IJPAM: Volume 20, No. 1 (2005)

LOCALLY ALGEBRAIC REDUCED SCHEMES
AND VECTOR BUNDLES:
UNIQUE FACTORIZATION THEOREMS

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Let $X$ be a connected and reduced locally algebraic scheme such that all its irreducible components are projective and $E$ a locally free sheaf on $X$ with finite rank. Here we prove that $E$ has a unique (up to permutations and isomorphisms of the factor) decomposition into irreducible locally free indecomposabole factors. Furthermore, the following conditions are equivalent:

(1) $E$ is indecomposable;

(2) there is a connected positive-dimensional reduced closed subscheme $Y$ such that $E\vert Y$ is indecomposable;

(3) $X =\cup _{\beta \in J} X_\beta$ with each $X_\beta$ union of finitely many irreducible components of $X$ and $E\vert X_\beta$ indecomposable for all $\beta \in J$.

Received: February 2, 2005

AMS Subject Classification: 14F05, 14J60

Key Words and Phrases: locally algebraic schemes, vector bundles, unique factorization for vector bundles

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 1