Holmstrom Old regulator notes

For arithmetic schemes, see 14G40

Complexes computing Motivic cohomology

• Voevodsky
• Suslin-Friedlander. Lecture 16 in MVW. Requires the base field to be perfect.
• Bloch?
• Polylog?
• Other explicit constructions for small cases?

Clarify the different maps between these complexes. In MVW, chapter 18, there is an inclusion

which is a quasi-isomorphism under certain hyps. This is a map from the SF complex to the Bloch complex, as I understand it. It seems also to be the case (although not sure about this) that there is a map from V complex to SF complex (rather than in the other direction). This would mean that a regulator map on the Bloch complex induces regulator maps on the other complexes. However,

• The description on V complexes might be simple
• The V complexes have products, which might (???) not be the case for the other ones.

Goncharov: Uses Bloch’s complex, defined as follows. Let $Z_m(X;n)$ be the free abelian group generated by irreducible codimension $n$ algebraic subvarieties in $X \times A^n$ which interesect properly all faces $X \times L_I$ (define this last thing). This is a homological complex, get a cohomological one by setting $Z_*(X;n) = Z_{2m-*}(X;n)$.

Gillet: $\mathbb{Z}^i_X(n) =$ (sheaf ass to???) the presheaf $U \mapsto Cor(U \times A^{n-i}, {G}_m^{\wedge n})$. The groups in this complex (over any open set) are zero for $i>n$. To compute motivic cohomology, we take Zariski hypercohomology of this complex.

Complexes computing Deligne cohomology

Goncharov: - For $X$ regular complex projective, we use the complex

where the first sheaf is in degree zero. Truncating the corresponding cohomology above $2n$ gives absolute Hodge cohomology.

• Can use instead a quasi-IMic complex, defined as the total complex associated with a certain bicomplex involving a complex of distributions (tensored with $\mathbb{R}(n-1)$, and the complex $\Omega^n_X \to \Omega^{n+1}_X \to \ldots$ (holomorphic de Rham complex)
• To compute the cohomology of the above complex, we replace the holomorphic de Rham complex by its Dolbeault resolution, taking the global sections of the obtained complex, and compute cohomology.
• Taking the canonical truncation of this complex in the degrees $[0,2n]$, we get a complex calculating the absolute Hodge cohomology of $X(\mathbb{C})$.
• We can define (following Deligne) yet another complex of abelian groups quasi-IMic to the latter complex. This is defined as the total complex of a double complex involving $D^{p,q}_X$, i.e. complex valued distributions of type $(p,q)$ on $X(\mathbb{C})$ for $p,q \in [0,n-1]$ and also $D^{n,n}_{cl}$. Here $D^{0,0}$ is placed in degree 1, so that the last term is in degree $2n$.
• We will work with a subcomplex of the previous complex, essentially restricting to the $\mathbb{R}(n-1)$-valued distributions. The cohomology of this complex is the absolute Hodge cohomology of $X(\mathbb{C})$. Ref to Goncharov preprint. Also involution-invariant subcomplex of this, for a variety defined over the reals, and corresponding cohomology.

Gillet: $D_X^*(p)$ is the subcomplex of the “standard complex of currents” where for $i \geq 2p$, we take $D_X^i(p)$ to consist of smooth forms and for $i = 2p-1$, we take $D_X^i(p)$ to consist of currents $g$ such that $\partial \ \bar{\partial}(g)$ is smooth. For the regulator map, we truncate $D_X^*(p)$ at $i = 2p-1$ (inclusive).

See chapter II of Lectures on Arakelov Geometry for currents and ordinary Arithmetic Chow groups (see notes under this CT page).

Previous definitions of the regulator map, on the level of complexes

Goncharov

Feliu?

Gillet (doesn’t make sense)

Our definition

Arithmetic Chow groups

At least for $X$ projective, Gillet says that the hypercohomology of the constructed single complex is arithmetic Chow groups (same indexing setup as for classical Chow groups).

Some notes from MathSciNet review of Jannsen’s ICM94: Finally the paper discusses arithmetic aspects of the theory. Integral motives over the ring of integers in a number field $K$ were introduced by Deninger and Nart extending ideas of Scholl. The group ${\rm Ext}^2 ({Q}, {Q} (1))$ in the resulting category is the first Arakelov-Chow group of ${\rm Spec} ({O}_K)$ as defined by Gillet and Soulé. This identification allows a description of the Bloch-Beĭlinson height pairing via a Yoneda pairing on Ext-groups.

Higher arithmetic Chow groups

To construct push-forwards for proper morphisms, maybe one could be inspired by Gillet: Homological descent for the K-theory of sheaves

Soule in http://www.ams.org/mathscinet-getitem?mr=991985 has a different approach to the regulator.

nLab page on Old regulator notes