Applications of duality theory to Cousin complexes

Abstract

We use the anti-equivalence between Cohen–Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S 2 condition and the lowest cohomology N of its “dual” complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite étale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex and a vector bundle of finite rank. We relate the various results in [P. Sastry, Duality for Cousin complexes, in: Contemp. Math., vol. 375, Amer. Math. Soc., Providence, RI, 2005, pp. 137–192] on Cousin complexes to dual results on coherent sheaves on formal schemes.