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In the mathematical field of 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
(in the integer coefficients or some other coefficients).
Let be the sheaf of differential k-forms on a manifold M. If n is the dimension of M, then the sheaf
is called the sheaf of (smooth) densities on M. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map:
If M is oriented; i.e., the orientation sheaf of the tangent bundle of M is literally trivial, then the above reduces to the usual integration of a differential form.
mathematical field of algebraic topology, the orientationsheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at...
sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology...
analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientationsheaf. The Serre duality...
especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties...
constant on each member of the stratification. A basic example is the orientationsheaf on a manifold since each point of the manifold admits an orientable...
{or}}_{M/X}} is the relative orientationsheaf. A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of Sato's hyperfunctions...
usual notion of an orientation coincides with a Z-orientation. The integration along the fiber Orientation bundle (or orientationsheaf) - this is used to...
The River Sheaf in Sheffield, South Yorkshire, England, flows northwards, past Dore, through Abbeydale and north of Heeley. It then passes into a culvert...
Alexander–Spanier cohomology or sheaf cohomology). (Here sheaf cohomology is considered only with coefficients in a constant sheaf.) These theories give different...
charts and atlases). Third, the sheaf OM is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of...
=\omega _{X,\Lambda }[d]} is the shifted Λ {\displaystyle \Lambda } -orientationsheaf. On the other hand, let X {\displaystyle X} be a smooth k {\displaystyle...
[ˈfaʃʃo]; pl.: fasci) is an Italian word literally meaning "a bundle" or "a sheaf", and figuratively "league", and which was used in the late 19th century...
translated this notion into the notion of the coherent (sheaf) (Especially, coherent analytic sheaf) in sheaf cohomology. This name comes from H. Cartan. Also...
corresponds to an orientation covering of a manifold X (cf. #covering.) 4. See also orientation of a vector bundle as well as orientationsheaf. pair 1. A...
_{X\setminus X_{n-2}}} is the constant sheaf on X ∖ X n − 2 {\displaystyle X\setminus X_{n-2}} . By replacing the constant sheaf on X ∖ X n − 2 {\displaystyle...
resolution of the constant sheaf R, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of R. Suppose that...
Burbage Moor to the west of the city to its mouth where it joins the River Sheaf in a culvert beneath Sheffield railway station. Like the other rivers in...
between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient...
of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic...
defined by the ideal sheaf ⨁ n = 0 ∞ I n + 1 {\displaystyle \textstyle \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n+1}} . This ideal sheaf is also the relative...
= 0 {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0} . The exponential sheaf sequence leads to the following exact sequence: H 1 ( X , O X ) ⟶ H 1 (...
general notion of a locally constant sheaf. Orientation character, a characteristic form related to the orientation line bundle, useful to formulate Twisted...
structure sheaf (i.e., the trivial line bundle), O C P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)} is Serre's twisting sheaf (i.e., the...
fixed dimension. Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or...
11%) Design C The winning design. 1,450 votes (34.46%) Design D The barley sheaf is from the historic arms of Aberdeenshire, and also represents the county's...
conditions, a local system can be equivalently described as a locally constant sheaf. The fundamental groupoid of the singleton space is the trivial groupoid...
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