In this post, I’ll be summarizing the basics of the correspondence between vector bundles, locally free sheaves and divisors on a smooth curve (defined over an algebraically closed field of characteristic zero) together with some of their individual properties.
Locally free sheaves and Vector bundles:
Proposition 1: a) A coherent sheaf on a curve is locally free the fibers (stalks) are free at every point (note that the statement is local.)
b) A subsheaf of a locally free sheaf is locally free. (Use the fact that local rings of a (smooth) curve are DVR, hence PID and a submodule of a finitely generated free is free)
c) A non-zero map from a rank one locally free sheaf to a locally free sheaf is injective. (If there is a non-zero kernel, by b) it is locally free of rank one, then the cokernel will be a torsion sheaf injecting to a locally free sheaf!)
d) Let be a locally free sheaf of rank and a subsheaf of rank There exists a subsheaf or rank containing s.t. is locally free. In particular, if is the maximal (w.r.t inclusion) the quotient is already locally free. (Think how to kill the torsion!)
Theorem: There is a natural one-to-one correspondence between vector bundles (or rank ) and locally free sheaves (of rank ) on
Idea: Given a vector bundle, the sheaf of sections is the corresponding sheaf. The converse is a little tricky. Given a locally free sheaf one can define a vector bundle as follows
where is the unique maximal ideal of the local ring
Remark: Subsheaves of a locally free sheaf do not necessarily correspond to the subbundles of its associated bundle. The point is that, injectivity of the map of locally free sheaves may fail to be injective when it is reduced modulo the maximal ideal of some point.
Lemma 1: A non-trivial global section of a vector bundle correspond to a non-zero map of sheaves where is the sheaf associated to
A Weil divisor on is a finite formal sum of points with integral multiplicities. If is a rational function on the divisor corresponding to is defined as
A Cartier divisor on is given by a covering of together with functions s.t. are invertible on A principal divisor is given by the cover consisting of alone and a function on (global section of )
Given a Cartier divisor, one can associate a Weil divisor to it by considering on each open set the zeros minus the poles of and this is well-defined, since is invertible on Conversely, given a Weil divisor one can construct a Cartier divisor by choosing open sets that contain at most one of the point on the support of and functions that vanish at these points with the assigned multiplicities. Therefore, these two seemingly different notions are equivalent over a smooth curve
Line Bundles (locally free sheaves of rank one) and Divisors:
Given a Cartier divisor, one can define a locally free sheaf of rank one by taking the trivial sheaf and gluing them by the isomorphisms on
Conversely, given an invertible sheaf and a trivialization one can define a Cartier divisor as follows; take an arbirtary open set, say and define a Cartier divisor with as and which is a unit on
Definition: Let be an effective divisor. Define the sheaf on the support of by Now, define the skyscraper sheaf as the extension by zero outside of
Lemma 2: A line bundle corresponds to an effective divisor if and only if has a non-zero global section (by lemma 1, there is a non-zero map of sheaves which is injective by Proposition 1, b)) In this case, one has the following short exact sequence of sheaves
one then write
Riemann-Roch Theorem: If is an invertible sheaf (line bundle), then the Euler-Poincare characteristic can be computed as
where is the genus of the curve.
Proof: First, assume that corresponds to an effective divisor then by lemma 2, there is a short exact sequence, thus the associated long exact sequence gives rise to Since is a skyscraper sheaf, its support is a zero-dimensional set, hence so the result.
In the general case, assume that corresponds to with both effective. Then where Indeed, is a locally free sheaf so is flat and tensoring the above short exact sequence with one obtains
Therefore, On the other hand, replacing by in the above sequence and tensoring with and do the same thing as above leads to the desired formula.
Lemma 3: Given a locally free sheaf of rank there exists a short exact sequence of sheaves
where is an invertible sheaf and a locally free sheaf of rank
Using lemma 3 and induction, one can prove the following formula for the calculation of where are locally free sheaves of rank respectively.
where the degree of a locally free sheaf of rank is defined by
Definition: A sheaf is said to be generated by global sections if the natural map
Proposition 2: If is a locally free sheaf on there exists a positive divisor on s.t. is generated by global sections.
Atiyah’s theorem: Given a locally free sheaf of rank generated by global sections, there exists an exact sequence
where is an invertible sheaf.