In mathematics, the stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point. Sheaves are defined on open...

S_{1}} is a coherent sheaf and locally generates S over S 0 {\displaystyle S_{0}} (that is, when we pass to the stalk of the sheaf S at a point x of X...

Look up sheaf in Wiktionary, the free dictionary. In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian...

stalk, a part of the brain Stalk (sheaf), a mathematical construction The Stalk, a 1994 science fiction novel by Chris Morris and Janet Morris Stalk (TV...

sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology...

{O}}_{X}} to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X {\displaystyle X} . The stalk at a point x {\displaystyle...

spirit: a coherent sheaf F {\displaystyle {\mathcal {F}}} on a scheme X {\displaystyle X} is a vector bundle if and only if its stalk F x {\displaystyle...

of OX (defined stalk-wise, or on open affine charts). For a morphism f: X → Y and a closed subscheme Y′ ⊆ Y defined by an ideal sheaf J, the preimage...

exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of...

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of...

In mathematics, the gluing axiom is introduced to define what a sheaf F {\displaystyle {\mathcal {F}}} on a topological space X {\displaystyle X} must...

image exhibits some relatively subtle features. Suppose we are given a sheaf G {\displaystyle {\mathcal {G}}} on Y {\displaystyle Y} and that we want...

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module...

group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with...

then verify that the sheaf axioms hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written Z...

In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally...

The sheaf of rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical...

constant sheaf on a topological space X {\displaystyle X} associated to a set A {\displaystyle A} is a sheaf of sets on X {\displaystyle X} whose stalks are...

algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is the local...

the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement. Given...

differentiable functions on U. As U varies, this determines a sheaf of rings on Rn. The stalk Op for p ∈ Rn consists of germs of functions near p, and is...

In the Zariski topology, the stalk of X at x is computed by taking a direct limit of the sections of the structure sheaf over all the Zariski open neighborhoods...

M} . If F is a quasicoherent sheaf on a scheme X, the support of F is the set of all points x in X such that the stalk Fx is nonzero. This definition...

effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk Ix is principal. It is equivalent to...

\operatorname {Spec} (R)} , that is, a prime ideal, then the stalk of the structure sheaf at p {\displaystyle {\mathfrak {p}}} equals the localization...

space factors through r. An analytic space is normal if every stalk of the structure sheaf is a normal ring (meaning an integrally closed integral domain)...

Stack Stalk is an outdoor 2001 sculpture by Ean Eldred and the architectural firm Rigga, located along the Eastbank Esplanade in Portland, Oregon. The...

(R_{j}/I_{j})} as schemes over U j {\displaystyle U_{j}} . There is a quasi-coherent sheaf of ideals I {\displaystyle {\mathcal {I}}} on X such that f ∗ O Z ≅ O X...

is a constructible sheaf over a genus g smooth projective curve C, of rank n outside a finite set X of points where it has stalk 0. Then χ ( C , F )...

that formation of stalks preserves finite limits. This implies that if T is a Lawvere theory and a sheaf F is a T-algebra, then any stalk Fx is also a T-algebra...