A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. Note that the same thing holds for the ordinary cohomology and the intersection cohomology, since the gaps between the compactly supported cohomology and these two cohomology groups are noncuspidal, as already explained. Computing with sheaves and sheaf cohomology in algebraic geometry. We will discuss the intersection cohomology complex as a replacement for the constant sheaf which rescues poincare duality on singular spaces, and prove its basic properties. The origins of sheaf theory, sheaf cohomology, and spectral sequences haynes miller february 23, 2000 jean leray november 7, 1906november 10, 1998 was con ned to an o cers prison camp \o ag in austria for the whole of world war ii. Pseudoeisenstein forms and cohomology of arithmetic groups ii. But what if the sheaf in question is far from being constant.
These are compared to ordinary cohomology and homology and the difference is measured by suitable cohomology and homology theories, called end cohomology and end homology. In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Representation theory and the cohomology of arithmetic groups. We compute the cohomology of k in two ways by means of the two spectral sequences e. Cohomology groups hax, e are defined, where x is a topological space and.
A gentle introduction to homology, cohomology, and sheaf. Computing with sheaves and sheaf cohomology in algebraic. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Equivalently, the complex of stalks at xis acyclic for all x2v. For local systems it is ordinary cohomology with twisted coefficients. Pdf using the additive property of compactly supported. James milne, section 18 of lectures on etale cohomology. If r z2z poincare duality holds for all manifolds, orientable or not. Compactly supported cohomology and nearby cycle cohomology of.
M is nite dimensional in this case, the kunneth formula is available only for the compactly supported cohomology. Let xbe a projective complex algebraic variety and let sbe a coherent sheaf on x. However, we can prove the results in cohomology and in compactly supported cohomology using the same techniques. Raoul bott, loring tu, differential forms in algebraic topology. Pseudoeisenstein forms and cohomology of arithmetic groups ii jurgen rohlfs. A topological space is locally euclidean if every p2mhas a neighborhood uand a homeomorphism u. Etale cohomology of curves siddharth venkatesh abstract. A finiteness result for the compactly supported cohomology of. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. K, a bounded sheaf complex of kvector spaces, such that h cx. There he took up algebraic topology, and the result was a spectacular. Topology, cohomology and sheaf theory tu june 16, 2010 1 lecture 1 1. In mathematics, the support of a realvalued function f is the subset of the domain containing those elements which are not mapped to zero. Poincare duality compatible with the definition of compactly supported cohomology in etale cohomology.
A sheaf cohomology theory for calgebras martin mathieu queens university belfast banach algebras 2017 at oulu on 6 july 2017 partially supported by uk engineering and physical sciences research council grant no. Cohomology with compact supports for coherent sheaves on an. Compactly supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. On the morsenovikov cohomology of blowing up complex. Inspired to provide fast algorithms for persistence, we prove that. The construction is sheaf theoretic and works both for topological spaces and for the etale cohomology of algebraic varieties. Compactlysupported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The theory is presented in a way that is meant to be read and appreciated by a broad audience. These groups generalize the ordinary cohomology groups of x with coefficients in an abelian sheaf, as well as the. In this manuscript we make a general study of the representations realized, for a reductive lie group of harishchandra class, on the compactly supported sheaf cohomology groups of an irreducible finiterank polarized homogeneous vector bundle defined in a generalized complex flag space. Sheaf cohomology and all vs nothing arguments giovanni caru supervisor.
With x one canassociatea taut separatedadic space ad locally of. In the ordinary theory, we used the pullback to prove the homotopy invariance and to construct the mv sequence. The interpretation via the espace etale is helpful here. These are notes for a talk on the etale cohomology of curves given in a grad student seminar on etale cohomology held in spring 2016 at mit.
The end homology and end cohomology of an open manifold. 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. Curry march, 20 abstract this note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex. This is motivated by work of baumfultonmacpherson 7, 8. They are being developed for lectures i am giving at the arizona winter school in tucson, march 1115, 2006. More generally we claim that for any f, the cech complex is acyclic on v s 2i u. In compactly supported cohomology cocycles and coboundaries on some space are required to have compact support. Compactly supported cohomology and nearby cycle cohomology.
The cech complex of a sheaf of abelian groups is the special case of this in which the sheaf of rings is the constant sheaf z with stalks z. This allows us to make sense of a section \over a closed subset. From now on we will assume that mis a sheaf of modules for some speci ed sheaf of rings on x. We will present the basic methods of computing sheaf cohomology. Intersection cohomology invariants of complex algebraic. Huber, a finiteness result for the compactly supported cohomology of rigid analytic varieties, j. February 28, 2006 1 introduction these notes are still in a preliminary form. So we are not able to apply the relative cohomological method via the compactly supported cohomology directly as 12. To compute compactly supported cohomology, we must first pick a local orientation of our space. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. We take the abelian category shx of sheaves of abelian groups on x. Compactly supported cohomology is the cohomology of the complex of compactly supported differential forms i. Poincare duality compatible with the definition of compactly.
The central work for the study of sheaf cohomology is grothendiecks 1957 tohoku paper. In his article 1 he developed more generally a theory of higher direct images with compact support for a morphism of schemes. This paper is a revised version of the authors doctoral dissertation submitted to the. To talk about derived functors, we need an abelian category to start with. Using the additive property of compactly supported cohomology. Abstract homotopy theory, generalized sheaf cohomology, homotopical algebra, sheaf of spectra, homotopy category, derived functor. Consider the example of the halfopen interval x 0, 1 decomposed as x 0 and a 0, 1. Martin mathieu queens university belfast a sheaf cohomology theory for calgebras. Homology, cohomology, and sheaf cohomology university of. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. In this paper we consider a somewhat di erent direction, which is related to the arithmetic genus. Lefschetz trace formula and rationality of zeta function 7 4. Can one still understand sheaf cohomology in some geometric way. We omit the verification that these maps are compatible with maps between short.
Sheaf theory omt lovering september 24, 2010 abstract in this essay we develop the basic idea of a sheaf, look at some simple examples and explore areas of mathematics which become more transparent and easier to think about in light of this new concept. This concept is used very widely in mathematical analysis. The euler characteristic computed using homology, or compactly supported cohomology is an additive invariant of a topological space. Representation theory and the cohomology of arithmetic groups birgit speh. Fourier, grenoble 57, 3 2007 9731017 a finiteness result for the compactly supported cohomology of rigid analytic varieties. It turns out that in the \ compactly supported theory, we can still do two rescues, one can be used to get. X y, this suggests defining the pushforward with compact support f. Though we attempt to avoid being too dependent on category theory and homological. Here eis the spectral sequnce we get by rst taking cohomology in the rst index, and e0is the spectral sequence we get by rst taking cohomology in the second index. Sep 05, 2012 this article uses basic homological methods for evaluating examples of compactly supported cohomology groups of line bundles over projective curve. Serre duality is a pretty property relating two cohomology classes a special case being the poincare duality which lets us compute cohomology classes in terms of homology classes. Pseudoeisenstein forms and cohomology of arithmetic. Several familiar spectral sequences arise as special cases. Let s denote the category whose objects are qcqs schemes over s and whose.
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