The Horrocks correspondence for coherent sheaves on projective spaces (Q985318)

Two locally free sheaves \(E\) and \(E'\) on \({\mathbb P}^n = {\mathbb P}(V)\) are stably equivalent if there exist fine direct sums of line bundles \(L\) and \(L'\) such that \(E \oplus L \simeq E' \oplus L'\). The graded modules \(H^i_*(E) = \bigoplus_{d \in {\mathbb Z}} H^i(E(d))\) are invariants for stable equivalence, but they not identify the stable equivalence class of \(E\). Denote by \(\Lambda\) the exterior algebra of \(V\). In [\textit{I. Coanda} and \textit{G. Trautmann}, Trans. Am. Math. Soc. 358, No. 3, 1015--1031 (2006; Zbl 1087.14013)], an equivalence between a certain subcategory of the homotopy category \(K(\Lambda)\), and the stable category of vector bundles on \({\mathbb{P}}(V)\) is described, and a bijection between HT-complexes and stable isomorphism classes is established. Moreover, a relation between the Tate resolution of a vector bundle as described by \textit{D. Eisenbud, G. Fløystad}, and \textit{F.-O. Schreyer} [Trans. Am. Math. Soc. 355, 4397--4426 (2003; Zbl 1063.14021)] and HT-complexes is described. In this paper, the author generalizes these results to the case of coherent sheaves, using the Bernstein-Gel'fand-Gel'fand correspondence and relating it to the Horrocks correspondence. He also gives direct proof for the BGG-correspondence and for a result by \textit{D. Eisenbud, G. Fløystad}, and \textit{F.-O. Schreyer} [loc. cit.] by using the comparison Lemma proved in the appendix B.