde (n). Nonetheless, there is an ideal (I) over tilde of (R) over tilde and an element delta of (R) over tilde so that IR5 is a Gorenstein ideal of (R) over tilde and (G') over tilde (n) is a resolution of (R) over tilde/(IR) over tilde (delta) by projective (R) over tilde (delta)-modules. The complex (G') over tilde (n) is obtained from a less complicated complex (G') over tilde (n) which is built directly, and in a polynomial manner, from the coefficients of a generic Macaulay inverse system Phi. Furthermore, (I) over tilde is the ideal of (R) over tilde determined by The modules of (G) over tilde (n) are Schur and Weyl modules corresponding to hooks.
The complex (G) over tilde (n) is bi-homogeneous and every entry of every matrix in (G) over tilde (n) is a monomial. If m(1),...,m(N) is a list of the monomials in x(1),..., x(d) of degree n - 1, then delta is the determinant of the N x N matrix (tm(i)m(j)). The previously listed results exhibit a flat family of k-algebras parameterized by I-n([d]) (k):
k[{t(M)}] -> (k circle times Z (R) over tilde/(I) over tilde)(delta) (*).
Every algebra P/I, with I is an element of I-n([d]), is a fiber of (*). We simultaneously resolve all of these algebras P/I.
The natural action of GL(d)(k) on P induces an action of GL(d)(k) on I-n([d]) (k). We prove that if d = 3, n >= 3, and the characteristic of k is zero, then I-n([d]) (k) decomposes into at least four disjoint, non-empty orbits under this group action.

• 出版日期2014-12-15