sequence of closed subsets such that each difference Xi−Xi−1 is a topological manifold of dimension i. A sheaf E of R-modules on X is constructible with respect to the given stratification if the restriction of E to each stratum Xi−Xi−1 is locally constant, with stalk a finitely generated R-module. A sheaf E on X that is constructible with respect to the given stratification has locally finitely generated cohomology. If X is compact, it follows that the cohomology groups Hj(X,E) of X with coefficients in a constructible sheaf are finitely generated. More generally, suppose that X is compactifiable, meaning that there is a compact stratified space W containing X as an open subset, with W–X a union of connected components of strata. Then, for any constructible sheaf E of R-modules on X, the R-modules Hj(X,E) and Hcj(X,E) are finitely generated. For example, any complex algebraic variety X, with its classical (Euclidean) topology, is compactifiable in this sense. Cohomology of coherent sheaves In algebraic geometry and complex analytic geometry, coherent sheaves are a class of sheaves of particular geometric importance. For example, an algebraic vector bundle (on a locally Noetherian scheme) or a holomorphic vector bundle (on a complex analytic space) can be viewed as a coherent sheaf, but coherent sheaves have the advantage over vector bundles that they form an abelian category. On a scheme, it is also useful to consider the quasi-coherent sheaves, which include the locally free sheaves of infinite rank. A great deal is known about the cohomology groups of a scheme or complex analytic space with coefficients in a coherent sheaf. This theory is a key technical tool in algebraic geometry. Among the main theorems are results on the vanishing of cohomology in various situations, results on finite-dimensionality of cohomology, comparisons between coherent sheaf cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem. Sheaves on a site In the 1960s, Grothendieck defined the notion of a site, meaning a category equipped with a Grothendieck topology. A site C axiomatizes the notion of a set of morphisms Vα → U in C being a covering of U. A topological space X determines a site in a natural way: the category C has objects the open subsets of X, with morphisms being inclusions, and with a set of morphisms Vα → U being called a covering