Presheaves formalise the situation common to the examples above: a presheaf ( of sets ) on a topological space is a structure that associates to each open set U of the space a set F ( U ) of sections on U, and to each open set V included in U a map F ( U ) → F ( V ) giving restrictions of sections over U to V. Each of the examples above defines a presheaf with restrictions of functions, vector fields and sections of a vector bundle having the obvious meaning.

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

Engine power is transmitted via a set of vee-belts that are slack when the engine is idling, but by means of a tensioner pulley can be tightened to increase friction between the belts and the sheaves.

The 39 categories of melakhah are: ploughing earth, sowing, reaping, binding sheaves, threshing, winnowing, selecting, grinding, sifting, kneading, baking, shearing wool, washing wool, beating wool, dyeing wool, spinning, weaving, making two loops, weaving two threads, separating two threads, tying, untying, sewing stitches, tearing, trapping, slaughtering, flaying, tanning, scraping hide, marking hides, cutting hide to shape, writing two or more letters, erasing two or more letters, building, demolishing, extinguishing a fire, kindling a fire, putting the finishing touch on an object and transporting an object between the private domain and the public domain, or for a distance of 4 cubits within the public domain.

In the language of schemes, finite type modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.

** A fiddle block has two or more sheaves in one block, each with its own axle, so the sheaves are aligned.

proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes.

There are also maps ( or morphisms ) from one sheaf to another ; sheaves ( of a specific type, such as sheaves of abelian groups ) with their morphisms on a fixed topological space form a category.

In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves.

It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is local in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.

If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms.

With this notion of morphism, there is a category of C-valued sheaves on X for any C. The objects are the C-valued sheaves, and the morphisms are morphisms of sheaves.

Because sheaves encode exactly the data needed to pass between local and global situations, there are many examples of sheaves occurring throughout mathematics.

Here are some additional examples of sheaves:

More generally, if f is a morphism of schemes from X to Y, it induces a map f < sub >*</ sub > from étale sheaves over X to étale sheaves over Y, and its right derived functors are denoted by

In mathematics, the Picard group of a ringed space X, denoted by Pic ( X ), is the group of isomorphism classes of invertible sheaves ( or line bundles ) on X, with the group operation being tensor product.

Over each open affine U, Proj S ( U ) bears an invertible sheaf O ( 1 ), and the assumption we have just made ensures that these sheaves may be glued just like the above ; the resulting sheaf on Proj S is also denoted O ( 1 ) and serves much the same purpose for Proj S as the twisting sheaf on the Proj of a ring does.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray.

However, this advantage is totally negated by the relatively large energy consumption required to simply move the cable over and under the numerous guide rollers and around the many sheaves.

Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

A belt drive is analogous to that of a chain drive, however a belt sheave may be smooth ( devoid of discrete interlocking members as would be found on a chain sprocket, spur gear, or timing belt ) so that the mechanical advantage is approximately given by the ratio of the pitch diameter of the sheaves only, not fixed exactly by the ratio of teeth as with gears and sprockets.

* ' Tohoku ': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept ( to include sheaves of abelian groups ).

The sheaves were replaced by a stylised SWFC logo that had been in use on club merchandise for several years prior to the introduction of the new crest.

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N ( x )< sup > β / 2 </ sup >, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.

There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules.

It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work ( the Tohoku paper ).

The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.

Typically bainite manifiests as aggregates, termed sheaves, of ferrite plates ( sub-units ) separated by retained austenite, martensite or cementite.

Bulk can be reduced by substituting blade sheaves for the plate, but then the rotary FP shutter essentially becomes a regular bladed FP shutter.

This shutter greatly improved efficiency over the typical Leica shutter by using stronger metal blade sheaves that were " fanned " much faster, vertically along the minor axis of the 24 × 36 mm frame.

Copal collaborated with Nippon Kogaku to improve the Compact Square shutter for the Nikon FM2 ( Japan ) of 1982 by using honeycomb pattern etched titanium foil, stronger and lighter than plain stainless steel, for its blade sheaves.

The lifting hook is operated by the crane operator using electric motors to manipulate wire rope cables through a system of sheaves.

While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot merely be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves, namely, the dualizing complex.

) In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups ( for coherent sheaves ) as the much finer complex topology.

Charles X Gustav ’ s Order of the Saviour took the form of a similar circular medallion bearing the letters IHS in diamonds surrounded by a border of diamonds in the center of a cross formed of four enameled Vasa sheaves and hanging from a pink ribbon worn around the neck, of which one example survives in the collections of the Royal Armory.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

It is called torsion if F ( U ) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion.

More generally, if F is an inverse system of étale sheaves F < sub > i </ sub >, then the

cohomology of F is defined to be the inverse limit of the cohomology of the sheaves F < sub > i </ sub >

An ℓ-adic sheaf is a special sort of inverse system of étale sheaves F < sub > i </ sub >, where

* Proper morphisms between locally noetherian schemes or complex analytic spaces preserve coherent sheaves, in the sense that the higher direct images R < sup > i </ sup > f < sub >∗</ sub >( F ) ( in particular the direct image f < sub >∗</ sub >( F )) of a coherent sheaf F are coherent ( EGA III, 3. 2. 1 ).

Theorem B is sharp in the sense that if H < sup > 1 </ sup >( X, F ) = 0 for all coherent sheaves F on a complex manifold X ( resp.

quasicoherent sheaves F on a noetherian scheme X ), then X is Stein ( resp.

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.

An injective sheaf F is just a sheaf that is an injective element of the category of abelian sheaves ; in other words, homomorphisms from A to F can always be lifted to any sheaf B containing A.

# Sheaves on topoi: If E is a topos and S is an object in E, the category E < sub > S </ sub > of S-objects is also a topos, interpreted as the category of sheaves on S. If f: T → S is a morphism in E, the inverse image functor f < sup >*</ sup > can be described as follows: for a sheaf F on E < sub > S </ sub > and an object p: U → T in E < sub > T </ sub > one has f < sup >*</ sup > F ( U )