As a consequence, we will derive some homotopytheoretic applications. We usually draw the rth stage of a spectral sequence in a tabular format with p increasing. For instance the leray spectral sequence and the exact sequence of low degrees. Derived functors and sheaf cohomology contemporary. An introduction to spectral sequences matt booth december 4, 2016. Using this we construct coboundary morphisms between grothendieck spectral sequences associated to objects in a short exact sequence. The associated grothendieck spectral sequence is the leray spectral sequence. Many other classes of spectral sequences are special cases of the grothendieck spectral sequence, for instance the. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. Thecoveringu is a leray covering relative to f if hju,f u0for all j0 and all u. Finally, it has been noted by a number of authors c. Ernie presented a version of this sequence on wednesday the version i give will look a little more general, as i will consider more general maps and will allow coe ecients in any sheaf. In this section, we construct the leray spectral sequence, an essential tool in modern. 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 has been suggested that the name spectral was given because, like spectres. 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. My main reference for this talk is the expository paper serre duality and appli cations by jun hou fung. The construction of the leray spectral sequence can be generalized to cohomology with support in specified families.
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 morphism of grothendieck sites. Then i claim that there is an adjointness isomorphism of the form. 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. For more nice explanations of spectral sequences, see 1 and 2. Verify the above claim, making concrete meaning of certain edge maps in grothendieckleray spectral sequences. 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. James milne, section 10 of lectures on etale cohomology. We consider cohomology of small categories with coefficients in a natural system in the sense of baues and wirsching.
Cohomology of the grothendieck construction springerlink. A relation or correspondence from p to q is a subset a c p x q. A grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on. Cartanleray spectral sequence for galois coverings of. There are essentially two situations in which spectral sequences arise.
His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its. In a sense there is really only one spectral sequence, just as there is only one concept of a long exact sequence although each object may originate in a variety of settings, but there are many di erent named uses. The spectral sequence associated to the composition of functors 21 7. With this background we can study the grothendieck spectral sequence in section 4.
An ath stage spectral cohomological sequence consists of the following. Leray spectral sequence encyclopedia of mathematics. We define the image and inverse image in the obvious way. I just would like to remark that many important spectral sequences are particular cases of the grothendieck spectral sequence for derived functor of the composition of two functor. In the case of locally compact spaces and cohomology with compact support, the leray spectral sequence was constructed by j. Introduction to spectral sequences ucb mathematics. The grothendieck spectral sequence minicourse on spectral. These are two simple conditions that force the leray spectral sequence to converge. 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.
Spectral sequences april 11, 2014 spectralsequences02. Successive spectral sequences benjamin matschke forschungsinstitut fur mathematik, eth zuric h benjamin. A coboundary morphism for the grothendieck spectral sequence. Sheaf cohomology on sites and the leray spectral sequence. The leray spectral sequence is now a special case of the grothendieck spectral sequence which can be found in most textbooks on homological algebra. Lerayserre to a particular setup in algebraic geometry, and will derive some. They have a reputation for being abstruse and difcult. Grothendieck let x be a topological space, withfx asheafofabeliangroups. The spectral sequence associated to a double complex 19 6. Some extremely brief notes on the leray spectral sequence. They were introduced by leray in the 1940s at the same time as he introduced sheaves. The equivariant serre spectral sequence 267 the category agx is in some sense the equivariant analogue of the category at associated to a topological space t.
In the special case that f aon z0, there is an evident map a. The leray spectral sequence, the way we proved it in lemma 20. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. K cat, we construct a spectral sequence abutting to the cohomology of the grothendieck construction of l in terms of the cohomology of k and of lk, for k. For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. For a comprehensive introduction to spectral sequences, see 3.
We also study the relationship between the graded centers of r and s. The grothendieck spectral sequence is a tool for actually computing some thing for the. On the leray spectral sequence and sheaf cohomology. In a sense there is really only one spectral sequence, just as there is only one concept of a long exact sequence although each object may originate in a variety of settings, but there are many di.
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