BGG correspondence and Roemer's theorem on an exterior algebra

Abstract

Let E = K< y_1, ..., y_n > be the exterior algebra. The ``(cohomological) distinguished pairs" of a graded E-module M describe the growth of a minimal graded injective resolution of M. Roemer gave a duality theorem between the distinguished pairs of M and those of its dual M^*. 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 Roemer's theorem.