Push forward sheaf cohomology pdf

For an integer q 0, we denote by hq z o x the qth local cohomology sheaf of o x, with support in z. Thus taking cohomology of a complex of amodules commutes. In this case, the cohomology sheaf hrs is the sheaf kerdimd. For any algebraic varieties xover any eld kor any scheme one can consider sheaf cohomology with respect to the zariski topology.

Now the claim follows from cohomology on sites, lemma 21. But that cohomology vanishing will imply that derived categories form a stack on the acyclic site. Shy, that sends a sheaf fof abelian groups on xto the push forward f fon y. Direct image sheaf, the pushforward of a sheaf by a map fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by integration on the fibres transfer operator, the pushforward on the space of measurable functions. Acomputation 03n8 how do we compute the cohomology of p1 c with coe. Intersection cohomology sheaf as an example of a perverse sheaf. So the complex s is exact meaning that it forms an exact sequence. Push forwards in oriented cohomology theories of algebraic varieties i. Properties of the category of perverse sheaves, including the classi cation of simples. Yare locally compact topological spaces and given a continuous map f. Sheaves on topological spaces constructible sheaves on.

Y is the espace etale of a sheaf gon y, then y y x. If f is a coherent sheaf on x, show that f f is a coherent sheaf. For a proper morphism in the sense of scheme theory, this functor sends coherent sheaves to coherent sheaves. Comparison of motivic and simplicial operations in mod. F and the cohomology hif of a complex of sheaves, we will from now on write hif for the latter. Note that specifying a sheaf on a onepoint topological space is the same as specifying a single abelian. Sheaf cohomology florian beck, jens eberhardt, natalie peternell contents 1. We restrict to smooth separated schemes of nite type over a eld kand lwill be a xed prime. Reduced, integral, noetherian schemes, mor phisms of finite type, finite morphisms, closed immersions, dimension, fibre product 4.

Therefore, with a view towards such applications, we construct simplicial cohomology operations that commute with proper push forwards and work out several examples of such push forward formulae. X is a coherent sheaf w on x endowed with a trace map t. Affine schemes, proj, structure sheaf, scheme associated to variety 3. Pushforward of coherent sheaves and field extensions. All other perverse sheaves are iterated extensions of objects like the above. F for higher pushforward sheaves, the r stands for. This is a quasicoherent o xmodule whose sections are annihilated by suitable powers of the ideal sheaf i. Existence of push forward maps in a cohomology the ory gives a powerful tool that allows to perform various computations and analyze properties o f the considered cohomology theory. And we have a powerful tool known as sheaf cohomology. In particular, a constant sheaf, the additive sheaf g a, the mulitiplicative sheaf g m etc. Xi this says that the push forward sheaf is a direct sum of perverse sheaves. If f is a sheaf on x, the push forward by f of f, also called its direct image, and denoted f. Chern classes in deligne cohomology for coherent analytic.

Every coherent sheaf f has a welldefined class in k 0 x, namely, the alternating sum of e i where e is any finite resolution of f by vector bundles locally free sheaves. In the sheaf of twisted integers example 2 each sx is isomorphic to the ring z, but this sheaf is not a sheaf of rings. Let f be a locally constant torsion sheaf on the etale ksitus and assume that chark is prime to the torsion of f. Given a coherent sheaf fon x, we can consider the sheaves f k fikf. On the other hand, if b is an abelian sheaf on u, the sheaf i b. A covering of p1 c is given by the two standard opens u 0,u 1, which are both isomorphic to a1 c, and whose intersection is isomorphic toa1 c \0 g m,c. Note that if a is an abelian sheaf on z then j a x a x. Reduced, integral, noetherian schemes, morphisms of finite type, finite morphisms, closed immersions, dimension, fibre products separated and proper morphisms. Sheaf cohomology mathematics and statistics mcgill university. K 1 in the category of sheaves on the etale site x et, where each k i is a zimodule. There is also a pushforward with proper support, f.

Y and is hinvariant where the action on y is meant to be trivial. The stalk of the cohomology sheaf coincides with the cohomology. In the kequivariant setting, the terminal object should be bk, i. Here we construct classical cohomology operations that commute with proper push forwards and work out several examples of such push forward formulae. A complex of sheaves is a complex in which each sr is a sheaf.

Indeed, the only reasonable known method of computing sheaf cohomology is the tech construction, and because of the direct limit involved, it is essential to have open coverings by cohomologically trivial subsets. X is its sheaf of rings, and an o xmodule is a sheaf which is a. However, well see below that cohomology with socalled twisted coe cients is also useful. Riemann surfaces can also be regarded as complex algebraic curves, i. Stein manifolds play this part in complex analytic geometry, and affine schemes in. A dualizing sheaf for x is a coherent sheaf w on x endowed with a trace map t. In order to avoid any possible confusion between sheaf cohomology hix. F is called the pushforward sheaf or the direct image of f. For a vector bundle e, we will typically also denote its sheaf of sections by e, as no confusion is likely to occur. This is explained for example in warners book on di erentiable manifolds. Pushing forward sheaves and the result on sheaf cohomology. Iff is a torsion sheaf, let z be the maximal closed hypersurface contained in suppf,letzii.

Sheaves, covering spaces, monodromy and an application. The pushforward f e is defined as the alternating sum of the classes in k 0 y of the higher direct images r i f e. Y is a continuous map between locally ringed spaces and fis a sheaf on x, then f f will denote the push forward. Affine schemes, proj, structure sheaf, scheme associated to a variety 3. A constructible sheaf f over x is an etale sheaf such that for some. X x a schematically dense open immersion of its underlying scheme into another scheme x. Note that specifying a sheaf on a onepoint topological space is the same as specifying a single abelian group. In this section, our goal is to understand what nonzero perverse cohomology. Itturnsoutthatthemayervietorissequenceholdsinetale. For example grothendiecks result concerns the functor rf or push forward, in sheaf cohomology.

For this reason, j a is said to be the extension by zero to x of the sheaf a on z. P 1, o n a d c o n a there is also a trivial extension of this to more general toric varieties. Zx is precisely singular cohomology with z coe cients. These cohomology groups again formed a cochain complex, so he took cohomology again. Once we dene cohomology of quasicoherent sheaves, we will quickly prove that if f is a coherent sheaf on pn k, then pn. Pushforwards in oriented cohomology theories of algebraic. Usually this will be assumed to be di erent from the characteristic of kand k. First, we need another result of serre hartshorne ii. It is the right derived functor of the direct image of a sheaf.

Normally, we can realize singular cohomology as the derived pushforward of the constant sheaf on xto a point the terminal \base object of the category we live in. Flasque sheaves are preserved by pushforward along morphism of sheaves. This says that it is a direct sum of ic sheaves of strata. Let a be a ring, let x pr be a closed immersion for some r. Smirnov abstract notions of a cohomology theory and an oriented cohomology theory for algebraic varieties are introduced, push forwards for projective morphisms are constructed and a riemannroch theorem for ring operations between oriented theories is proved. We use a witten type deformation of the dolbeault complex of l, introduced by tian and zhang, to show, that the cohomology of the sheaf of holomorphic sections. Recall that in the category shx of sheaves of abelian groups, we have a notion of exactness which amount to exactness at stalks. Let a be an abelian category, that is, roughly, an additive category in which there exist wellbehaved kernels and cokernels for each morphism, so that, for example, the notion of an exact sequence in a makes sense. Let l be a positive gequivariant line bundle over x. One part of this is the pullback construction for coefficients. These cohomology groups can also be obtained as sheaf cohomology of a constant sheaf.

Higher direct images of the structure sheaf in positive characteristic. Feb 11, 2020 fiber integration or push forward is a process that sends generalized cohomology classes on a bundle e b e \to b of manifolds to cohomology classes on the base b b of the bundle, by evaluating them on each fiber in some sense. Therefore, none of the material presented here is an original work of mine and no such assumption. The serre spectral sequence of a noncommutative fibration for. Cohomology, derived categories, and derived functors in the di erential graded setting. Let x be a smooth projective variety acted on by a reductive group g. The following proposition shows that the category of quasicoherent sheaves on a noetherian scheme xhas enough injectives. Given the reversal of arrows which often occurs in considering algebras rather than spaces, this becomes a push forward construction in noncommutative geometry. Homology, cohomology, and sheaf cohomology university of. Asking for help, clarification, or responding to other answers. Character sums, automorphic forms, equidistribution, and. We use a witten type deformation of the dolbeault complex of l, introduced by tian and zhang, to show, that the cohomology of the sheaf of holomorphic sections of the induced. The sheaf cohomology we will see later measures howx, is di. For our purposes, we can pretend as if we are working with lim k, although the homological algebra is far from straight forward.

Thanks for contributing an answer to mathematics stack exchange. The perverse tstructure on the constructible derived category. These notes began with a course given at brown university in the. If x2xis a point, then locally xis cut out by a maximal ideal m x of functions vanishing on x. Cohomology of the mumford quotient maxim braverman abstract. It is wellknown that this functor admits a left adjoint f, which can be geometrically described in terms of the espace etale as follows. Properties of the category of perverse sheaves, including the. He found that the cohomology groups of the pushforward sheaf formed a cochain complex, so he took the cohomology. Eventually, after taking cohomology continuously, the groups converged to the cohomology of the sheaf. In particular, each summand enjoys all the remarkable. Independence of embedding for higher sheaf cohomology of local cohomology on projective space.