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. It is constructed from the following filtration of the total space e. A condition of this sort does not seem to hold in my setting, so perhaps there is no spectral sequence of the type i want. Alexandre grothendieck, on certain aspects of homological algebra, trans.
The classical grothendieck spectral sequence gr, thm 2. The grothendieck spectral sequence is a tool for actually computing some thing for the composition of true derived functors, where we have r g f. Thus, analogous to, there is a weak ghomotopy equivalence 16 x hocolimpx. Cbe additive functors between abelian categories where a,bhave enough injectives and cis complete. The respective e 2terms are a priori seen to be isomorphic. When one applies spectral sequences successively it fairly often happens that there are several ways to do this.
We will see why grothendieck wrote to serre on february 18, 1955. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians. Using this we construct coboundary morphisms between grothendieck spectral sequences associated to objects. A coboundary morphism for the grothendieck spectral sequence.
In mathematics, in the field of homological algebra, the grothendieck spectral sequence, introduced by alexander grothendieck in his tohoku paper, is a spectral sequence that computes the derived functors of the composition of two functors. The grothendieck spectral sequence is a special case of this, arising as the spectral sequence of a double complex. These are two simple conditions that force the leray spectral sequence to converge. Lecture notes in mathematics a collection of informal reports and seminars edited by a. Serre spectral sequences, the adamsnovikov spectral sequence following the chromatic spectral sequence, successive grothendieck spectral sequences, and successive eilenbergmoore spectral sequences. With this background we can study the grothendieck spectral sequence in section 4. On the yoga of motives and the degeneration of leray spectral sequence. The purpose of this paper is to prove the existence of the grothendieck spectral sequence gss, which is the chain rule for derived functors. The grothendieck spectral sequence compares the composition of the derived functors with the derived functors of the composition. The grothendieck spectral sequence minicourse on spectral. 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. Given a topological space x with known homology groups, we can consider the eilenbergmoore spectral sequence to compute h x. K cat, we construct a spectral sequence abutting to the cohomology of the grothendieck construction of l in terms of the cohomology of k.
Let g0 be a connected real semisimple lie group with. 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. Let a be a chain complex of objects in an abelian category c. An essay on spectral sequences university of washington. A b is called left exact if it takes short exact sequences ses in a. 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. Grothendieck let x be a topological space, withfx asheafofabeliangroups. If you dont mind violating grothendiecks stated wishes, you can now read this translation.
For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. Jinhyun park, personal notes on grothendieck spectral sequence pdf. We construct a generalized grothendieck spectral sequence for com puting the derived functors of a composite functor t. 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. This has been done in the past i, when x is a gwcomplex, by considering the filtration ofx by its skeletons. We begin by recalling that a functor between abelian categories f. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. It is a bit more purely categorical than what weve. A grothendieck spectral sequence is a spectral sequence that. The spectral sequence whose existence is asserted in the above theorem is an example of a. The grothendieck spectral sequence in algebraic geometry george torres abstract. I give a construction which makes no such assumption on x.
However, there were certain difficulties in making the presentation of the fields medal since the 1966 international congress of mathematicians at which the presentation was to. 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. The conley index and spectral sequences bartlomiejczyk, piotr, topological methods in nonlinear analysis, 2005. Bis called left exact if it takes short exact sequences ses in 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. Cohomology of the grothendieck construction springerlink. That is, suppose c bt as are additive functors between abelian categories with enough projectives, and tp is sacyclic for projective. Jun 14, 2005 pdf file 640 kb article info and citation. Let a,b and c be abelian categories, such that a and b have enough injectives and c is cocomplete i. C 00 be left exact additive functors of abelian categories, and let c. Given the above setup, suppose further that for any injective object i of a, rrfgi 0 for all r0. The key to proving the existence of the leray spectral sequence is the following lemma. On the way, we will develop some necessary tools in the homological algebra of bicomplexes in abelian categories.
On degeneration of the spectral sequence for the composition of zuckerman functors dragan mili. The grothendieck spectral sequence minicourse on spectral sequences, ut austin, may 2017 richard hughes may 12, 2017 1 preliminaries on derived functors. Vakils spectral sequence proofs of the snake lemma despite the typo, fixed in the version of this article that appears as 1. We usually draw the rth stage of a spectral sequence in a tabular format with p increasing. But we view the serre spectral sequence as a special case of the spectral sequence of a ltered space or ltered chain. In mathematics, in the field of homological algebra, the grothendieck spectral sequence. 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. For this lecture, were going to discuss the grothendieck spectral sequence for composition of derived functors. Using this we construct coboundary morphisms between grothendieck spectral sequences associated to objects in a short exact sequence. The language of derived categories provides a very elegant albeit quite abstract way of formulating the existence and convergence of grothendiecks spectral sequence among other things.
But we view the serre spectral sequence as a special case of the spectral sequence of a ltered space or ltered chain complex, which has many other important cases. Homology of small categories, spectral sequences 327 x. Then for any object a 2athere is a spectral sequence starting on. Mar 21, 2006 we consider cohomology of small categories with coefficients in a natural system in the sense of baues and wirsching. The latter approach is the best choice when trying to prove general statements, but.
The adams spectral sequence appeared in the 60s trying to determine the homotopy groups of some topological spaces. This is just the grothendieck spectral sequence derived categories, lemma. Clarificatons and corrections for grothendieck duality and base change 3 local isomorphism d which is a variant of 2. The tohoko paper referred to in this citation is about abelian categories, sheaves of modules, resolutions, derived functors, and the grothendieck spectral sequence. The double complex we will use in question is called a cartaneilenberg resolution. We consider cohomology of small categories with coefficients in a natural system in the sense of baues and wirsching. The lerayserre spectral sequence is a spectral sequence with.
An essay on spectral sequences may 15, 2016 this is primarily an essay on the serre spectral sequence for the homologycohomology of a bration. Introduction to spectral sequences ucb mathematics. I realized that i forgot that the usual grothendieck spectral sequence has a condition that the first functor maps injectives to acyclics for the second functor. Its about abelian categories, sheaves of modules, resolutions, derived functors, and the grothendieck spectral sequence. 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 an object in k. Grothendieck is also well known for his \tohoko paper in which he talks about numerous mathematical concepts including abelian categories, sheaves of modules, resolutions, derived functors, and the grothendieck spectral sequence. For more nice explanations of spectral sequences, see 1 and 2. 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. 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. Nov, 2014 the tohoko paper referred to in this citation is about abelian categories, sheaves of modules, resolutions, derived functors, and the grothendieck spectral sequence. In fact he was even rewarded the fields medal for the advances in algebraic topology he showed in that paper. Furthermore, this spectral sequence is functorial in a. For a comprehensive introduction to spectral sequences, see 3. A grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on categories of chain complexes.
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