In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties.
Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions.
Cohomology provides computable tools for producing sections, or explaining why they do not exist.
Unlike vector bundles, coherent sheaves (in the analytic or algebraic case) form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels.
These results generalize a large body of older work about the construction of complex analytic functions with given singularities or other properties.
[2] This is related to the fact that the category of quasi-coherent sheaves on an affine scheme
are isomorphic to the Čech cohomology groups with respect to the open covering
is a very special case of Grothendieck's vanishing theorem: for any sheaf of abelian groups
The sequence reads as which can be simplified using the previous computations on projective space.
There is an analogue of the Künneth formula in coherent sheaf cohomology for products of varieties.
is the genus of the curve, we can use the Künneth formula to compute its Betti numbers.
, this is proved by reducing to the case of line bundles on projective space, discussed above.
The finite-dimensionality of cohomology also holds in the analogous situation of coherent analytic sheaves on any compact complex space, by a very different argument.
Cartan and Serre proved finite-dimensionality in this analytic situation using a theorem of Schwartz on compact operators in Fréchet spaces.
Relative versions of this result for a proper morphism were proved by Grothendieck (for locally Noetherian schemes) and by Grauert (for complex analytic spaces).
The finite-dimensionality of cohomology leads to many numerical invariants for projective varieties.
Among many possible higher-dimensional generalizations, the geometric genus of a smooth projective variety
Grothendieck duality theory includes generalizations to any coherent sheaf and any proper morphism of schemes, although the statements become less elementary.
GAGA theorems relate algebraic varieties over the complex numbers to the corresponding analytic spaces.
Moreover, for every coherent algebraic sheaf E on a proper scheme X over C, the natural map of (finite-dimensional) complex vector spaces is an isomorphism for all i.
For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CPn is algebraic.
Serre's vanishing theorem says that for any ample line bundle
Note that Serre's theorem guarantees the same vanishing for large powers of
Kodaira vanishing and its generalizations are fundamental to the classification of algebraic varieties and the minimal model program.
Kodaira vanishing fails over fields of positive characteristic.
For example, the Hodge theorem implies that the definition of the genus of a smooth projective curve
Hodge theory has inspired a large body of work on the topological properties of complex algebraic varieties.
For example, if L is a line bundle on a smooth proper geometrically connected curve X over a field k, then where deg(L) denotes the degree of L. When combined with a vanishing theorem, the Riemann–Roch theorem can often be used to determine the dimension of the vector space of sections of a line bundle.
Knowing that a line bundle on X has enough sections, in turn, can be used to define a map from X to projective space, perhaps a closed immersion.
of dual numbers, examines whether there is a scheme XR over Spec R such that the special fiber is isomorphic to the given X.