Regularity and Cohomology of Determinantal Thickenings

We consider the ring S = C[xij ] of polynomial functions on the vector space C m×n of complex m × n matrices. We let GL = GLm(C) × GLn(C) and consider its action via row and column operations on C m×n (and the induced action on S). For every GL-invariant ideal I ⊆ S and every j ≥ 0, we describe the decomposition of the modules ExtjS(S/I, S) into irreducible GL-representations. For any inclusion I ⊇ J of GLinvariant ideals we determine the kernels and cokernels of the induced maps ExtjS(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 ⊆ S for which the induced maps ExtjS(S/I, S) −→ H j I (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.