Coherent algebraic sheaf

A coherent sheaf of modules on an algebraic variety or scheme. The structure sheaf of a Noetherian scheme and, in particular, of an algebraic variety is coherent.

Coherent algebraic sheaves are a convenient tool of investigating algebraic varieties. Intuitively, a coherent algebraic sheaf can be regarded as a continuous algebraic system of linear spaces on a variety (see Vector bundle on an algebraic variety) and arises in the consideration of linear and algebraic families of divisors, imbeddings of varieties in a projective space, differential forms, vector fields and automorphisms, deformations of varieties and subvarieties, in one word, in the linearization of all kinds of problems in algebraic geometry (see [3]). Here the results are stated in terms of the cohomology of coherent algebraic sheaves. The cohomology theory of coherent algebraic sheaves includes: a) finiteness theorems (in algebraic geometry), asserting the finiteness of the dimensions of the cohomology spaces $H^i(X,\mathcal{F})$, $i \ge 0$, of a coherent sheaf $\mathcal F$ on a complete variety $X$; b) the Riemann–Roch theorem, which calculates the Euler–Poincaré characteristic of a coherent algebraic sheaf; c) theorems of Serre type (see Affine scheme) or Kodaira vanishing theorems (cf. Kodaira theorem) (see [4], [5]); d) duality theorems (see Duality in algebraic geometry) relating the $i$-dimensional and the $(n-i)$-dimensional cohomology spaces of sheaves on a smooth variety of dimension $n$; e) the Künneth formula, giving an expression for the homology spaces of certain sheaves on a product of varieties; f) comparison of theorems in algebraic geometry with other theorems on cohomology — analytic, formal, étale; and g) a theory of local cohomology, useful in the study of coherent algebraic sheaves on incomplete varieties. One of the most important of its applications relates to the Lefschetz theorem, which compares the properties of a variety and its hyperplane section.

Many results generalize to the case when the single variety $X$ is replaced by a family of varieties, that is, to the case of a morphism $f : X \to Y$. In this case, the cohomology spaces are replaced by the sheaves $R^n f_*$ of the derived direct image functor $f_*$; here an important role is played by the behaviour of these sheaves under a base change.