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,{tM}], where M roams over all monomials of degree 2n−2 in {x1,…,xd}. 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[{tM}]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.