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. Niveau spectral sequences on singular schemes and failure of generalized gersten conjecture paul balmer abstract. The hypercohomology groups of a boundedbelow complex generalize the ordinary cohomology groups on an object. As usual, hypercohomology and hyperhomology are essentially the same. 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 defined using the cdh hypercohomology. Applying these observations in section 3 to two natural. We cover essentially only that part of the theory needed in algebraic geometry.
Bis called left exact if it takes short exact sequences ses in a. For example, for the spectral sequence of a filtered complex, described below, r 0 0, but for the grothendieck spectral sequence, r 0 2. Since ris a pid, it follows from the universal coe. The adams spectral sequence lecture 8 april 27, 2010 recall that our goal this week is to prove the following result. We know that if we have a short exact sequence of chain complexes. 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. As a motivating example, consider the long exact homology sequence. Some spectral sequences for filtered complexes and. Some extremely brief notes on the leray spectral sequence.
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. 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. Homotopy spectral sequences and obstructions springerlink. X p of the derived categories of perfect complexes of o x,xmodules with homology supported on the closed point x. H0e of a rank r vector bundle e, the exact sequence.
Introduction to spectral sequences ucb mathematics. Spectral sequences daniel murfet notesspectralsequences. As we explain in this paper, a version of this theorem is true in positive characteristic as well. Spectral sequences also crop up when trying to generalize the universal coe cient or.
M0k degenerates and hnj m0k agrees with h nmc,c when k c. You can also prove these facts directly without using the spectral sequence which is a good exercise in cohomology of sheaves. In this context, one should repeat the naive and the natural ltration of complexes. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful.
Hypercohomology and the associated spectral sequence come up in 4. 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 grothendieck spectral sequence minicourse on spectral. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of x. This sequence converges because each object is a fixed distance away from the edge of the nonzero region. For filtered complexes, we relate the different model structures. X \to y between topological spaces or more generally the direct image of a morphism of sites, followed by the pushforward. Hypercohomology spectra and thomasons descent theorem. 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. Some spectral sequences for filtered complexes and applications. 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. The spectral sequence for the hypercohomology of b. 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. The grothendieck spectral sequence minicourse on spectral sequences, ut austin, may 2017 richard hughes may 12, 2017 1 preliminaries on derived functors.
They were introduced by leray in the 1940s at the same time as he introduced sheaves. The notion of derived functor gives us a sequence of functors rif. We construct a new localglobal spectral sequence for thomasons nonconnective ktheory, generalizing the quillen spectral sequence to possibly nonregular schemes. The spectral sequence whose existence is asserted in the above theorem is an example of a.
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. Kleine ag degeneration of the hodgetode rham spectral. As a consequence, we establish an isomorphism between. But once the initial hurdle of believing in spectral sequences is surmounted, one cannot help but be amazed at their power. 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. Hypercohomology of a complex via cech cohomology mathoverflow.
Over more general bases, we are computing the etale version of weibels homotopy ktheory. Question about hypercohomology spectral sequence of a. These are two simple conditions that force the leray spectral sequence to converge. If s is a complex of sheaves on x, the hypercohomology sequence with ep. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians. Grothendieck spectral sequence from the hypercohomology. A spectral sequence is a recipe for computing some information about the cohomology of the double complex. Notes on the adams spectral sequence harvard university. Spectral sequences, closed model cate gories, filtered colimits. This can be a serious obstacle to understanding spectral sequences on. Surprisingly, this fragmentary bit of information is sufcent to prove lots of things. Recall the following general result in the computation of hypercohomology groups. We prove that the hodgede rham spectral sequences abutting to the. A special example of the hypercohomology spectral sequence is the hodgede rham.
It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf. Introduction we construct a series of spectral sequences for the hypercohomology of. 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. A spectral sequence with rightward orientation is a sequence of tables or pages e p. 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. The mayervietoris sequence of relative hypercohomology lubkin, s. If p 0 and q 0 can be chosen to be zero, this is called a firstquadrant spectral sequence. We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic tspectra, especially the motivic cobordism spectrum. Homotopy spectral sequences and obstructions homotopy limits.
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. Derived functors and sheaf cohomology contemporary. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians. 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. Before illustrating our package, we mention that we were motivated in part by phrases in the literature, along the lines of.
Fix a prime, which we will assume is odd in order to simplify the. Ravi vakil spectral sequences are a powerful bookkeeping tool for proving things involving complicated commutative diagrams. Its proof will concern the next two sections, as will the. 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. Whitehead, but he got them just after he wrote the paper whitehead 56. 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. In these notes we introduce the notion of a spectral sequence and give some basic prop erties. 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. The spectral sequence also converges if, vanishes for all p less than some p 0 and for all q less than some q 0. A second leray spectral sequence of relative hypercohomology is constructed. Some extremely brief notes on the leray spectral sequence greg friedman intro. We give the definition for hypercohomology as this is more common. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order.
The adamsnovikov spectral sequence and the homotopy. Assuming the correct answer to question 0 is yes, i thought the hypercohomology spectral sequence should read. Some extremely brief notes on the leray spectral sequence intro. The hypercohomology can also be defined using derived categories. The universal complex orientation of the complex bordism spectrum mu determines. 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.
Hypercohomology let c be an abelian category with enough injectives, d another abelian category, and f. For a pointed cosimplicial spacex, the author and kan developed a spectral sequence abutting to the homotopy of the total space totx. Of course, in degenerate cases, one hopes that there are techniques for getting at the hypercohomology rapidly. Second leray spectral sequence of relative hypercohomology. This will be useful in showing that a spectral sequence degenerates at e2, just by showing that the displaced spectral sequence e satis. This also has the advantage that it works to define the cup product for hypercohomology on any topos insert future reference here.
Grothendieck spectral sequence from the hypercohomology spectral sequence. More sepci cally, the sinvariant sk of a knot kgives a lower bound on the slice 4ball genus of. The motivic spectral sequence 7 beilinson asked 4, see also 5, p. We begin by recalling that a functor between abelian categories f. In this paper,x is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Such outputs are represented by respective data types spectralsequencepage and spectralsequencepagemap. 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 construct a hierarchy of spectral sequences for a filtered complex under a leftexact functor. 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. The long exact hypercohomology sequence associated to a short exact. A users guide to spectral sequences second edition john mccleary vassar college cambridge university press.
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