Holmstrom 4 Memo notes Grayson

Memo notes from Grayson: The Motivic Spectral Sequence

Introduction

Atiyah-Hirzeburch spectral sequence (AHSS) relating topological K-theory to singular cohomology. Want motivic version. Some results:

• Bloch-Lichtenbaum spectral sequence for the spectrum of a field
• Friedlander-Suslin and Levine extensions of the above, to the global case for a smooth variety over a field
• Goodwillie-Lichtenbaum idea involving tuples of commuting automorphisms, this gives something for affine regular noetherian schemes
• Voevodsky’s approach via slice filtration
• “Recent preprint” of Levine clarifying which formal properties one needs

Applications to computing algebraic K-theory. Main open question: How handle a general noetherian regular scheme.

Algebraic K-theory and cohomology

Riemann-Roch for a complete nonsingular alg curve relates the degree of a line bundle to the dimension of its space of global sections, or rather its Euler characteristic. The Euler char is additive on short exact seqs of locally free coherent sheaves on $X$. The universal target for additive functions is the Grothendieck ring $K_0(X)$ of locally free coherent sheaves. Notation: $K'_0(X)$ for the Grothendieck group on all coherent sheaves (it is a module over the former, and for $X$ nonsingular quasiprojective, they agree). The group $K'_0(X)$ has a filtration whose $i-th$ member $F^i$ is generated by coherent sheaves whose support has codim at least $i$. The ring $K_0(X)$ has a more complicated filtration, the $\gamma$-filtration (formerly the $\lambda$-filtration), defined using exterior powers. When $X$ is nonsingular quasiprojective, the map from $K_0$ to $K'_0$ respects the filtrations, and induces an isomorphism on $Gr^i \otimes \mathbb{Q}$ (why do we have to tensor with the rationals here?).

Grothendieck used $K_0$ and the Chow ring to extend RR to nonsingular varieties of any dimension (the GRR thm). A consequence of GRR is that for $X$ nonsingular quasiprojective, the algebraic cycles of codim at least $i$ account for all the classes of coherent sheaves of codim at least $i$, up to torsion. (For a subvar $Z$, consider the coherent sheaf \mathcal(O)_Z; it gives a class in $\mathcal(K'_0(X)$. Get a map $CH^iX \to Gr^i K'_0(X)$ which is an isomorphism after tensoring with $\mathbb{Q}$. Proof uses Chern classes to define an inverse map. Grothendieck’s Chern character is an isomorphism

Quillen defined higher algebraic K-groups $K_n(X)$ and $K'_n(X)$. Again, have map as above, which is an isomorphism for $X$ nonsingular quasiprojective. Also ahve filtrations, and $F^i_{\gamma} K_n(X)$ lands in $F^{i-n} K'_n(X)$, but the filtrations may disagree rationally. One reason for this: The higher K-groups harbour elements whose Chern characters involve higher cohomology groups. E.g. a field, for which the etale cohomology dim can be bigger than $0$.

Question: Analogue of the above Chern character isomorphism?

Topological K-theory and cohomology

AHSS, coming from the skeletal filtration of the finite cell complex studied. Let $\mathbb{C}(X^{top})$ be the topological ring of continuous functions on $X$. There is a way to take the topology of a ring into account when defining K-theory. Get topological K-groups $K_n(X^{top}) = K_n(\mathbb{C}(X^{top})) = \pi_n(K(X^{top})$ for a certain infinite loop space $K(X^{top})$. Computations for the one-point space, Bott periodicity, the $\beta$ element, the $\Omega$-spectrum $BU$. Have a homotopy equivalence between the mapping space $K(*^{top})^X$ and $K(X^{top})$, which implies representability for topological K-theory.

Thm: Have the following isomorphism:

This follows from the AHSS. The AHSS is constructed either through the skeletal filtration of $X$, or from the Postnikov tower of $BU$. Long exact seeq ass to a cofibration sequence and an $\Omega$-spectrum. Details of the AHSS. The odd-numbered rows (and even-numbered differentials) are all zero. Reindexing to remove all these zeros, gives the AHSS:

Here $\mathbb{Z}(-q)$ ($=K_{-2q}(*^{top})$, I think) is simply the group $\mathbb{Z}$, destined to for the i-th stage of the weight filtration. The initial term would have been $H^p(X, \mathbb{Z}(-q/2))$ without reindexing.

The motivic spectral sequence

Let $X$ be a nonsingular algebraic variety, or more generally, a regular scheme. Consider a f.g. ring $A$ and a prime number $\ell$. Quillen asked if there could be a spectral sequence of the form

This would degenerate in case $A$ is a ring of integers in a number field and either $\ell$ is odd or $A$ is totally imaginary. Beilinson asked (around 1982) if there could be an integral version:

where the initial term would be so called motivic cohomology. Comparing with the Chern character formula, one would expect $H^{2i}(X, i) = CH^i(X)$. Beilinson also suggested that this should be defined by the Zariski hypercohomology of a certain complex, concentrated in the “right” degrees. (Lichtenbaum suggested that this complex should be derived from a complex in the etale topology?).

Since odd-numbered rows are zero anyway one can reindex in exactly the same way as in the topological setting (same indices in the spectral sequence).

Filtrations

Spectral sequences can be constructed from (in increased order of generality) a homological bicomplex, a filtered chain complex, or an exact couple. Nice discussion of these things, omitted here. Filtration on a spectrum, layers hopefully coning from complexes of abelian groups. Remark of Goodwillie, saying that tensor product over a commutative ring makes the K-theory spectrum of the ring into a ring spectrum. The zero’th layer will also be a ring spectrum, and the higher layers will be modules over it. Consequence: Our search for a filtration can be limited to those compatible with products (?)

Commuting automorphisms

Goodwillie-Lichtenbaum idea: tuples of commuting automorphisms. Get a spectral sequence starting at the cohomology of certain complexes $\mathbb{Z}^{\oplus}$ and converging to K-theory, for any spectrum of a noetherian regular ring. Cancellation theorem for the space of stable IMs between two projective modules.

See Homotopy invariance for a description of a standard way of converting a functor into a homotopy invariant one.

For schemes $X,Y$, consider the exact category $P(X,Y)$ of those coherent sheaves over $X \times Y$ which are flat over $X$ and whose support is finite over $X$. Various interpretations of special cases of this.

Def of the algebraic circle $S$. For any functor $F$ from schemes to abelian groups, can define $F(S)$ by the “obvious” decomposition $F(G_m) = F(Spec (\mathbb{Z})) \oplus F(S)$. More nice details one this, and on iterated smash products of $S$.

More details on commuting automorpisms.

Cancellation and comparison with motivic cohomology

Description of cancellation thm. It is used by Suslin to prove that $\mathbb{Z}^{\oplus}(t)(X) \to \mathbb{Z}(t)(X)$ is an qis locally on a smooth variety $X$.

Def of correspondences. Explicit representation of correspondences.

More details related to fundamental stuff about motivic cohomology, omitted.

Higher Chow groups and a motivic spectral sequence

Brief, but very nice, intro to higher Chow groups.

Sketch of motivic spectral sequence for the spectrum of a field, following Bloch and Lichtenbaum. Def: A cube in any category is a diagram indexed by the partially ordered set of subsets of $\{1, 2, \ldots n \}$. Def of homotopy fiber of a cube of spaces or spectra. Given a set of subschemes of a scheme, their intersections form a cube of schemes. Applying $K$ gives a cube of spectra. When the subschemes are closed, get multirelative K-theory $K(X; Y_1, \ldots, Y_n)$ by taking the homotopy fiber of this cube. Also version with supports. Can use this to get the following theorem:

Thm: If $X = Spec(k)$ for a field $k$, then there is a motivic spectral sequence of the following form:

Gillet and Soule showed that the filtration of the abutment provided by this spectral sequence is the $\gamma$-filtration, and the spectral sequence degenerates rationally.

Extension to the global case

Will sketch ideas of Suslin and Friedlander for the global case, e.g. for a nonsingular variety. First step: Show that the Bloch-Lichtenbaum spectral sequence arises from a filtration of the K-theory spectrum, and that the successive quotients are Eilenberg-MacLane spectra. For the subsequent steps, Levine has a different approach, see below. The paper by Suslin and Friedlander is “long, with many foundational matters spelled out in detail”.

Some ingredients in Friedlander-Suslin:

Relative K-groups. For spectra, homotopy fibers and homotopy cofibers (mapping cones) amount to the same thing (via a degree shift). Cubes. Freudenthal suspension thm, Blakers-Massey excision thm. A homotopy invariance result and a result of Landsburg gives the motivic spectral sequence for affine space. A moving lemma of Suslin. Voevodsky’s globalization thm for pretheories. Get the case of smooth affine semilocal variety over a field. Globalization techniques of Brown and Gersten involving hypercohomology of sheaves of spaces leads to the case of a smooth affine variety.

Levine’s approach:

Instead of transfer maps, use G-theory so that singular varieties can be handled. A localization thm, using a very general moving lemma. The final result is more general than Friedlander-Suslin: it gives a motivic spectral sequence for any smooth scheme over a regular noetherian scheme of dimension 1.

The slice filtration

Half-page intro to motivic homotopy theory. Voevodsky’s idea: Build a filtration of the motivic spectrum $KGL$ such that each $F^p KGL$ is also a motivic spectrum. The filtration is to be compatible with products. The cofiber $F^{0/1} KGL$ is to be equivalent to the motivic analogue $H_{\mathbb{Z}}$ of the EilenbergMacLand spectrum. Hence each quotient $F^{p/p+1} KGL$ will be an $H_{\mathbb{Z}}$-module. Modulo convergence problems, filtrations of motivic spectra yield spectral sequences as before.

The slice filtration (ref: Voevodsky: Open problems … I) of a motivic spectrum $Y$. The spectrum $F^q Y$ is the part of $Y$ that can be constructed from $(2q,q)$-fold suspensions of suspension spectra of smooth varieties. Smashing two motivic spheres amounts to adding the indices, so the filtration is compatible with any multiplication on $Y$. The layer $s_p(Y) = F^{p/p+1} Y$ is called the $p$-th slice of $Y$. Voevodsky’s proposes a number of related conjectures. Conjecture 1: The slice filtration of $H_{\mathbb{Z}}$ is trivial, i.e. $s_0(H_{\mathbb{Z}})$ includes the whole thing. The main Conjecture 10 says: $s_0(1) = H_{\mathbb{Z}}$ where $1$ denotes the sphere spectrum. A corollary would be that the slices of any motivic spectrum are modules over $H_{\mathbb{Z}}$. Conjecture 7: $s_0(KGL) = H_{\mathbb{Z}}$. Since $KGL$ is $(2,1)$-periodic, this implies $s_q(KGL)= \Sigma^{2q,q} H_{\mathbb{Z}}$, thereby identifying the $E_2$ term of the spectral sequence, giving the desired motivic spectral sequence. In Voevodsky, it is shown that Conjecture 7 follows from Conjecture 10 together with a seemingly simpler conjecture. In another paper, he also proves Conjecture 10 over fields of characteristic zero.

Filtrations for general cohomology theories

Sketch of some ideas from Levine: The homotopy coniveau filtration.

Can define the homotopy coniveau filtration for a contravariant functor $E$ from the category of smooth schemes over a noetherian separated scheme $S$ of finite dimension to the category of spectra. We’ll assume here that $S$ is the $Spec$ of an infinite field. The filtration will be of the form

for any smooth variety $X$.

Can define version of $E$ with supports…

Def of filtration.

Levine introduces some axioms for $E$. Axiom 1 is homotopy invariance. It ensures that $E(X) = F^0 E(X)$ is an equivalence, so that the filtration is actually a filtration of $E(X)$. Taking homotopy groups and exact couples leads to a spectral sequence. The homotopy groups of $E$ should be bounded below if the spectral sequence is to converge.

The terms $F^n E$ and the layers $F^{n/n+1} E$ are contravariant functors from the category of smooth varieties (and equidimensional maps) to the category of spectra, so we can repeat the above processes, obtaining versions with support $(F^n E)^V(X)$ and a filtration $F^m F^n E(X)$.

We say that $E$ satisfies Nisnevich excision if for any etale map $f: X' \to X$ and for any closed subset $V \subseteq X$ for which $f$ restricts to an isomorphism from $f^{-1}(V)$ to $V$, it follows that the map $E^V(X) \to E^{V'}(X)$ is a weak homotopy equivalence. This is Levine’s Axiom 2.

The first main conequence of Axiom 2 is a localization thm…

Def: p-fold T-loop space of $E$. Asiom 3: There is a functor $E'$ satisfying axioms 1 and 2, and a natural weak equivalence from $E$ to the 2-fold T-loop space of $E'$. Axiom 3 implies axioms 1 and 2.

The first main consequence of axiom 3 is a moving lemma…

We say that $E$ is well-connected if is satisfies Axiom 3 and two other conditions: (1) for every closed subset $W$ of a smooth variety $X$, the spectrum $E^W(X)$ is well-connected, and (2) for every finitely generated field extension $L/k$, for every $d \geq 0$, and every $n \neq 0$, we have $\pi_n(F^{0/1}(\Omega^d_T E)(L) ) = 0$.

A well-connected functor “allows computation in terms of cycles”.

Levine shows that his coniveau filtration is the same as the slice filtration. He is able to construct a homotopy coniveau spectral sequence analogous to the Atiyah-Hirzebruch spectral sequence, converging to the homotopy groups of $E(X)$. In the case $E=K$, recover the sequence given by Friedlander-Suslin above.

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