On quasicoherent sheaves

On an arbitrary ringed space, one can define sheaves of modules (easily) and coherent sheaves (in general, very painfully). 

On a scheme X, there is a third option, which is defining quasicoherent sheaves.

If X=Spec R is affine, the functor (sheaves of modules on X) to (R modules) sending F to F(X) has a left adjoint, denoted by M maps to tilde M.  If M=R, then \tilde M is canonically isomorphic to the structure sheaf. 

The left adjoint is of course unique up to canonical isomorphism, or one can choose an explicit construction and get a functor; we then define a sheaf of modules F as being quasicoherent if the natural map from tilde (F(X)) to F is an isomorphism.

I much prefer to say that a sheaf of modules on X is quasicoherent if for every principal open D(f), the natural map F(X)_f\to F(D(f)), which exists for every sheaf of modules, is an isomorphism.


Exercise Let X = Spec R be an affine scheme. Show that global sections define an equivalence of categories from Qcoh(X) to R-Mod.

We then define quasicoherent sheaves among sheaves of modules on arbitrary schemes X by any of the following equivalent properties:

  1. For each U open affine in X, the restriction of F to U is quasicoherent;
  2. For some open affine cover {U_i} of X, the restriction of F to U_i is quasicoherent for every i;
  3. For each affine open U in X, and each regular function f on U, the natural map F(U)_f\to F(D(f)) is an isomorphism.

Exercise Let X be a quasicompact scheme, f a regular function on X, V=D(f) the associated principal open, and s in F(V). Then there exists r>0 and a global section t of F such that t restricted to V is equal to f^r\cdot s.

Exercise Let X in P^N be quasiprojective, and F quasicoherent on X; let f a global section of O_X(d) for some d>0, V:=d(f) the associated principal open, and s in F(V). Then there exists r>0 and a global section t of F(rd) such that t restricted to V is equal to f^r\cdot s.




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