MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

Thank you in advance!

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.