BGG correspondence and Romer's theorem on an exterior algebra

摘要

Let E = K[y(1),..., y(n)] be the exterior algebra. The (cohomological) distinguished pairs of a graded E-module N describe the growth of a minimal graded injective resolution of N. Romer gave a duality theorem between the distinguished pairs of N and those of its dual N*. In this paper, we show that under Bernstein-Gel'fand-Gel'fand correspondence, his theorem is translated into a natural corollary of local duality for ( complexes of) graded S = K[x(1),..., x(n)]-modules. Using this idea, we also give a Z(n)-graded version of Romer's theorem.