The mayervietoris sequence of relative hypercohomology lubkin, s. More precisely, the second term of the leray spectral sequence has the form and its limit is the bigraded group associated with a filtration of the graded group. The spectral sequence whose existence is asserted in the above theorem is an example of a. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. A relation or correspondence from p to q is a subset a c p x q. In mathematics, the leray spectral sequence was a pioneering example in homological. Dan petersen, leray spectral sequence, november 2010 pdf.
The second page of of the corresponding atiyahhirzebruch spectral sequence see above for f f \ast hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine ktheory classes, hence to genuine dbrane. Some extremely brief notes on the leray spectral sequence. This spectral sequence specializes to the ordinary leray spectral sequence in sheaf cohomology theory, but may also be used for generalized ale cohomology theories such as ale ktheory. Leray number lx is a natural combinatorialtopological measure of the complexity of x. By a spectral sequence in an abelian category a, we mean a collection of bigraded objects e. Hodge numbers and leray spectral sequence mathoverflow. A leray spectral sequence for noncommutative differential fibrations article pdf available in international journal of geometric methods in modern physics 105. The leray spectral sequence is the special case of the grothendieck spectral sequence for the case where the two functors being composed are a pushforward of sheaves of abelian groups along a continuous map f. It can be thought of as the hereditary homological dimension of x.
Second leray spectral sequence of relative hypercohomology. Let abe an abelian category with enough injectives, and t. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians. The exact sequence in homotopy groups, and the leray serre spectral sequence for homology groups of a.
A spectral sequence is a computational tool they are more compiex analogues of long exact sequences. Jardine mathematics department, university of western ontario, london, ontario n6a 5b7, canada communicated by e. Springer nature is making coronavirus research free. For a comprehensive introduction to spectral sequences, see 3. This can be a serious obstacle to understanding spectral sequences on. Introduction to spectral sequences ucb mathematics. Recall the following general result in the computation of hypercohomology groups. Galois connections and the leray spectral sequence. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians. These are two simple conditions that force the leray spectral sequence to converge. We know that if we have a short exact sequence of chain complexes 0.
Xby means of he kunneth theorem, namely that there is a split short exact sequence 0. Remark note on terminology often the terminology atiyahhirzebruch spectral sequence is taken to refer to only this case with f f \ast, while the general case is then referred to as serre spectral sequence for generalized cohomology or similar. From the exact couple formulation of spectral sequences, we introduce a special case of the leray serre spectral sequence and use it to compute h cpn. Some extremely brief notes on the leray spectral sequence intro. A very brief outline of the proof of the main result theorem 3. If x happens to be an eilenbergmac lane object of the form ka, n for some sheaf of abelian groups a on g and olsln, then ru,x is the sheaf associated. Leray spectral sequence encyclopedia of mathematics.
Some extremely brief notes on the leray spectral sequence greg friedman intro. As a motivating example, consider the long exact homology sequence. Dec 22, 2018 classical leray spectral sequence of a. The leray spectral sequence of a map f computes the homology of the domain of f from the fibers of f. It can be thought of as a mild generalization of the kunneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. The leray spectral sequence is now a special case of the grothendieck spectral sequence which can be found in most textbooks on homological algebra. This is skew in generality to an earlier one constructed by s. We first show that the spectral sequence associated to a filtration by.
It is named after jean leray and guy hirsch, who independently proved it in the late 1940s. For a proof of the existence of a spectral sequence under the conditions alluded to above, see grothendieck spectral sequence. We already know how to do this in case of the trivial bre sequence f. Sometimes the spectral sequence can be identi ed with the leray spectral sequence of the embedding j. A spectral sequence is a collection of three sequences er,dr. Adiabatic limits, nonmultiplicativity of signature, and. Fibrations and the lerayserre spectral sequence 4 3. He was born in chantenaysurloire today part of nantes.
You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. A spectral sequence connecting the cohomology with values in a sheaf of abelian groups on a topological space with the cohomology of its direct images under a continuous mapping. Journal of pure and applied algebra 61 1989 189196 189 northholland the leray spectral sequence j. Fibrations and the leray serre spectral sequence 4 3.
In these notes we will cover the following instances of grothendieck spectral sequences. If a bounded collapsing spectral sequence converges to h, we can read the h n off. Sheaf cohomology on sites and the leray spectral sequence. Aug 25, 2011 a leray spectral sequence for noncommutative differential fibrations article pdf available in international journal of geometric methods in modern physics 105 august 2011 with 76 reads. He was a member of the institute for advanced study in princeton every fall semester from. Leray gave a hint of how he came to the notion of a spectral sequence in a later paper 47, p. In fact, the result is basically obvious if you use czech cohomology on the base. We usually draw the rth stage of a spectral sequence in a tabular format with. The leray spectral sequence, the way we proved it in lemma 20.
Leray complexes combinatorics and geometry roy meshulam. In this note we give a minimal presentation of spectral sequences following ega. X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. It has been suggested that the name spectral was given because, like spectres.
Eg just as there is o les of a pair xa in homology. The construction of the leray spectral sequence can be generalized to cohomology with support in specified families. Once this lie group version is established, various consequences of the classical cartanleray spectral sequence theorem carry over. If uu i is an open cover of y, form the cech complex with respect to cover f. But once the initial hurdle of believing in spectral sequences is surmounted, one cannot help but be amazed at their power. By a spectral sequence in an abelian category a, we mean a collection of bi. There are essentially two situations in which spectral sequences arise. These were subsequently developed by many others, 5 each separately becoming an important tool in homological algebra. Adiabatic limits, nonmultiplicativity of signature, and leray spectral sequence authors. A generalization of the cartanleray spectral sequence. The conclusions we draw about level set persistent homology corollary4. Leray s work of this period proved seminal to the development of spectral sequences and sheaves. The leray spectral sequence is motivic springerlink.
We define the image and inverse image in the obvious way. The purpose of this note is to construct a leraytype spectral sequence for homotopy classes of maps of simplicial presheaves, both stably and unstably, for any. In 1934 leray published an important paper that founded the study of weak solutions of the navierstokes equations. A short exact sequence of chain complexes gives rise to a long exact sequence in homology, which is a fundamental tool for computing homology in a number of situations. In the same year, he and juliusz schauder discovered a topological invariant, now called the lerayschauder. By this, we are highly interested in collapsing spectral sequences at stage 1 or 2.
Is the serre spectral sequence a special case of the leray. X of an open subset to a compact complex variety ep. We shall not be able to avoid using spectral sequences see pp 307309 of my book on etale cohomology for a brief summary of spectral sequences and chapter 5 of weibels book for a complete treatment. The associated grothendieck spectral sequence is the leray spectral sequence. As a consequence, we will derive some homotopytheoretic applications. Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. Spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. Friedlander received 26 may 1988 revised 7 november 1988 the purpose of this note is to construct a leraytype spectral sequence for homotopy classes of maps of. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. It is impossible to describe everything about spectral sequences in the duration of a single course, so we will focus on a special and important example. It is a very special case of the leray spectral sequence. In this section, we construct the leray spectral sequence, an essential tool in. Constructing generalised leray spectral sequences imma g alvez preliminary report, with f neumann and a tonks motivation construction of the first spectral sequence this project arose as a spino of an earlier one with the same collaborators to extend to some categories of stacks several classical results from geometry and topology. Algebraic topology lectures by haynes miller notes based on livetexed record made by sanath devalapurkar images created by john ni march 4, 2018 i.
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