A spectral sequence is a recipe for computing some information about the cohomology of the double complex. A spectral sequence with rightward orientation is a sequence of tables or pages e p. These were subsequently developed by many others, 5 each separately becoming an important tool in homological algebra. X, so the second hypercohomology spectral sequence now reads ei,j 1 h jx. Fibrations and the lerayserre spectral sequence 4 3. Galois connections and the leray spectral sequence 203 1 cn is the module of functions on chains ao sheaf cohomology on sites. 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.
Originally, sheaf cohomology was constructed as a cohomology theory on the category of sheaves on topological spaces. For a comprehensive introduction to spectral sequences, see 3. Actually i guess that these cases of the leray spectral sequence may be replaced by more direct arguments. We cover essentially only that part of the theory needed in algebraic geometry. Talking about weight ltration in the cohomology of complex algebraic variety one obviously uses deligne construction 5, 6. Dec 22, 2018 there is a compariso n theorem between the leray spectral sequence and the spectral sequence as sociated to the blochogus complex. Fibrations and the leray serre spectral sequence 4 3. Sheaf cohomology on sites and the leray spectral sequence. On the leray spectral sequence and sheaf cohomology. Vector bundle extensions, sheaf cohomology, and the. Introduction to spectral sequences ucb mathematics. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense.
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. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of. By a spectral sequence in an abelian category a, we mean a collection of bi. Today we would divide his discoveries into three parts. Therefore it is rarely the case that leray s result is quoted in its original form.
The leray spectral sequence, the way we proved it in lemma. 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. The cohomological serre spectral sequence associated to a serre fibration follows from the lerre spectral sequence and in algebraic topology there. This is a grothendieck spectral sequence, by taking categories a abx, b aby and c abgp.
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. Or at least for some good sheaves like the sheaf of smooth functions or holomorphic functions. Leray s work of this period proved seminal to the development of spectral sequences and sheaves. Vector bundle extensions, sheaf cohomology, and the heterotic. In other words, sheaf cohomology of the holomorphic objects. At the time of leray s work, neither of the two concepts involved spectral sequence, sheaf cohomology had reached anything like a definitive state. 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. 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.
Surprisingly, this fragmentary bit of information is sufcent to prove lots of things. X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. Galois connections and the leray spectral sequence 195 2 pzu. Lerayhirsch theorem for dolbeault cohomology mathoverflow. The central work for the study of sheaf cohomology is grothendiecks 1957 tohoku paper. Higher derived functors cohomology from a new viewpoint. Lerays work of this period proved seminal to the development of spectral sequences and sheaves. 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 in order to compute sheaf cohomology, they were used. An introduction to spectral sequences matt booth december 4, 2016. These are two simple conditions that force the leray spectral sequence to converge. Notes on spectral sequences amar abdelmoubine henni1 abstract. Sheaves, sheaf cohomology, and spectral sequences were invented by jean leray at the prisonerofwar camp oflag xviia in.
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. For xed r, er pq is called the rth page of the spectral sequence 2fixed maps d r pq. For more nice explanations of spectral sequences, see 1 and 2.
In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of the necessary vector bundle extensions. An ath stage spectral cohomological sequence consists of the following data. There he took up algebraic topology, and the result was a spectacular. 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. Leray spectral sequence encyclopedia of mathematics. There is a compariso n theorem between the leray spectral sequence and the spectral sequence as sociated to the blochogus complex. We compute the cohomology of k in two ways by means of the two spectral sequences e. 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. Does the same formula apply for sheaf cohomology in general.
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, the way we proved it in lemma 20. 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. For the algebraic and categorical significance of cohomology see one of the.
More precisely, the second term of the leray spectral sequence has the form. Galois connections and the leray spectral sequence. Aug 25, 2011 sheaf cohomology, the leray spectral sequence. I am asking this because i know that lerayhirsch theorem is a very particular case of leray spectral. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The grothendieck spectral sequence minicourse on spectral.
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. Derived functors and sheaf cohomology contemporary. Our goal is to prove that the leray spectral sequence associated to a. Spectral sequences are a powerful technical tool in algebraic topology and algebraic geometry to compute homologycocomology, and homotopy. In chapter 3, we redevelop the cohomology theory of a space with coe cients in a sheaf, including lerays classical spectral sequences. The most important structure determined by a diagram d is its cohomology. 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. By a spectral sequence in an abelian category a, we mean a collection of bigraded objects e. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. A homological spectral sequence in some abelian category ais the following data. 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.
107 1219 467 68 1469 859 1302 1286 1345 320 767 1109 1248 128 280 1047 510 761 1029 525 462 420 601 1484 1219 1399 271 1056 1225 1358 879 49 1099 863 714 993 1149 1070 639 787 893 623 521 811 1168 1145