Artinian Gorenstein algebras with linear resolutions

Abstract

For each pair of positive integers n, d, we construct a complex G∼'(n) of modules over the bi-graded polynomial ring R∼=Z[x1,. . .,xd,], where M roams over all monomials of degree 2n-2 in . The complex G∼'(n) has the following universal property. Let P be the polynomial ring k[x1, . . ., xd], where k is a field, and let In[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 In[d](k), then there exists a homomorphism R∼→P, so that P⊗R∼G∼'(n) is a minimal homogeneous resolution of P/I by free P-modules.The construction of G∼'(n) is equivariant and explicit. We give the differentials of G∼'(n) as well as the modules. On the other hand, the homology of G∼'(n) is unknown as are the properties of the modules that comprise G∼'(n). Nonetheless, there is an ideal I∼ of R∼ and an element δ of R∼ so that I∼R∼δ is a Gorenstein ideal of R∼δ and G∼'(n)δ is a resolution of R∼δ/I∼R∼δ by projective R∼δ-modules.The complex G∼'(n) is obtained from a less complicated complex G∼(n) which is built directly, and in a polynomial manner, from the coefficients of a generic Macaulay inverse system Φ. Furthermore, I∼ is the ideal of R∼ determined by Φ. The modules of G∼(n) are Schur and Weyl modules corresponding to hooks. The complex G∼(n) is bi-homogeneous and every entry of every matrix in G∼(n) is a monomial.If m1, . . ., mN is a list of the monomials in x1, . . ., xd of degree n-1, then δ is the determinant of the N×N matrix (tmimj). The previously listed results exhibit a flat family of k-algebras parameterized by In[d](k):()k[]δ→(k⊗ZR∼I∼)δ. Every algebra P/I, with I∈In[d](k), is a fiber of (). We simultaneously resolve all of these algebras P/I.The natural action of GLd(k) on P induces an action of GLd(k) on In[d](k). We prove that if d=3, n≥3, and the characteristic of k is zero, then In[d](k) decomposes into at least four disjoint, non-empty orbits under this group action. © 2014 Elsevier Inc.