Second leray spectral sequence of relative hypercohomology. Derived functors and sheaf cohomology contemporary. Some spectral sequences for filtered complexes and applications. H0e of a rank r vector bundle e, the exact sequence. For filtered complexes, we relate the different model structures. 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. Hypercohomology spectra and thomasons descent theorem. The adams spectral sequence lecture 8 april 27, 2010 recall that our goal this week is to prove the following result.
Hypercohomology spectra and thomasons descent theorem stephen a. Question about hypercohomology spectral sequence of a. This can be a serious obstacle to understanding spectral sequences on. A similar spectral sequence for the connective ktheory leads to a cycle class. Introduction we construct a series of spectral sequences for the hypercohomology of a. Grothendieck spectral sequence from the hypercohomology. Notes on the adams spectral sequence harvard university. As usual, hypercohomology and hyperhomology are essentially the same. For a pointed cosimplicial spacex, the author and kan developed a spectral sequence abutting to the homotopy of the total space totx. The mayervietoris sequence of relative hypercohomology lubkin, s. Niveau spectral sequences on singular schemes and failure of generalized gersten conjecture paul balmer abstract.
Some extremely brief notes on the leray spectral sequence greg friedman intro. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. This leads to a physical interpretation of the hypercohomology groups from the perspective of the coulomb branch, as well as an interpretation for the spectral sequences used to compute. We know that if we have a short exact sequence of chain complexes. In this work, we build a spectral sequence in motivic homotopy that is analogous to both the serre spectral sequence in algebraic topology and the leray spectral sequence in algebraic geometry. I do it because i asked what i should do to obtain the cohomology of the complex of global sections in case the only piece or one of the few information i had is that the sheaves are. We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic tspectra, especially the motivic cobordism spectrum. X p of the derived categories of perfect complexes of o x,xmodules with homology supported on the closed point x. Ravi vakil spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. This also has the advantage that it works to define the cup product for hypercohomology on any topos insert future reference here. Some spectral sequences for filtered complexes and. Spectral sequences, closed model cate gories, filtered colimits. The long exact hypercohomology sequence associated to a short exact.
If s is a complex of sheaves on x, the hypercohomology sequence with ep. For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. A users guide to spectral sequences second edition john mccleary vassar college cambridge university press. 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. When the base field k admits resolution of singularities and x is a scheme of finite type over k, we show that voevodskys slice filtration leads to a spectral sequence for mglx whose terms are the motivic cohomology groups of x. When the base field k admits resolution of singularities and x is a scheme of finite type over k, we show that voevodskys slice filtration leads to a spectral sequence for mglx whose terms are the motivic cohomology groups of x defined using the cdh hypercohomology. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. For complexes that vanish for negative indices, the hypercohomology can be defined as the derived functors of h 0 fh 0 h 0 f. The spectral sequence itself is available over regular schemes since it is an incarnation of the slice spectral sequence in motivic homotopy theory. In particular, the ktheory of a scheme x maps to its associated hypercohomology spectrum h et x. Niveau spectral sequences on singular schemes 3 between the idempotent completion of the quotient d p d p.
If p 0 and q 0 can be chosen to be zero, this is called a firstquadrant spectral sequence. Kleine ag degeneration of the hodgetode rham spectral. Surprisingly, this fragmentary bit of information is sufcent to prove lots of things. This sequence converges because each object is a fixed distance away from the edge of the nonzero region. Such outputs are represented by respective data types spectralsequencepage and spectralsequencepagemap. Although spectral sequences often appear formidable to the uninitiated, they provide an invaluable tool to the working topologist, and show their faces throughout algebraic geometry and beyond.
As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of x. The notion of derived functor gives us a sequence of functors rif. More sepci cally, the sinvariant sk of a knot kgives a lower bound on the slice 4ball genus of. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians.
As a motivating example, consider the long exact homology sequence. Introduction we construct a series of spectral sequences for the hypercohomology of. The spectral sequence also converges if, vanishes for all p less than some p 0 and for all q less than some q 0. Whitehead, but he got them just after he wrote the paper whitehead 56. The slice spectral sequence for singular schemes and applications. In these notes we introduce the notion of a spectral sequence and give some basic prop erties. Fix a prime, which we will assume is odd in order to simplify the. But once the initial hurdle of believing in spectral sequences is surmounted, one cannot help but be amazed at their power. We endow the categories of filtered complexes and of bicomplexes of rmodules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasiisomorphism at a certain fixed stage of the associated spectral sequence. Some extremely brief notes on the leray spectral sequence intro. The associated spectral sequence is highly degenerate.
Spectral sequences also crop up when trying to generalize the universal coe cient or. Assuming the correct answer to question 0 is yes, i thought the hypercohomology spectral sequence should read. A second leray spectral sequence of relative hypercohomology is constructed. The grothendieck spectral sequence minicourse on spectral sequences, ut austin, may 2017 richard hughes may 12, 2017 1 preliminaries on derived functors. Of course, in degenerate cases, one hopes that there are techniques for getting at the hypercohomology rapidly. Its proof will concern the next two sections, as will the.
Applying these observations in section 3 to two natural. There is one case in which we do obtain very striking and easily statable results. Spectral sequences daniel murfet notesspectralsequences. A special example of the hypercohomology spectral sequence is the hodgede rham. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians.
The spectral sequence whose existence is asserted in the above theorem is an example of a. Grothendieck spectral sequence from the hypercohomology spectral sequence. We cover essentially only that part of the theory needed in algebraic geometry. This will be useful in showing that a spectral sequence degenerates at e2, just by showing that the displaced spectral sequence e satis. Before we jump into the interesting issue of when the hodgetoderham spectral sequence degenerates, id like to begin by recalling the basic definitions and facts about hypercohomology and spectral sequences especially as the latter pertains to the former. The motivic spectral sequence 7 beilinson asked 4, see also 5, p. Introduction to spectral sequences ucb mathematics. 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. We construct a hierarchy of spectral sequences for a filtered complex under a leftexact functor. We construct a new localglobal spectral sequence for thomasons nonconnective ktheory, generalizing the quillen spectral sequence to possibly nonregular schemes. It plays a somewhat analogue role to the complex conjugate of the hodge spectral sequence for varieties over c and therefore we call it the conjugate spectral sequence following katz ka. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order. This is skew in generality to an earlier one constructed by s. A spectral sequence with rightward orientation is a sequence of tables or pages e p.
Hypercohomology of a complex via cech cohomology mathoverflow. This approach to lie algebroid hypercohomology permits us to look at the spectral sequence thatonecanattach, asshown in1, toanextension ofliealgebroids ahochschildserre spectral sequence, as the grothendieck spectral sequence associated with the right. The atiyahhirzebruch spectral sequence ahss is a type of spectral sequence that generalizes the serre spectral sequence from ordinary cohomology h h\bullet to any generalized eilenbergsteenrod cohomology theory e e\bullet. 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. As a consequence, we establish an isomorphism between. Recall the following general result in the computation of hypercohomology groups. In the most degenerate case, wherein every sheaf in the complex is acyclic, then the hypercohomology of the complex is the cohomology of the spaces of global sections with. This is explained in 4, where we also determine enough about the differentials in the sequence to begin to understand what is happening theorem 5. Over more general bases, we are computing the etale version of weibels homotopy ktheory. As we explain in this paper, a version of this theorem is true in positive characteristic as well. Bis called left exact if it takes short exact sequences ses in a.
X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. M0k degenerates and hnj m0k agrees with h nmc,c when k c. Hypercohomology let c be an abelian category with enough injectives, d another abelian category, and f. Some extremely brief notes on the leray spectral sequence. In this context, one should repeat the naive and the natural ltration of complexes. Homotopy spectral sequences and obstructions springerlink. They were introduced by leray in the 1940s at the same time as he introduced sheaves. The hypercohomology can also be defined using derived categories. A source for the adams spectral sequence and the adamsnovikov spectral sequence is 1, although the point of view im adopting here is from the. The hypercohomology groups of a boundedbelow complex generalize the ordinary cohomology groups on an object. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf. We study the behaviour of supersymmetric ground states in a class of onedimensional n 2 abelian gauged linear sigma models, including theories for which the target space is a complete intersection in projective space, and more generally, models with an interaction term introduced by herbst, hori and page in which the vacua correspond to elements of hypercohomology groups of complexes of. We give the definition for hypercohomology as this is more common. We begin by recalling that a functor between abelian categories f.
The spectral sequence for the hypercohomology of b. Hypercohomology and the associated spectral sequence come up in 4. The adamsnovikov spectral sequence and the homotopy. Mitchell revised october 1996 dedicated to the memory of bob thomason the celebrated lichtenbaumquillen conjectures predict that for a su ciently nice scheme and given prime,theadic algebraic kgroups of x are closely related to the adic etale cohomology groups of x. Since ris a pid, it follows from the universal coe. The grothendieck spectral sequence minicourse on spectral. In this paper,x is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. We know that if we have a short exact sequence of chain complexes 0.
137 1259 1161 178 253 438 459 31 1477 61 165 1296 796 1332 1016 361 1551 861 707 1434 228 1080 202 548 1020 320 1225 603 1043 1481 1027 1006 256 17 112 911 872 952 112 959 823 1225 1321 1157 729