Regularity and cohomology of determinantal thickenings

We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I in S for which the induced maps Ext^j_S(S/I,S) -> H_I^j(S) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt-Blickle-Lyubeznik-Singh-Zhang, holds for determinantal thickenings.