Holmstrom 6 Memo notes Kahn

Intro

In alg geom, have VBs and cycles. This leads to K-theory and motivic cohomology, resp. They are related via the Chern character and Atiyah-Hirzebruch type spectral seqs.

It is often easier and more powerful to work with VBs. E.g. Quillen’s localisation thm is much easier to prove than the localisation thm for higher Chow groups. In short, no moving lemmas needed for VBs. VBs can often be classified by moduli spaces, which underlies finiteness thms like Tate’s thm or the Mordell conjecture, or Quillen’s fin gen proof for K-groups of curves over a finite field. They also have better functoriality than cycles, which has been used for example by Saito to establish functoriality properties for the weight spectra sequences for smooth projective varieties over $\mathbb{Q}_p$.

On the other hand, it is “fundamental” to work with mot cohomology. Since there is an spectral sequence from MC to KT (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory) (K-theory), the MC groups contains finer torsion information than K-groups. They also appear naturally as Hom groups in triangulated cats of motives, and they appear naturally, rather than K-groups, in the conjectures of Lichtenbaum on special values of zeta functions.

Some highlights of the chapter: Conjectures of Soulé and Lichtenbaum on on the order of zeroes and special values of zeta functions of schemes of finite type over $Spec(\mathbb{Z})$, and an approach to prove them in characteristic $p$ (we don’t touch the much more delicate Beilinson conjectures and their refinements).

Discussion of references.

Note: It is perhaps surprising that the Bloch-Kato conjecture and the Beilinson-Lichtenbaum conjecture seem irrelevant for the conjectures on special values of zeta functions after all.

Algebraic topology

Outline of singular cohomology and complex K-theory, and how these are related via the Chern character and the Atiyah-H. spectral seq. Also Adams operation, and degeneration of AH after tensoring with rationals.

Some details: Complex K-theory groups $K^i(X)$ may be defined as $[X, \mathbb{Z} \times BU]$ if $i$ is even, and $[X, U]$ if $i$ is odd. If $X$ is compact, $K^0(X)$ stably classifies complex VBs on $X$.

We have the AH spectral sequence

where $E_2^{p,q} = H^p(X , \mathbb{Z})$ if $q$ is even and zero otherwise.

The Chern character has the form

and

These are isomorphisms for $X$ finite-dimensional, and can be used to prove that AH above degenerates after tensoring with $\mathbb{Q}$.

Another proof of this uses the Adams operations $\Psi^k: K^i(X) \to K^i(X)$. One shows that $\Psi^k$ acts on the spectral sequence and induces multiplication by $q^k$ on $E_2^{p, 2q}$. Hence all differentials in AH are torsion, with explicit upper bounds; this yields in particular the formula

up to groups of finite exponent. Here the LHS stands for the common eigenspace of weight $j$ for all the Adams operations.

Can prove finite generation of singular cohomology of a finite CW complex, for example by induction on the number fo cells. Finite generation of cohomology implies finite generation of K-theory by AH. Conversely, if we know finite gen of K-theory for a space, we can show that integral cohomology is f.g. up to some small torsion. We do not get finite generation on the nose though, unless $X$ has small cohomological dimension.

Algebraic geometry

1: Algebraic K-theory

Quillens def: Let $X$ be a noetherian scheme. Consider the abelian cat $M(X)$ of coherent sheaves, and the full exact subcat $P(X)$ of locally free sheaves. Then

and

where $B$ is the classifying space, or nerve, and $Q$ is Quillen’s Q-construction on an exact cat. There is always a map $K_*(X) \to K'_*(X)$ which is an isomorphism when $X$ is regular (Poincaré duality (?)).

Later work of Waldhausen and Thomason-Trobaugh. The latter slightly modifies Quillen’s def as to obtain functoriality missing in Quillen’s def. There is always a map

and this is an iso as soon as $X$ has an ample family of VBs, for example if $X$ is quasi-projective over an affine base, or is regular.

Milnor introduced Milnor K-theory of a field (details omitted). Because algebraic K-theory has a product structure, there are natural homomorphisms

which are IMs for $i \leq 2$ but not otherwise. Surprisingly, Milnor K-groups are not ad hoc.

Conjecture 1 (Quillen conjecture): Let $A$ be a finitely generated regular $\mathbb{Z}$-algebra of Krull dimension $d$ and $\ell$ a prime number. Then there exists a spectral sequence with

for $q = -2 i$, and zero for $q$ odd, and whose abutment is isomorphic to $K_{-p-q}(A) \otimes \mathbb{Z}_{\ell}$, at least for $-p -q \geq 1+d$.

In this conjecture, étale cohomology groups may be defined as inverse limits of étale cohomology groups with finite coeffs; they coincide with the continuous étale cohomology groups of Dwyer-Friedlander and Jannsen, by Deligne’s Theoreme de finitude. Note: When $A$ is the ROI in a number field, such a spectral sequence must degenerate for cohomological dimension reasons when $\ell$ is odd or $A$ is totally imaginary.

There is a complementary conjecture of Lichtenbaum (Conjecture 46 below) relating algebraic K-theory and the Dedekind zeta function when $A$ is the ROI of a number field. These two conjectures has inspired lots of work over the past decades. Here is a brief historical survey:

1. Quillen: K-groups of rings of algebraic integers, or smooth curves over a field, are f.gen.
2. Soulé constructed higher Chern classes from algebraic K-theory with finite coeffs to étale cohomology. He proved that the corresponding $\ell$-adic homomorphisms are surjective up to finite torsion in the case of rings of algebraic integers in a global field. A consequence: finiteness of certain $\ell$-adic cohomology groups.
3. Friedlander introduced étale K-theory in the geometric case, inspired by Artin-Mazur, and formulated the Quillen-Lichtenbaum conjecture.
4. Dwyer and Friedlander defined continuous étale cohomology and [continuous] étale K-theory in full generality. They did two things: (1) Contructed a spectral sequence with $E_2$ terms the former, converging to the latter, for any $\mathbb{Z}[\ell^{-1}]$-scheme of finite étale cohomological dimension. (2) Defined a natural transf $K_i(X) \otimes \mathbb{Z}_{\ell} \to \hat{K_i^{et}(X)}$, where the last term is $\ell$-adic étale K-theory, and proved that this map is surjective when $X$ is the spectrum of a ring of integers of a global field. This refines the consequence of Soulé above. They also reinterpreted Quillen’s conjecture by conjecturing that the finite coeff version of the above natural transf is an iso for $i$ large enough (Lichtenbaum-Quillen conjecture).
5. Dwyer, Friedlander, Snaith and Thomason introduced algebraic K-theory with the Bott element inverted, proved that it maps to a version of étale K-theory and that, under some assumptions, this map is surjective. So “algebraic K-theory eventually surjects onto étale K-theory”.
6. Thomason went a step further by showing that, at least for nice enough schemes, étale K-theory is obtained from algebraic K-theory by “inverting the Bott element”. Hence, in this case, the spectral seq in (4) above converges to something close to algebraic K-theory.
7. The Milnor and Bloch-Kato conjectures showed up. Early proofs by Merkurjev-Suslin, and Rost of certain cases.
8. Merjurjev-Suslin and Levine independently studied the indecomposable part $K_3(F)/K_3^M(F)$ of $K_3$ of a field. This was the first idea in the direction “Bloch-Kato implies Beilinson-Lichtenbaum”.
9. Levine proved that a form of Bloch-Kato for semilocal rings implies a form of Q-L, expressed in terms of Soulé’s higher Chern classes.
10. Kahn introduced anti-Chern classes going from étale cohomology to algebraic K-theory and étale K-theory, defined when Bloch-Kato is true.
11. Hoobler (unpublished) proved that Bloch-Kato for regular semilocal rings implies Bloch-Kato for arbitrary semilocal rings. A previous argument of Lichtenbaum, relying on Gersten’s conjecture, showed that Bloch-Kato conjecture for regular semi-local rings of geometric origin follows from B-K for fields.
12. Motivic cohomology introduced: Bloch, Friedlander, Suslin, Voevodsky. Motivic spectral sequences: various names.
13. Suslin and Voevodsky formulated Beilinson-Lichtenbaum conjecture for motivic cohomology (Conj 17 below) and proved that under RoS, it follows from Bloch-Kato. Geisser and Levine removed the RoS hypothesis and also covered the case where the coefficients are a power of the characteristic.
14. Voevodsky proved BK at the prime 2 and conditionally at any prime.
15. The Q-L conjecture at 2 was proved by various authors. Conditionally, the same proof works at any prime. Note: If one had finite generation results for mot cohomology, Conjecture 1 would actually follow.

2: Bloch’s cycle complexes

Refs also to Geisser, Levine and Grayson in same volume.

2 nice pages: incorporate.

3: Suslin-Voevodsky motivic cohomology

Incorporate, including Conjecture 4.

4:Beilinson-Soulé conjecture

Incorporate under motivic cohomology.

Review of Motives

Incorporate: Pure motives, mixed motives, and some other cats of motives (Hodge cycles etc).

Some stable homotopy theory of schemes and functors to other cats.

Comparisons

1: Etale topology

Incorporate under étale motivic cohomology.

2: Zariski topology

Incorporate this section and next under motivic cohomology and Milnor K-theory.

A uniqueness thm for motivic cohomology; see also footnote.

3: Back to the B-S conjecture

4: Borel-Moore étale motivic homology.

Create new CT for this.

Applications: Local structure of algebraic K-groups and finiteness theorems

Incorporate under algebraic K-theory. Also some stuff for $K^{TT}$ and $K'$.

Things for motivic cohomology:

For $X$ regular (perhaps a $\mathbb{Z}[1/2]$-scheme, the map $H^i(X, \mathbb{Z}/2^v(n)) \to H^i_{et}(X, \mu^{\otimes n}_{2^v})$ is injective for $i \leq n+1$, and even an iso for $i \leq n$, so the first group is finite by Thm de finitude and arithmetic finiteness theorems.

Open question 35: Let $X$ be regular of finite type over $Spec \mathbb{Z}[1/m]$. Is $H^i( X, \mathbb{Z} / m (n) )$ finite for all $i$?

Remark: This is false over $Spec \bar{\mathbb{Q}}$ by an example of Schoen of the form $CH^2(E^3)/ \ell$. Kahn now tends to doubt whether it is true even over $\mathbb{Z}$.

The picture in arithmetic geometry

Bass conjecture for K-theory and motivic cohomology. Some stuff on motivic cohomology (incorporate).

Many pages on zeta functions of arithmetic schemes (not touching L-functions of var’s over number fields): incorporate!

Then from page 396, the main theme is the Tate-Beilinson conjecture. Possible page for this: Algebraic cycles, algebraic K-theory, motivic cohomology, Milnor K-theory, Lichtenbaum cohomology, étale motivic cohomology, Voevodsky motives (related to pure numerical motives), Chow groups.

From page 404, some pages on Lichtenbaum cohomology.

More on motivic cohomology (finiteness conjectures).

Unconditional results: Varieties of abelian type over finite fields

Results for motivic cohomology, formulated in terms of: Pure rational motives, Kimura finite-dimensional motives.

Questions and speculations

Lictenbaum cohomology, motivic cohomology, Lichtenbaum K-theory.

Thm: Let $X$ be smooth projective of dimension $d$ over $\mathbb{F}_p$. Then the map $CH^d(X) \to H^{2d}_{et}(X, \mathbb{Z}(d) )$ is bijective. (I think this is étale motivic cohomology).

Some stuff on the char zero case: much more complicated. Mention of Lichtenbaum’s theory, which will be for Arakelov varieties. Come back to this section for the Arakelov project.

References: Go through!

nLab page on 6 Memo notes Kahn