# Definition of torsion sheaf on reducible spaces

I need to discuss torsion-free sheaves on reduced, but possibly reducible spaces. Here "torsion" means "element is annihilated by a non-zero-divisor". The standard references (EGA, Hartshorne, ...) restrict themselves to normal varieties or irreducible spaces and do not seem to cover the definition in this level of generality.

I saw some papers and survey articles that use torsion-free sheaves on reducibly spaces, but mostly without any discussion of the definition. Some papers even seem to give the "wrong" definition where "torsion" = "annihilated arbitrary non-zero element", which is probably not what the authors had in mind.

Is anyone aware of a reliable reference for the definition and for basic properties of torsion-free sheaves that I could possibly use?

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EGA is not as restrictive as you claim. See I.8.4 (1970 edition) for the case of reduced schemes and IV.20.1 for the case of ringed spaces. –  Fred Rohrer Sep 25 '12 at 9:43

Hi Stefan! Torsion-free (coherent) sheaves are pure sheaves of codimension 0; I suggest that you try "The Geometry of Moduli Spaces of Sheaves", by Daniel Huybrechts and Manfred Lehn, where pure sheaves are discussed in detail.

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