Differences Between Reflexive and Projective Modules

ac.commutative-algebraalgebraic-k-theoryhomological-algebravector-bundles

Let R be a normal noetherian domain.

What is the difference between a finitely generated reflexive module and a finitely generated projective module?

Can anybody recommend any references or make any suggestions about this?

Finitely generated projective modules can be identified with idempotents matrix…

Finitely generated projective modules correspond with vector bundles over topological space…

Are there similar results about reflexives modules?

Best Answer

Well, the answer is well known of course. For a finitely generated module over a commutative normal Noetherian domain TFAE

  1. M is reflexive
  2. M is torsion-free and equals the intersection of its localizations at the codimension 1 primes
  3. M satisfies Serre's condition S2 and its support is Spec R.
  4. M is the dual of a f.g. module N

As you say, a finite projective module is the same as a locally free sheaf on Spec R. Similarly, a finite reflexive module is the same as the push forward of a locally free sheaf from an open subset U of Spec R whose complement has codimension $\ge2$.

So for an easy example take a Weil divisor D which is not Cartier, the associated divisorial sheaf (corresponding to an R-module of rank 1) is reflexive, not projective. Your example with a line on a quadratic cone is of this form.

This stuff is standard and used all the time in higher-dimensional algebraic geometry around the Minimal Model Program. For an old reference covering some of this, see e.g. Bourbaki, Chap.7 Algebre commutative, VII.

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