Holmstrom 5 Memo notes Weibel

Memo notes from Weibel in K-theory handbook

Problem: Compute K-theory of number fields and their rings of integers. Have known for long that K-groups of ROIs are f.g. with known ranks. Ref: Borel: Cohomologie reelle stable…, and LNM 341. Recent work by Suslin, Voevodsky, Rost etc have given us (nearly complete???) torsion information.

Thm: On the odd K-groups.

Discussion: K-groups of $\mathbb{Z}$; Vandiver’s conjecture is equivalent to $K_{4m}(\mathbb{Z}) = 0$ for $m \geq 2$. It is known that these groups have odd order, with no prime factor smaller than $10^7$.

Will also survey K-groups of local fields and global function fields.

Section 2: $K_n$ for $n \leq 3$, this material is classical - the groups have presentations by generators and relations.

Section 3: For odd groups, have cyclic summand which is understood through a construction by Harris and Segal: a variant of Adams e-invariant.

Section 4: Canonical free summands related to units (Borel). Other (almost periodic) summands related to $Pic(R)$ and $Br(R)$, detected by etale Chern classes (Soule).

Section 5: Global function fields. In this case, there is a smooth projective curve $X$ whose higher K-groups are finite, and are related to the action of $Frob$ on $Jac(X)$. The orders of these groups are related to the values of the zeta function of $X$ at negative integers.

Section 6: Local fields. For an extension of $\mathbb{Q}_p$, we understand the $p$-completion, but not the actual K-groups.

Section 7: Odd torsion, in the case of a number field. Also 2-primary torsion in totally imaginary number fields. This uses the Voevodsky-Rost theorem.

Section 8: 2-primary torsion in real number fields. Uses a theorem of Voevodsky.

Section 9: Odd torsion in $K_{2i}(\mathbb{Z})$: Irregular primes, Vandiver (Soule).

Key technical tool: Motivic spectral sequence. Details. Established by Levine over a Dedekind domain. With finite coeffs, motivic cohomology groups are etale cohomology (for a field). For a Dedekind domain, have a similar but more complicated result for the $E_2$ terms. Pic, role of etale cohomological dimension.

Also a spectral sequence from terms based mainly on etale cohomology (does he mean $\ell$-adic? to $K_{-p-q}(X; \mathbb{Z}/ \ell^{\infty} )$ (what does he mean by this?).

Section 2:

Class groups. Bass-Milnor-Serre (1967) on $K_2$. Localization sequence involving tame symbol/tame kernel. Relation to etale cohomology (Tate, Merkurjev, Suslin). Under some hyp, have mod $m$ short exact sequence with $Pic$, $K_2$ , $Br$.

Quillen and Borel: Homological techniques for the rank of higher K-groups.

Classical results for $K_3$.

Section 3: The e-invariant

Def and details. “Harris-Segal map, Harris-Segal summand”.

It is an isomorphism for finite fields.

Def: Exceptional field.

Example: Relation to the Bott map.

Remark: Exlains homotopy-theoretic motivation. Talks about the K-theory space of a symmetric monoidal category.

Remark: Bernoulli number, relation to the Riemann zeta function and to the J-homomorphism.

Example: On the image of the natural map $\pi_n^s \to K_n(\mathbb{Z})$ (this must mean the n-th stable homotopy group of spheres, because he later talks about the image of $J$ in this group).

Birch-Tate conjecture: Let $F$ be a number field. Then it is known that the $\zeta_F$ has a pole of order $r_2$ at $s = -1$. B-T conjecture says, that for totally real number fields ($r_2 = 0$), we have

The odd part of this conjecture was proven by Wiles, using Tate’s Theorem 4. The 2-primary part would follow from the 2-adic Main Conjecture of Iwasawa theory. Hence the full Birch-Tate conjecture is known for abelian extensions of $\mathbb{Q}$.

Section 4: Etale Chern classes

Want to relate etale cohomology and K-groups. In one direction, can use etale Chern classes (ref: Soule: K-theories des anneaux d’entiers…) in the form given in Dwyer-Friedlander: Algebraic and etale K-theory (1985).

In this section, we describe maps in the other direction: Kahn maps, or anti-Chern classes. These are an efficient reorganization of the maps above. There are also higher Kahn maps, omitted here. Refs to Kahn: Deux theoremes de comparaison en cohomologie etale: applications; On the Lichtenbaum-Quillen conjecture; K-theory of semi-local rings with finite coeffs and etale cohomology.

If $F$ is a field in which $\ell$ is invertible, there is a canonical map

called the first etale Chern class. It is equal to a map to etale K-theory composed with an edge map in the Atiyah-Hirzebruch spectral sequence for etale K-theory. For $i=1$ it is the Kummer isomorphism.

For each $i$ and $v$, can construct a splitting of the first etale Chern class, at least of $\ell$ is odd. This is the Kahn map. Definition. It is compatible with the Harris-Segal map (diagram). Theorem on injectivity. Compatibility with change of coeffs, so get maps

and

Corollary: For ring of $S$-integers.

We also have the second etale Chern class

and a Kahn map in the other direction. For $i=1$, this is just Tate’s map, which is an isomorphism for all fields by Merkurjev-Suslin.

Theorems similar to the above (compatibility etc).

Many details omitted.

Example: The Main Conjecture of Iwasawa theory implies that for odd $\ell$, the order of the finite group $H^2_{et}( \mathbb{Z}[1/ \ell], \mathbb{Z}_{\ell}(2k)) is the$\ell$-primary part of the numerator of$\zeta(1-2k)$. By Euler's formula, this is also the$\ell$-primary part of the numerator of$B_k / 2k`.

Theorem: For every number field $F$, and all $i$, the Adams operation $\psi^k$ acts on $K_{2i-1}(F) \otimes \mathbb{Q}$ as multiplication by $k^i$.

Proof: This follows from an isomorphism with an etale cohomology group, which commutes with the Adams op.

Section 5: Function fields

A global field of characteristic $p>0$ is a finitely generated field $F$ of transcendence degree 1 over $\mathbf{F}_p$. The algebraic closure of $\mathbf{F}_p$ in $F$ is a finite field $\mathbf{F}_q$ of char $p$. There is a unique smooth projective curve $X$ over $\mathbf{F}_q$ whose function field is $F$. If $S$ is a nonempty set of closed points of $X$, then $X-S$ is affine, and its coordinate ring is called the ring of $S$-integers in $F$. Will discuss K-groups of $X$, of $F$, and of the $S$-integers.

Thm: For $n \geq 1$, the groups $K_n(X)$ are finite of order prime to $p$.

Proof uses localization sequence, Weil reciprocity, and more…

More details on the above groups.

K-groups of $\bar{X}$, with action of geometric Frob. Explicit description of these Galois modules, proof uses Suslin’s theorem saying that for $i \geq 1$ and $\ell \neq p$, we have

Thm describing $K_n(X) = K_n(\bar{X})^G$.

The zeta function of a curve. Definition: $\zeta_X(s) = Z(X, q^{-s})$ where

Weil’s result for curves, explanation in terms of action of algebraic Frob on $\ell$-adic cohomology.

Formula for $| \zeta_X(1-i) |$ in terms of $\ell$-adic cohomology, or equivalently in terms of K-groups.

Iwasawa modules: alternative interpretation of some etale cohomology groups. Ref to Dwyer-Mitchell: On the K-theory spectrum of a ring of algebraic integers, and to Mitchell in the Kth handboook for the number field case.

Section 6: Local fields

Details omitted. Mention of Leopoldt’s conjecture, in the following form (possibly valid only for totally real fields): The torsion free part $\mathbb{Z}_p^{d-1}$ of $\mathcal{O}_F^{\times} \otimes \mathbb{Z}_p$ injects into the torsion free part of $\mathbb{Z}_p^{d}$ of $\prod \mathcal{O}_{E_j}^{\times}$ (the completions of the number field $F$ above $p$).

Section 7: Number fields at primes where $cd=2$

Here we obtain a cohomological description of the odd torsion in the K-groups of a number field, and also the 2-primary torsion in the K-groups of a totally imaginary number field. These are the cases where $cd_{\ell}(\mathcal{O}_S) = 2$, something which forces the motivic spectral sequence to degenerate completely.

Details omitted.

Theorem of Wiles describing all the odd special values of the zeta function of a totally real field.

Section 8: Real number fields at the prime 2

Infinite cohomological dimension: this complicates descent methods. Also other problems with this case.

Details omitted.

Section 9: The odd torsion in $K_*(Z)$

Some details omitted.

Vandiver’s conjecture in terms of the class group; the Herbrand-Ribet thm.

Vandiver’s conjecture for $\ell$ is equivalent to non-existence of $\ell$-torsion in $K_{4k}(\mathbb{Z})$ for all $k$.

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