Derived functors and sheaf cohomology contemporary. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. 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. X, so the second hypercohomology spectral sequence now reads ei,j 1 h jx.
Does the same formula apply for sheaf cohomology in general. Surprisingly, this fragmentary bit of information is sufcent to prove lots of things. An introduction to spectral sequences matt booth december 4, 2016. The leray spectral sequence, the way we proved it in lemma. 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. Leray in order to compute sheaf cohomology, they were used. An ath stage spectral cohomological sequence consists of the following data. For a comprehensive introduction to spectral sequences, see 3. I am asking this because i know that lerayhirsch theorem is a very particular case of leray spectral. X \to y followed by the pushforward x x \to to the point. The leray spectral sequence, the way we proved it in lemma 20. Fibrations and the leray serre spectral sequence 4 3. A spectral sequence with rightward orientation is a sequence of tables or pages e p.
Spectral sequences are a powerful technical tool in algebraic topology and algebraic geometry to compute homologycocomology, and homotopy. The associated grothendieck spectral sequence is the leray spectral sequence of a sheaf f. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. There is a compariso n theorem between the leray spectral sequence and the spectral sequence as sociated to the blochogus complex. Sheaf cohomology on sites and the leray spectral sequence. For the algebraic and categorical significance of cohomology see one of the. Here eis the spectral sequnce we get by rst taking cohomology in the rst index, and e0is the spectral sequence we get by rst taking cohomology in the second index. The physical properties of these vacua were discussed previously. We explain the structure of this complex for a smooth scheme over a complete discrete valuation ring of mixed characteristic 0,p and. Pn the spectral sequence degenerates and we conclude that hodd dr p n 0 and h2i dr p n is 1dimensional for 0 i n. Lerays work of this period proved seminal to the development of spectral sequences and sheaves. Much of the material in these notes parallels that in, for example. We cover essentially only that part of the theory needed in algebraic geometry.
By a spectral sequence in an abelian category a, we mean a collection of bigraded objects e. Originally, sheaf cohomology was constructed as a cohomology theory on the category of sheaves on topological spaces. In particular, i study explicitly how the theory works for the. Aug 25, 2011 sheaf cohomology, the leray spectral sequence. For xed r, er pq is called the rth page of the spectral sequence 2fixed maps d r pq. On the leray spectral sequence and sheaf cohomology. X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. From the exact couple formulation of spectral sequences, we introduce a special case of the leray serre spectral sequence and use it to compute h cpn.
We compute the cohomology of k in two ways by means of the two spectral sequences e. A spectral sequence is a recipe for computing some information about the cohomology of the double complex. 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. 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. Galois connections and the leray spectral sequence 203 1 cn is the module of functions on chains ao sheaf cohomology on sites. More precisely, the second term of the leray spectral sequence has the form.
The origins of sheaf theory, sheaf cohomology, and spectral sequences haynes miller february 23, 2000 jean leray november 7, 1906november 10, 1998 was con ned to an o cers prison camp \o ag in austria for the whole of world war ii. Our goal is to prove that the leray spectral sequence associated to a. Vector bundle extensions, sheaf cohomology, and the. There he took up algebraic topology, and the result was a spectacular. The cohomological serre spectral sequence associated to a serre fibration follows from the lerre spectral sequence and in algebraic topology there. Sheaves, sheaf cohomology, and spectral sequences were invented by jean leray at the prisonerofwar camp oflag xviia in. 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. Introduction to spectral sequences ucb mathematics.
The most important structure determined by a diagram d is its cohomology. At the time of leray s work, neither of the two concepts involved spectral sequence, sheaf cohomology had reached anything like a definitive state. I am asking this because i know that leray hirsch theorem is a very particular case of leray spectral sequence which is valid for sheaf cohomology in general. Dec 22, 2018 there is a compariso n theorem between the leray spectral sequence and the spectral sequence as sociated to the blochogus complex.
For more nice explanations of spectral sequences, see 1 and 2. Therefore it is rarely the case that leray s result is quoted in its original form. We usually draw the rth stage of a spectral sequence in a tabular format with. In chapter 3, we redevelop the cohomology theory of a space with coe cients in a sheaf, including lerays classical spectral sequences. Actually i guess that these cases of the leray spectral sequence may be replaced by more direct arguments.
Or at least for some good sheaves like the sheaf of smooth functions or holomorphic functions. Higher derived functors cohomology from a new viewpoint. A homological spectral sequence in some abelian category ais the following data. Leray s work of this period proved seminal to the development of spectral sequences and sheaves. These were subsequently developed by many others, 5 each separately becoming an important tool in homological algebra. Lerayhirsch theorem for dolbeault cohomology mathoverflow. The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences.
If x happens to be an eilenbergmac lane object of the form ka, n for some sheaf of abelian groups a on g and olsln, then ru,x is the sheaf associated. Galois connections and the leray spectral sequence 195 2 pzu. Galois connections and the leray spectral sequence. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is grothendiecks 1957 tohoku paper. Vector bundle extensions, sheaf cohomology, and the heterotic. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of the necessary vector bundle extensions. Group cohomology lecture notes university of washington. The cohomological serre spectral sequence associated to a serre fibration follows from the lerre spectral sequence and in algebraic topology there are lots of calculations with the serre spectral sequence. These are two simple conditions that force the leray spectral sequence to converge. By a spectral sequence in an abelian category a, we mean a collection of bi.
This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp. In other words, sheaf cohomology of the holomorphic objects. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf theory, and also from algebra. Today we would divide his discoveries into three parts. Fibrations and the lerayserre spectral sequence 4 3. A spectral sequence connecting the cohomology with values in a sheaf of abelian groups on a topological space with the cohomology of its direct images under a continuous mapping. 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. Leray spectral sequence encyclopedia of mathematics. The grothendieck spectral sequence minicourse on spectral. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of.
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