Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. Finally, it has been noted by a number of authors c. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its. 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. They were introduced by leray in the 1940s at the same time as he introduced sheaves. The spectral sequence whose existence is asserted in the above theorem is an example of a. For instance the leray spectral sequence and the exact sequence of low degrees. Verify the above claim, making concrete meaning of certain edge maps in grothendieckleray spectral sequences. An ath stage spectral cohomological sequence consists of the following. The grothendieck spectral sequence is a tool for actually computing some thing for the. Many other classes of spectral sequences are special cases of the grothendieck spectral sequence, for instance the.
Grothendieck let x be a topological space, withfx asheafofabeliangroups. Then i claim that there is an adjointness isomorphism of the form. In the special case that f aon z0, there is an evident map a. For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. Leray spectral sequence encyclopedia of mathematics.
These are two simple conditions that force the leray spectral sequence to converge. A relation or correspondence from p to q is a subset a c p x q. 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. The leray spectral sequence is now a special case of the grothendieck spectral sequence which can be found in most textbooks on homological algebra. 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. A coboundary morphism for the grothendieck spectral sequence. We consider cohomology of small categories with coefficients in a natural system in the sense of baues and wirsching. There are essentially two situations in which spectral sequences arise. The conclusions we draw about level set persistent homology corollary4. 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. 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. 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.
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. An introduction to spectral sequences matt booth december 4, 2016. 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. Some extremely brief notes on the leray spectral sequence. 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. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. We also study the relationship between the graded centers of r and s. In the case of locally compact spaces and cohomology with compact support, the leray spectral sequence was constructed by j. They have a reputation for being abstruse and difcult. 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. My main reference for this talk is the expository paper serre duality and appli cations by jun hou fung.
Spectral sequences april 11, 2014 spectralsequences02. You can supposedly also look at grothendiecks famous. 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. Successive spectral sequences benjamin matschke forschungsinstitut fur mathematik, eth zuric h benjamin. On the leray spectral sequence and sheaf cohomology. The construction of the leray spectral sequence can be generalized to cohomology with support in specified families. Derived functors and sheaf cohomology contemporary. James milne, section 10 of lectures on etale cohomology. The leray spectral sequence, the way we proved it in lemma 20. The spectral sequence associated to the composition of functors 21 7. In this section, we construct the leray spectral sequence, an essential tool in modern. Lerayserre to a particular setup in algebraic geometry, and will derive some. Many spectral sequences in algebraic geometry are instances of the grothendieck spectral sequence, for example the leray spectral sequence.
For more nice explanations of spectral sequences, see 1 and 2. Cartanleray spectral sequence for galois coverings of. 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. We usually draw the rth stage of a spectral sequence in a tabular format with p increasing. Thecoveringu is a leray covering relative to f if hju,f u0for all j0 and all u. 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.
Cohomology of the grothendieck construction springerlink. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. For a comprehensive introduction to spectral sequences, see 3. We define the image and inverse image in the obvious way. With this background we can study the grothendieck spectral sequence in section 4. 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. 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. The spectral sequence associated to a double complex 19 6. The grothendieck spectral sequence minicourse on spectral. Spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. Introduction to spectral sequences ucb mathematics.
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