Grothendieck leray spectral sequence pdf

You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. An ath stage spectral cohomological sequence consists of the following. 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. Spectral sequences april 11, 2014 spectralsequences02. Grothendieck let x be a topological space, withfx asheafofabeliangroups. The grothendieck spectral sequence minicourse on spectral. It has been suggested that the name spectral was given because, like spectres. Many spectral sequences in algebraic geometry are instances of the grothendieck spectral sequence, for example the leray spectral sequence. Cartanleray spectral sequence for galois coverings of.

The spectral sequence whose existence is asserted in the above theorem is an example of a. 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. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its. 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. For instance the leray spectral sequence and the exact sequence of low degrees. As a consequence, we will derive some homotopytheoretic applications.

The spectral sequence associated to the composition of functors 21 7. 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. Cohomology of the grothendieck construction springerlink. These are two simple conditions that force the leray spectral sequence to converge. Leray spectral sequence encyclopedia of mathematics. Lerayserre to a particular setup in algebraic geometry, and will derive some. Thecoveringu is a leray covering relative to f if hju,f u0for all j0 and all u.

My main reference for this talk is the expository paper serre duality and appli cations by jun hou fung. They have a reputation for being abstruse and difcult. In the special case that f aon z0, there is an evident map a. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. 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. 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.

Many other classes of spectral sequences are special cases of the grothendieck spectral sequence, for instance the. James milne, section 10 of lectures on etale cohomology. We also study the relationship between the graded centers of r and s. They were introduced by leray in the 1940s at the same time as he introduced sheaves. 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. Introduction to spectral sequences ucb mathematics. We consider cohomology of small categories with coefficients in a natural system in the sense of baues and wirsching. The leray spectral sequence is now a special case of the grothendieck spectral sequence which can be found in most textbooks on homological algebra. Sheaf cohomology on sites and the leray spectral sequence. We define the image and inverse image in the obvious way. Then i claim that there is an adjointness isomorphism of the form.

A grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on. Using this we construct coboundary morphisms between grothendieck spectral sequences associated to objects in a short exact sequence. The conclusions we draw about level set persistent homology corollary4. A relation or correspondence from p to q is a subset a c p x q. An introduction to spectral sequences matt booth december 4, 2016. A coboundary morphism for the grothendieck spectral sequence. For a comprehensive introduction to spectral sequences, see 3. In the case of locally compact spaces and cohomology with compact support, the leray spectral sequence was constructed by j. The leray spectral sequence, the way we proved it in lemma 20. The associated grothendieck spectral sequence is the leray spectral sequence. Successive spectral sequences benjamin matschke forschungsinstitut fur mathematik, eth zuric h benjamin. You can supposedly also look at grothendiecks famous.

We usually draw the rth stage of a spectral sequence in a tabular format with p increasing. 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. X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. The construction of the leray spectral sequence can be generalized to cohomology with support in specified families. For more nice explanations of spectral sequences, see 1 and 2.

For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. On the leray spectral sequence and sheaf cohomology. Finally, it has been noted by a number of authors c. With this background we can study the grothendieck spectral sequence in section 4. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. 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 this section, we construct the leray spectral sequence, an essential tool in modern. Derived functors and sheaf cohomology contemporary. 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. Verify the above claim, making concrete meaning of certain edge maps in grothendieckleray spectral sequences. 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 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.