We check that the associated sheaf cohomology apparatus sati es mayervietoris and kunneth formulae. The corresponding long exact sequence in cohomology as discussed above is what is traditionally called the mayervietoris sequence of the cover of x x by u u and v v in a acohomology. A gentle introduction to homology, cohomology, and sheaf. A coarse mayervietoris principle pennsylvania state university. Licata, carlo angiuli, ed morehouse 1 introduction an eilenbergsteenrod cohomology theory consists of a family of contravariant functors cn n.
Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models. Witt cohomology, mayervietoris, homotopy invariance, and the gersten. A covering of p1 c is given by the two standard opens u 0. Axioms for cohomology, and the mayervietoris sequence in. The hilbert scheme of elliptic curves and reflexive sheaves on fano 3folds vermeire, peter, osaka journal of mathematics, 2008. In algebraic topology, we learn about the mayervietoris sequence, which allows us to break apart a topological space into smaller pieces whose cohomology we hopefully understand. A special case of that spectral sequence is the mayer vietoris long exact sequence. A cochain complex kis a sequence of homomorphisms of abelian groups d. Level sets persistence studies the homology groups of preimages. Homology, cohomology, and sheaf cohomology university of. A special case of that spectral sequence is the mayervietoris long exact sequence. Below will construct the cechtocohomology spectral sequence, see lemma 20. Pdf the generalized mayervietoris principles and spectral. Mare subcomplexes of k, then we can form a long exact sequence of homology groups and homeomorphisms between them.
The generalized mayervietoris principles and spectral sequences. In this thread angelo explained beautifully how presheaf cohomology cech cohomology relates to sheaf cohomology. Computational algebraic geometry meets string theory rigid. We introduce homological algebra and sheaf theory as required. Delignes mixed hodge structure for projective varieties with. Topology, cohomology and sheaf theory tu june 16, 2010 1 lecture 1 1. From exactness of the mayervietoris sequence it follows that hc1e. R, the mayervietoris exact sequence, and the kunneth formula see below. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. Quantum sheaf cohomology is a 0,2 deformation of the ordinary quantum cohomology ring, realized as the ope ring in a2twisted theories. Below will construct the cechto cohomology spectral sequence, see lemma 20. These four chapters do not have to be read in this order, however. I will compute some simple examples and also use the mayer vietoris sequence to provide motivation for the existence of such a structure. A coarse mayervietoris principle 87 the complexes c form an inverse sequence under the obvious surjective restriction maps, and we define c limc.
We check that the associated sheaf cohomology apparatus sati es mayer vietoris and kunneth formulae. Anyway, it is clear that they dont really use that hypothesis, and it works perfectly for any sheaf of abelian groups. Cohomology theorems for asymptotic sheaves jorge mozo fernandez received december 8, 1997, revised november 18, 1998. One could begin with homology and perhaps continue with cohomology before turning to homotopy. This sequence extends a sixterm mayervietoris ktheory sequence for the category pic of milnor and bass, and when br is isomorphic to the second etale cohomology group with coefficients in the sheaf of units multiplicative group the extended sequence. R, the mayer vietoris exact sequence, and the kunneth formula see below. Chapters 1 and 4, and homology and its mirror variant cohomology in chapters 2 and 3. U 1, which are both isomorphic to a1 c, and which intersection is isomorphic to a 1 c f 0g g m. The mayervietoris long exact sequence is a consequence of these two results. Cech cohomology as a cohomology theory mit opencourseware. Excision property and mayer vietoris sequence conversely, let us assume that we have an element c0 n 1 such that i 1c 0 n 1 0 f 0 n 1 c 0 n 1. In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. We introduce homological algebra and sheaf theory as re quired.
Reduced cohomology groups and their properties 145 2. Base change and projection formula discuss the mayervietoris theorems, the compatibility of inverse images and derived tensor products, and the interaction between derived direct images and derived hom. Cohomology gaps for sheaves on threefolds ballico, edoardo and gasparim, elizabeth, journal of geometry and symmetry in physics, 2011. Other readers will always be interested in your opinion of the books youve read. In this note we give two mayer vietoris spectral sequences of the localization of certain o xmodules over the open complement of a closed subvariety y s i y i of an algebraic. A topological space is locally euclidean if every p2mhas a neighborhood uand a homeomorphism u. Since it is such a basic, useful and easy to understand variant of the spectral sequence we treat it here separately. The derivation of the mayervietoris sequence from the eilenbergsteenrod axioms does not require the dimension axiom, so in addition to existing in ordinary cohomology theories, it holds in extraordinary cohomology theories such as topological ktheory and cobordism. Note that this also makes sense if u and v are disjoint, if we.
X, i to be the ith cohomology group of the abelian sheaf f. We strongly urge the reader to read this online at instead of reading the old material. Delignes mixed hodge structure for projective varieties. Local cohomology, arrangements of subspaces and monomial ideals. Ii sheaf cohomology 33 1 differential sheaves and resolutions 34. Licata, carlo angiuli, ed morehouse 1 introduction an eilenbergsteenrod cohomology theory consists of a. A holomorphic line bundle is defined by transition functions relative to open sets u. In addition we show in complete generality that hochschild cohomology satis. Excision property and mayervietoris sequence conversely, let us assume that we have an element c0 n 1 such that i 1c 0 n 1 0 f 0 n 1 c 0 n 1. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. Wx is homotopy invariant and all its nisnevich cohomology. I will compute some simple examples and also use the mayervietoris sequence to provide motivation for the existence of such a structure. The central work for the study of sheaf cohomology is grothendiecks 1957 tohoku paper. In the other direction, one could postpone homology and cohomology until after parts of chapter 4.
The corresponding long exact sequence in cohomology as discussed above is what is traditionally called the mayer vietoris sequence of the cover of x x by u u and v v in a a cohomology. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. A coarse mayer vietoris principle 87 the complexes c form an inverse sequence under the obvious surjective restriction maps, and we define c limc. This is a nontrivial fact that can be shown for example by combining the computation of h0x. Morels proof of the conjecture is based among others on the the following result.
When this construction is applied to the presheaf l1, we obtain the sheaf of locally l1 functions. As a consequence, new proofs of classical results for brations are obtained. Hochschild cohomology of abelian categories and ringed spaces. International school for advanced studies trieste u. Axioms for cohomology, and the mayer vietoris sequence in. By duality see universal coefficient theorem an analogous statement holds for the homology of x x, u u and v v. Given a cochain complex we can compute its cohomology groups. In algebraic topology, we learn about the mayer vietoris sequence, which allows us to break apart a topological space into smaller pieces whose cohomology we hopefully understand.
Homotopy invariance of the sheaf wnis and of its cohomology. The derivation of the mayer vietoris sequence from the eilenbergsteenrod axioms does not require the dimension axiom, so in addition to existing in ordinary cohomology theories, it holds in extraordinary cohomology theories such as topological ktheory and cobordism. In this note we give two mayervietoris spectral sequences of the localization of certain o xmodules over the open complement of a closed subvariety y s i y i of an algebraic. Exactness of the mayervietoris sequence in homotopy type. In this note we give two mayervietoris spectral sequences of the localization of certain. How do we compute the cohomology of p1 c with coef. Sheaves and homotopy theory daniel dugger the purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of thomason 14 and jardine 7, 8, 9. More speci cally, if kis a simplicial complex and l.
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