The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

Nevertheless, unlike the case of ordinary cohomology theories, the coefficients alone do not determine the theory in the sense that there might be more then one theory with given coefficients.

To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2 dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve.

If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.

In this case classifying maps give rise to the first Chern class of X, in H < sup > 2 </ sup >( X ) ( integral cohomology ).

As an important special case when G is the group of real numbers R and the underlying topological space has the additional structure of a smooth manifold, the Mayer – Vietoris sequence for de Rham cohomology is

In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC ( and even in much weaker theories ).

In the case that V is a non-singular algebraic curve and i = 1, H < sup > 1 </ sup > is a free Z < sub > l </ sub >- module of rank 2g, dual to the Tate module of the Jacobian variety of V, where g is the genus of V. Since the first Betti number of a Riemann surface of genus g is 2g, this is isomorphic to the usual singular cohomology with Z < sub > ℓ </ sub > coefficients for complex algebraic curves.

This is the simplest case of a much more general relationship between homology and integration, which is most efficiently formulated in terms of differential forms and de Rham cohomology.

Furthermore, in this case the structure group of the normal bundles is the circle group ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from to the circle, which in turn equals the first integral cohomology group.

In mathematics, particularly in algebraic topology, Alexander – Spanier cohomology is a cohomology theory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D. Wallace.

In the case of orientable surfaces, this is the action on first cohomology H < sup > 1 </ sup >( Σ ) ≅ Z < sup > 2g </ sup >.

In the case of X the multiplication of the cochains associated to the copies of is degenerate, whereas in T multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to Z ( more generally M where this is the base module ).

In de Rham cohomology, the cup product of differential forms is also known as the wedge product, and in this sense is a special case of Grassmann's exterior product.

Another way is that the cohomology of the complement supports a non-trivial Massey product, which is not the case for the unlink.

This begins with Ext groups calculated over the ring of cohomology operations, which is the Steenrod algebra in the classical case.

The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients.

In case X is defined over the complex numbers, the latter group maps to the topological cohomology group

Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch (—) td ( X ) and to the non-proper case by considering cohomology with compact support.

In this case, commonsense statements are statements that are so crucial to keeping the account coherent that they are all but impossible to deny.

As in the case of myth, these narratives are believed because they construct and reinforce the worldview of the group within which they are told, or “ because they provide us with coherent and convincing explanations of complex events ”.

In fact, this was not the case as new conventions developed, which were themselves internally coherent.

In further extensions to the design logic, George and Bock ( 2012 ) use case studies and the IBM survey data on business models in large companies to describe how CEOs and entrepreneurs create narratives or stories in a coherent manner to move the business from one opportunity to another.

First published in 1947 in the aftermath of the crushing Conservative election defeat of 1945, and aimed at the mass market and the layman, it presented a well-written and coherent case for Conservatism.

In this case a coherentist would need to explain how special relativity is more coherent than both Newtonian mechanics and the Lorentz ether theory, which explanation would lead us on from simple inconsistency.

* In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.

In the case of The Library of Babel meaning is hard to find as any coherent works are rare.

Here the condition corresponds to case when the dualizing object, which a priori lies in a derived category, is represented by a single module ( coherent sheaf ).

In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules.

It was argued that not all elements may be present in every case, but the picture is sufficiently regular and coherent to permit clinical recognition.

In the case of the electrons in the tubulin subunits of the microtubules, Hameroff has recently suggested that these are part of a Frohlich condensate, which is a coherent oscillation of dipolar molecules.

supporters make a good case for a coherent understanding of the nature of the cosmos,

The advantage to discretionary review is that it enables an appellate court to focus its limited resources on developing a coherent body of case law, or at least it is able to focus on making decisions in consistent fashion ( in jurisdictions where case law is not recognized ).

# The adjudication of the case in a federal forum " would be disruptive of state efforts to establish a coherent policy with respect to a matter of substantial public concern.

( Unless in case of a phase conjugated mirror or if there is some sort of adaptive optics that is matched to the system, and this is only possible if the light is reasonably temporally coherent.

To create an image with the VRD a photon source ( or three sources in the case of a color display ) is used to generate a coherent beam of light.

In the case of actual multiple-wavelength colour holograms of this type, the colour information is recorded and reproduced just as in the Lippmann process, except that the highly coherent laser light passing through the recording medium and reflected back from the subject generates the required distinct standing waves throughout a relatively large volume of space, eliminating the need for reflection to occur immediately adjacent to the recording medium.

pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.

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 ).

This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds.

With increasing level of generality, it turns out, an increasing amount of technical background is helpful or necessary to understand these theorems: the modern formulation of both these dualities can be done using derived categories and certain direct and inverse image functors of sheaves, applied to locally constant sheaves ( with respect to the classical analytical topology in the first case, and with respect to the étale topology in the second case ).

One also has isomorphisms of all higher direct image sheaves in this case.

The definition in that case is usually said with greater care ( using invertible sheaves or holomorphic line bundles ); see below.

Almost everything in this section generalizes mutatis mutandis to the case of local sections of a fibre bundle, a Banach bundle over a Banach manifold, a fibered manifold, or quasi-coherent sheaves over schemes.

This is the case, for example, when looking at the category of sheaves on projective space in the Zariski topology.

The category in the case is chosen to be the category of all quasicoherent sheaves on the ringed space which reduces to the category of all modules over some ring R in case of affine schemes.

and expect this “ twisted ” sheaf to contain grading information about N. In particular, if N is the sheaf associated to a graded S-module M we likewise expect it to contain lost grading information about M. This suggests, though erroneously, that S can in fact be reconstructed from these sheaves ; however, this is true in the case that S is a polynomial ring, below.