Holmstrom 4 Memo notes Geisser

Introduction

Will survey motivic cohomology, algebraic K-theory, and topological cyclic homology. Focus on arithmetic geometry applications; only commutative stuff. Main focus: Sheaf theoretic results for smooth schemes, which give global results using local-to-global methods.

We discuss motivic cohomology for smooth varieties over a field or a Dedekind ring. Suslin-Voevodsky’s construction has several technical advantages over Bloch’s, e.g. gives correct def for singular schemes. But it is only well understood over a field, and it does not give well-behaved etale motivic cohomology, so will use Bloch’s higher Chow groups instead, since we want Dedekind rings as bases to be included in discussion.

For algebraic K-theory, etale K-theory and topological cyclic homology, we discuss definitions and sheaf-theoretic properties, and relationships between the theories. In many cases, etale K-theory with p-adic coeffs agree with topological cyclic homology.

Appendix: Intersection theory for higher Chow groups.

Motivic cohomology

The existence of a complex of sheaves whose cohomology groups are related to special values of L-functions was first conjectured by Beilinson (for Zariski top) and Lichtenbaum (etale top). Refs: Letter to Soulé (1982), Height pairings between algebraic cycles; Values of zeta-functions at non-negative integers (1983), Motivic complexes (in Motives vol). The conjectural relationship between these complexes is called the Beilinson-Lichtenbaum conjecture.

Constructions of S-V and Bloch (Algebraic cycles and higher K-theory, 1986). Bloch’s higher Chow groups are defined for any scheme, but have properties analogous to a Borel-Moore homology theory in topology. In particular, they behave like a cohomology theory only for smooth schemes over field. Voevodsky’s groups have good properties for non-smooth schemes, but their basic properties are only established for schemes of finite type over a field, and they do not give a good etale theory (etale hypercohomology vanishes with mod-$p$ coefficients over a field of char $p$). The two definitions agree for smooth varieties over a field.

Definition

Definition of higher Chow groups (p 196): very nice, incorporate into CT page later. Have a long exact sequence

where the first map is $\times m$.

Hypercohomology

One can view $\mathbb{Z}(n) : = z^n( - , 2n - *)$ as a complex of presheaves on $X$. It is in fact a sheaf for Zariski, Nisnevich and etale topology. If $X$ is integral, there is a canonical quasi-isomorphism $\mathbb{Z}(0) \to \mathbb{Z}$ of complexes of Zariski sheaves. Since etale covers of a normal scheme is normal, the same qis holds in Nisnevich and etale tops, if $X$ is normal. For $X$ smooth of finite type over field or Dedekind ring, there is a qis $\mathbb{Z}(1) \cong \mathbf{G}_m[-1]$ for all three topologies. This was proved by Bloch in the above ref for $X$ quasi-projective over a field, and follows for general $X$ by localization. For any of these topologies, define motivic cohomology with coeffs in an abelian group $A$ as the hypercohomology of $\mathbb{Z}(n) \otimes A$.

If $X$ is of finite type over a finite field, then it is worthwhile to consider the Weil-etale topology. A nice paragraph on this (p 197), incorporate into Weil-etale cohomology.

The following properties are mainly due to Bloch (above ref) and Levine: K-theory and motivic cohomology of schemes I (Preprint 2002). The whole exposition below on motivic cohomology is super-nice; incorporate it together with its references into the mot. cohomology. page.

Functoriality

Localization

Gersten resolution

Products

Affine and projective bundles, blow-ups

Milnor K-theory

Definition and basic properties. Incorporate (p 201).

Beilinson-Lichtenbaum conjecture

Conjecture: Let $X$ be a smooth scheme over a field, and let $\varepsilon: X_{et} \to X_{Zar}$ be the canonical map of sites. Then the canonical map

is a qis, or more concretely: For every smooth scheme $X$ over a field, we have

for $i \leq n+1$.

Note that this is not true for higher $i$ (example)$. The Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture with mod$m` coeffs (references).

Cycle map

Let $H^i(X, n)$ be a bigraded cohomology theory which is the hypercohomology of a complex of sheaves $C(n)$; important examples are etale cohomology $H^i_{et}(X, \mu_m^{\otimes n})$ and Deligne cohomology $H^i_D(X, \mathbb{Z}(n))$. Assume that the $C(n)$ are contravariantly functorial as objects of the derived category, i.e. for $f: X \to Y$ there exists a map $f^* C(n)_X \to C(n)_Y$ in the derived category, compatible with composition. Assume furthermore that $C(n)$ admits a cycle class map $CH^n(X) \to H^{2n}(X,n)$, is homotopy invariant, and satisfies a weak form of purity. Then Bloch constructs (Bl: Algebraic cycles and the Beilinson conjectures (1986); see also Geisser-Levine: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky) a natural map

Unfortunately, this construction does not work for CTs which satisfy the projective bundle formula but are not homotopy invariant, like crystalline, de Rham, or syntomic cohomology.

Rational coefficients

Nice discussion of Parshin’s conjecture. Incorporate.

Sheaf theoretic properties

(of the motivic complexes)

Invertible coefficients

Let $X$ be smooth over a field $k$, and $m$ prime to $char(k)$. Then there is a qis in the etale topology

Explanation of reason for this (nice!) and reformulation of the Beilinson-Lichtenbaum conjecture.

Characteristic coefficients

The de Rham-Witt complex, relation to crystalline cohomology. Very nice. Frobenius and Verschiebungm slope spectral sequence, the logarithmic de-Rham-Witt sheaf. Gersten resolution. Relation to Milnor K-theory. Discussion, ending with a map to crystalline cohomology.

Projective bundle and blow-up

Formulas for etale motivic cohomology of blow-ups and projective bundles. Uses logarithmic de Rham-Witt cohomology.

Add refs to de Rham-Witt stuff from tha above.

Mixed characteristic

Let $X$ be an essentially smooth scheme over the spectrum of a Dedekind rings. The Bloch-Kato conjecture implies the following sheaf theoretic properties of the motivic complex: Purity, Beilinson-Lichtenbaum, Rigidity, Etale sheaf property, Gersten resolution. Explanation of all this.

Syntomic complex (ref to Fontaine-Messing: p-adic periods and p-adic etale cohomology). See the whole paragraph for syntomic cohomology (page 207-208) including more references.

K-theory

Recall: Quillen’s Q-construction. Nerve. $K$ and $K'$. “The functor $K'$ has properties analogous to the properties of Bloch’s higher Chow groups”.

For a ring, also have the plus-construction. For an affine scheme, these agree. The Q-construction has better functoriality properties, but the plus construction is more accessible to calculations. Quillen, for example, calculated the K-theory of finite fields. This was the only ring with completely known K-theory, until the calculation for truncated polynomial algebras over finite fields.

Waldhausen improved upon the Q-construction by his S-construction (W: Algebraic K-theory of spaces). It gives a symmetric spectrum in the sense of Hovey-Shipley-Smith; see the appendix of Geisser and Hesselholt: Topological cyclic homology of schemes. It also allows categories with more general WEs than IMs as input, for example cats of complexes and quasi-IMs. Using this and results from SGA (LNM 225), Thomason-Trobaugh gave a better behaved definition of K-theory (of a scheme). For simplicity, assume $X$ noetherian. Then the $K'$-theory of $X$ is the Waldhausen K-theory of the category of complexes which are quasi-IMic to a bounded complex of coherent $\mathcal{O}_X$-modules. This gives the same homotopy groups as Quillen’s construction. The K-theory of $X$ is the Waldhausen K-theory of the category of perfect complexes, i.e. complexes quasi-IMic to bounded complex of locally free $\mathcal{O}_X$-modules of finite rank. If $X$ has an ample line bundle (for example if $X$ is quasi-projective over an affine scheme, or separated, regular and noetherian), then this agrees with Quillen’s def.

The K-groups with coefficients are the homotopy groups of the smash product $K/m(X) = K(X) \wedge M_m$ of the K-theory spectrum with the Moore spectrum. There is a long exact sequence:

where the first map is multiplication by $m$. Similarly for $K'$-theory.

We let $K_i(X, \mathbb{Z}_p)$ be the homotopy groups of the homotopy limit $holim_n K/p^n(X)$. Then the homotopy groups are related by the Milnor exact sequence:

Ref to Bousfield-Kan: LNM304. The K-groups with coefficients satisfy all the properties given below for K-groups, except for product structure in the case where $m$ is divisible with 2 but not 4, or 3 but not 9.

Basic properties

Functoriality properties, product structures, homotopy invariance, blow-up formula (all made very precise).

Localization

Localization sequence, Mayer-Vietoris, Brown-Gersten spectral sequence (also for Bass K-theory).

Gersten resolution

Coniveau filtration, gives a spectral sequence. For $X$ smooth over a field, this degenerates to give the Gersten resolution.

The product structure induces a canonical map from Milnor K-theory of fields to the corresponding Quillen K-groups. Via the Gersten resolutions, we can generalize this to any regular semi-local ring essentially of finite type over a field.

Motivic cohomology and K-theory

If $X$ is of finite type over a DVR, then we have a spectral sequence

By Gillet-Soulé: Filtrations… there are Adams operators acting on the $E_r$-terms, compatible with the action on the abutment. By Suslin: Higher Chow groups and etale cohomology, the spectral sequence degenerates after tensoring with $\mathbb{Q}$ and the induced filtration agrees with the gamma-filtration.

References for different approaches to the motivic spectral sequence.

Using the spectral sequence, can translate results from motivic cohomlogy to K-theory. For example, Parshin’s conjecture states that for a smooth projective variety over a finite field, $K_i(X)$ is torsion for $i>0$, and this implies that for a field $F$ of char $p$, Milnor and Quillen K-groups agree after tensoring with the rationals.

Sheaf theoretic properties

Details omitted, include later in K-theory.

Etale K-theory and Topological Cyclic Homology

Hypercohomology spectra (see glossary).

Continuous hypercohomology

See Continuous hypercohomology. Ref to Jannsen.

Hypercohomology of K-theory

If $X_{Zar}$ is the Zariski site of a noetherian scheme of finite dimension, then by Thomason, the augmentation map $\eta: K'(X) \to \mathbb{H}^{\bullet} (X_{Zar}, K')$ is a homotopy equivalence. Same for Bass groups and the Nisnevich topology. Hence two spectral sequences coincide (details omitted).

If $X_{et}$ is the small etale site of the scheme $X$, then we write $K^{et}(X)$ for $\mathbb{H}^{\bullet} (X_{et}, K)$ and $K^{et}(X, \mathbb{Z}_p)$ for $holim_r \ \mathbb{H}^{\bullet} (X_{et}, K/p^r)$. Dwyer and Friedlander gave a different construction of etale K-theory, but by Thomason they agree whenever $X$ is separated, noetherian, regular, of finite Krull dim, with $\ell$ invertible on $X$ (does he mean $p$ here??). Various spectral sequences computing etale K-theory.

The Lichtenbaum-Quillen conjecture

This is the K-theory version of the Beilinson-Lichtenbaum conjecture. It states that on a regular scheme, the canonical map $K_i(X) \to K^{et}_i(X)$ from K-theory to etale K-theory is an isomorphism for sufficiently large $i$ (expected: above the cohomological dimension of $X$). Rationally, etale and algebraic K-theory agree, so we can restrict attention to finite coeffs. Some results (omitted).

Levine has announced an “etale motivic spectral sequence”, from etale motivic cohomology to etale K-theory. More details, omitted.

Topological cyclic homology

Bokstedt, Hsiang and Madsen defined this for a ring. Using Thomason’s hypercohomology construction, one can extend it to schemes (Geisser and Hesselholt, 1999).

Many nice details here, look at this later. Including cyclotomic trace map.

Comparison

Etale K-theory and TC. Many details omitted.

Appendix: Basic intersection theory

All this should be incorporated later under Higher Chow groups. Very briefly: Def of proper intersection, Intersection with divisors, regular sequence, pullback along a regular embedding, flat pullback, proper push-forward

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