The spectral sequence whose existence is asserted in the above theorem is an example of a. Surprisingly, this fragmentary bit of information is sufcent to prove lots of things. The leray spectral sequence, the way we proved it in lemma 20. Most spectral sequences we will encounter will be biregular. More on the approximation of e 1by e rand degeneration of a spectral sequence 4. The serre spectral sequence is a way to compute the cohomology of y in terms of the cohomology of xand f. Let ebe a spectral sequence, and suppose that for some r. In mathematics, the leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by jean leray. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. The mayervietoris spectral sequence not yet included 5. A spectral sequence with rightward orientation is a sequence of tables or pages e p. Introduction the frolicher spectral sequence consists of vector spaces efcm that are assigned to each complex manifold m.
In this scheme the sun is designated a type g2 see appendix i, page 22. Sunder institute of mathematical sciences madras 6001 india july 31, 2000. Eilenbergmoore spectral sequences following the geometric constructions due to larry smith and luke hodgkin, without any. Rham models, and indicates an interplay between models and spectral. Last time we defined cohomology in homotopy type theory. Like most explanations of spectral sequences, it focuses on a particular spectral sequence. The groups ph are the subgroups of the group h, induced bythe filtration of t. The bockstein spectral sequence not yet included 5. In the presence of additional conditions that depend on data internal to the spectral sequence, they lead to satisfactory convergence and. In mathematics, the frolicher spectral sequence often misspelled as frohlicher is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for kahler manifolds. The arrangement of spectral classes based on temperature is known as the spectral sequence. Fibrations and the lerayserre spectral sequence 4 3. A general description of the terms in the frolicher.
I have heard others make similar complaints about the proliferation of subscripts and superscripts. Pdf another proof of the persistence of serre symmetry in the. Xby means of he kunneth theorem, namely that there is a split short exact sequence 0. In mathematics, the frolicher spectral sequence often misspelled as frohlicher is a tool in the theory of complex manifolds, for expressing the potential failure. Pdf on the degeneration of the fr\\olicher spectral. Pdf the frolicher spectral sequence of certain solvmanifolds. If is a ring, then every term is a bigraded ring, is differentiation in, and the multiplication in is induced by that in. The differential of the spectral sequence coincides with the transgression this homology lerayserre spectral sequence is dual to the cohomology lerayserre spectral sequence, with differentials of bidegree, for which. For instance, if you have a space filtered by subspaces, or a chain complex filtered by subcomplexes, or some arbitrary sequence of composable maps in a triangulated category. The first part, in which i attempted to motivate the notion of spectral sequence, and constructed the basic example that well be using, is at the ncategory cafe here. A cohomologically graded spectral sequence e er consists of zbigraded rmodules er ep,q rr. The arrangement of a stars position in the temperature classification system o, b, a, f, g, k, m, ranging from hot to cool. The general idea is that one starts with interesting groups ep. On the other hand, if n 1 mod 4 there is no pair of everywhere independent vector elds on sn.
A plot of the intensity of light at different frequencies. The differentials on each page of the frolicher spectral sequence can be explicitly. Spectral sequences will be introduced in this seminar, but let us mention for the moment that this is a tool that tells us how to compute the cohomologyontherightside,derhamcohomologyh. Pdf serres duality theorem implies a symmetry between the hodge numbers, hp,q. Thus the spectral sequence would somehow codify a relationship between these two families of. The early spectral classification system was based on the appearance of the spectra, but the. Traditionally this was proven frolicher 55 by way of harmonic differential forms. The subject of spectral sequences is elementary, but the notion of the spectral sequence of a double complex involves so many objects and indices that it seems at first repulsive. Complex manifolds, compact nilmanifolds, dolbeault cohomology, frolicher spectral sequence. Vakils spectral sequence proofs of the snake lemma despite the typo, fixed in the version of this article that appears as 1. The fr\ olicher spectral sequence can be arbitrarily non degenerate.
A general description of the terms in the frijlicher spectral sequence luis a. Alfred frolicher, relations between the cohomology groups of dolbeault and topological invariants, proceedings of the national academy. The spectral sequence, arranged from hottest to coldest, is as follows. We say that a spectral sequence degenerates at er0 if the di. The spectral sequence associated to a ltred di erential module 4 3. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians. On the spectral sequence associated to a multicomplex. Michael boardman march 1999 abstract convergence criteria for spectral sequences are developed that apply more widely than the traditional concepts. A general description of the terms in the frijlicher. We use the description to exhibit examples of compact complex nilmanifolds of complex dimension 3 for which the frolicher spectral sequence does not. Whiteheads exact sequence a nice application of exact couples 5. Some calculations with the frolicher spectral sequence.
The frolicher spectral sequence of certain solvmanifolds article pdf available in journal of geometric analysis 251 october 2012 with 31 reads how we measure reads. The temperature classification system for stars ranging from hot o and b to cool k and m. Thegroup hnis nowbydefinition just thenthderhamgroup, whichis filtered asfollows. The main content of this short note is to give explicit formulas for. In particular on a compact kahler manifold x the frolicher spectral. Continuing, if n 8e 1 7 mod 8, then there are 7 independent vector elds on sn.
To describe the spectral sequence associated to such an object we assume, for the moment. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \spectral sequence strike fear into the hearts of many hardened mathematicians. If the sequence eventually stabilizes on both ends or many variants of weaker hypotheses, the spectral sequence detects the difference between the limit and the colimit. Or the distribution of wavelengths and frequencies spectral sequence. The frolicher spectral sequence consists of vector spaces ejm that are assigned to each complex manifold m.
Dolbeault cohomology and frolicher spectral sequence. 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. A general description of the terms in the frolicher spectral sequence. A spectral sequence is a recipe for computing some information about the cohomology of the double complex. Roughly speaking, a spectral sequence is all of the data in the above construction. We already know how to do this in case of the trivial bre sequence f. Spectral sequences in this document we gather together facts and calculations about spectral sequences. We will present the example in an elementary way without going into the general theory of nilmanifolds with leftinvariant complex structure.
A spectral sequence is set up, the degeneration of which would give the results of hodge theory and. We go over the basic tools necessary to work with the frolicher spectral sequence, and do some example computations with complex nilmanifolds and. Frobenius and the hodge spectral sequence arthur ogus department of mathematics, university of california, berkeley, california 94720 emailogus math. It follows from 1 and 2 that the entry of every subsequent page of the spectral sequence is also zero. It is usually seen nowadays as a special case of the grothendieck spectral sequence definition. In deligneillusie 87 this is proven instead by reduction to positive characteristic. The spectral sequence was studied by boardman, and by hurtubise, who noted that the differentials of the spectral sequence differ from the maps induced by the higher differentials of the multicomplex. The frolicher spectral sequence is the spectral sequence of a double complex of.
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