Abstract
Let \(E = K{\left\langle {y_{1} ,...,y_{n} } \right\rangle }\) 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. Römer 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 \(S=K[x_1, \ldots, x_n]\)-modules. Using this idea, we also give a \(\mathbb{Z}^{n} \)-graded version of Römer’s theorem.
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To the memory of Professor Tetsushi Ogoma.
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Yanagawa, K. BGG Correspondence and Römer’s Theorem on an Exterior Algebra. Algebr Represent Theor 9, 569–579 (2006). https://doi.org/10.1007/s10468-006-9037-y
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DOI: https://doi.org/10.1007/s10468-006-9037-y