# From coherent sheaves to curves in P^3

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- Dipartimento di Matematica e Informatica
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LE MATEMATICHEVol. LV (2000) Fasc. II, pp. 271 283 FROMCOHERENT SHEAVES TO CURVES IN P3 MIREILLE MARTIN-DESCHAMPS Dedicated to Silvio Greco in occasion of his 60-th birthday. 0. Introduction. Let k be an algebraically closed �eld, and let P = P3k be the projec-tive 3-space over k. We denote by R the associated polynomial ring R =k[X, Y, Z , T ].The aim of this paper is to precise the correspondences between equiva-lence classes of some objects related to P: vector bundles (i.e. locally free co-herent sheaves), coherent sheaves, �nite length graded R-modules and locallyCohen-Macaulay curves (i.e. closed subscheme of pure dimension one with noembedded points). Some of these correspondences are well-known, others arenew. The notion of pseudo-isomorphismwas introduced in [5]: De�nition 0.1. Let A be a ring, let N and N � be two coherent sheaves on P 3A, �at over A and let f be a morphism from N to N � . We say that f is apseudo-isomorphism if it induces: i) an isomorphism of functors (from the category of �nite type A-modules toitself) H 0(N (n)⊗A .) → H 0(N �(n)⊗A .) for all n � 0,ii) an isomorphism of functors (from the category of �nite type A-modulesto the category of the graded A[X, Y, Z , T ]-modules) H 1∗ (N ⊗A .) →H 1∗ (N � ⊗A .), 272 MIREILLE MARTIN-DESCHAMPS iii) a monomorphism of functors H 2∗ (N ⊗A .) → H 2∗ (N � ⊗A .). Two coherent sheaves on P3A, �at over A, are pseudo-isomorphic if thereexists a chain of pseudo-isomorphisms connecting them: N = N0 → N1 ← N2 → N3 ← · · · → N2p−1 ← N2p = N �. This de�nes an equivalence relation called psi-equivalence. It turns outthat when A is a �eld the psi-equivalence is an extension to the set of coherentsheaves of (local) projective dimension ≤ 1 of the stable equivalence for vectorbundles with H 2∗ = 0 (cf. 2.4).Now, using the bijective correspondence between the set of isomorphismclasses of �nite length graded R-modules and the set of stable equivalenceclasses of vector bundles with H 2∗ = 0 ([4], see also 2.3), and a well-knowntheorem o

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