For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r. Namely, if xis a topological space, u fu igis an indexed open covering, and f is an abelian sheaf on x, then we get a cech to derived functor spectral sequence ep. By this, we are highly interested in collapsing spectral sequences at stage 1 or 2. It expresses, in the language of homological algebra, the singular cohomology of the total space x of a serre fibration in terms of the cohomology of the base space b and the fiber f. This is quite similar to the homology spectral sequence except that differentials go in the opposite.
Furthermore, this spectral sequence is functorial in a. Computing the general slice spectral sequence is usually hard. If uu i is an open cover of y, form the cech complex with respect to cover f. From the exact couple formulation of spectral sequences, we introduce a special case of the lerayserre spectral sequence and use it to compute h cpn. Group cohomology lecture notes university of washington. A spectral sequence is a recipe for computing some information about the cohomology of the double complex. An adams spectral sequence primer june 2, 2009 835. Degeneration of cohomological spectral sequences39 6. However in speci c cases, the structure of cohomological mackey functor will shed light on it. It is a reformulation using homological algebra, and an extension, of a technique called killing homotopy groups applied by the french school of henri cartan and jean. The fundamental example of a spectral sequence, from which essentially all the other examples arise as special cases, is the spectral sequence of a filtered complex. Multiplicative structure in the cohomological lerayserre.
Notes on the adams spectral sequence harvard university. A primer on spectral sequences university of chicago. Cohomological spectral sequences associated to inverse systems32 5. Secondary calculus and cohomological physics wikipedia. Depending upon the spectral sequence, the boundary map on the first sheet can have a degree which corresponds to r 0, r 1, or r 2. Homological spectral sequences associated to inverse systems43 6. When the number of independent variables is zero, i. Some extremely brief notes on the leray spectral sequence.
This way we consider it as a cohomological spectral sequence in the category of abelian groups. Moores original paper addresses this for singular homology. Multiplicative structure in the cohomological lerayserre spectral sequence please elucidate a proof. Extension problems for cohomological spectral sequences associated to inverse systems 54 8. Pdf an introduction to spectral sequences semantic scholar. Convergence of spectral sequences of cohomological type. In mathematics, the adams spectral sequence is a spectral sequence introduced by j.
Vakil uses cohomological indexing, but weibel prefers. Let us begin by recalling the major conclusions of the last lecture. Construction of cohomological spectral sequences33 5. Before getting into some of the details, its important to be aware that there are two different conventions for spectral sequences. On the other hand, if n 1 mod 4 there is no pair of everywhere independent vector elds on sn. The lemma youre quoting does not seem to depend on the nonnegativity of the filtration, so the same proof will work. Spectral sequence an overview sciencedirect topics. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces.
If a bounded collapsing spectral sequence converges to h, we can read the h n off. I will also try to brie y touch the topics of local coe cients and convergence. Like all spectral sequences, it is a computational tool. Secondary calculus acts on the space of solutions of a system of partial differential equations usually nonlinear equations. Extension problems for cohomological spectral sequences. We consider other spectral sequences, constructed by means of the spectra s mu, and some others. Pdf on a spectral sequence for twisted cohomologies. Convergence is clear since ep,q 1 is concentrated in the band. In homological algebra and algebraic topology, a spectral sequence is a means of computing. In mathematics, the serre spectral sequence sometimes lerayserre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology.
An ath stage spectral cohomological sequence consists. In chapter 4, we use the slice spectral sequence, which was developed in hhr16 and modi ed by ullman in his thesis ull, to compute the slice spectral sequence for v hz. We equip everything with products, yielding graded algebras and bigraded algebras. We can deduce some bounds on the cohomological dimension of schemes and on the cohomological dimension of fields using the results in section 57. The spectral sequence whose existence is asserted in the above theorem is an example of a. This spectral sequence has the wanted form by 7 and by agreement of the negative ktheory of 11 with thomasons, see 11, thm. Maps of complexes, snake lemma, long exact sequence. Spectral sequence abstractsupplement in this talk i plan to introduce spectral sequences by explaining how they arise from a ltration of an abelian group and then performing a couple of instructive applications of the lerayserre spectral sequence. Product structure on cohomological serre spectral sequence. They appear essentially everywhere where homology appears.
The cohomological version is simpler to construct, but requires stronger assumptions. For a proof of the existence of a spectral sequence under the conditions alluded to above, see grothendieck spectral sequence. Notes on spectral sequence northeastern university. Extension problems for homological spectral sequences associated to modules 52 8. Any triangle isomorphic to a distinguished triangle is distinguished. A spectral sequence is a sequence of complexes which, by taking successive cohomologies converges to the cohomology of a given complex or to some graded version thereof. In contrast to the lerayserre spectral sequence, the atiyahhirzebruch spectral sequence for the trivial fibration is in general noncollapsing. We will give the most important general family of examples. A fundamental lemma in homological algebra asserts that there is then a long exact sequence in homology.
A cohomological spectral sequence is like the above, but dpq r. Spectral sequences are computational devices in homological algebra. Continuing, if n 8e 1 7 mod 8, then there are 7 independent vector elds on sn. The hochschildserre spectral sequence it is well past time for an example. Some extremely brief notes on the leray spectral sequence intro. Well always indicate whether a spectral sequence is homological or cohomological by the placementofthepqsubscript. In mathematics, in the field of algebraic topology, the eilenbergmoore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. By the way, the authoritative reference for convergence questions of spectral sequences, imho, is boardman, conditionally convergent spectral sequences. The spectral sequences are completely di erent, but they have the same rst page and the same last page.
Of course these two notions are the same up to some sign conventions, but sign conventions are the bane of every homological algebraist. Since the 2nd page of these spectral sequences involves the right derived functors of the inverse limit, we refer to them as cohomological spectral sequences. An introduction to spectral sequences matt booth december 4, 2016. Its proof will concern the next two sections, as will the. Alan hatcher, spectral sequences in algebraic topology web. These are some notes for my talk in the bell show at the conference structured ring spectra tng, on august 4th, 2011, in hamburg, germany. As a motivating example, consider the long exact homology sequence. Higher derived functors cohomology from a new viewpoint.
Fibrations and the lerayserre spectral sequence 4 3. Criteria for cohomological descent 43 references 67 introduction in classical cech theory, we \compute or better. By this, we are highly interested in collapsing spectral sequences. A triangulated category is an additive category c equipped with an additive selfequivalence and a class of distinguished triangles x. Spectral sequences via examples the graduate journal of. A special example of the hypercohomology spectral sequence is the hodgede rham spectral sequence on a complex manifold. Some aspects of homological algebra alexandre grothendieck1 november 11, 2011 1the essential content of chapters 1, 2, and 4, and part of chapter 3 was developed in the spring of 1955 during a seminar in homological algebra at the university of kansas. For every a in a there is a spectral sequence e2 p. Any of the standard texts on homological algebra or cohomology of groups contains an account of the theory of spectral sequences. A homological type spectral sequence consists of a sequence of. The 2nd goal is to study the degeneration of these spectral sequences at their corresponding 2nd pages and the filtration that this degeneration provides. We introduce cohomological type spectral sequences and say a few more words about the extension problem.
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