This page is supposed to be a review of the seminal article
introducing Beilinson-Deligne cohomology, Beilinson regulators, higher regulators, and the Beilinson conjectures.
Please note that it is currently in a very preliminary state, having been prepared quickly as notes for a seminar talk on the first section of the paper. It follows the paper very closely, and an interested reader might like to rewrite from the nPOV.
By analytic space we will mean real analytic space. Let $An$ denote the real analytic site. Consider the category $Ab(Sh(An))$ of abelian sheaves on $An$. Beilinson denotes by $D^+(An)$ its bounded above derived category, i.e. the category of connective cochain complexes up to quasi-isomorphism. Given an analytic space $X \in An$, we will also consider its petit topos $X^\sim$ of sheaves on the site $Ouv(X)$ of open subsets.
In $D^+(An)$ we have the complex $\Omega^\bullet$, of de Rham complexes of holomorphic forms. Let $\Omega^{\ge i}$ denote the “stupid” filtration.
Fix a subring $A \subset \mathbf{R}$ and let $A(p) = A \cdot (2\pi i)^p \subset \mathbf{C}$ for $p \in \mathbf{Z}$. We will abuse notation and write $\mathbf{C} \in D^+(An)$ also for the constant sheaf valued in the complex concentrated in degree zero. Similarly we will view $A(p)$ as an object of $D^+(An)$.
We will write $H^*_B(X, \mathbf{C})$ for the Betti cohomology of $X \in An$.
For each $p \in \mathbf{Z}$, the inclusions $\Omega^{\ge p} \hookrightarrow \Omega^\bullet$ and $A(p) \hookrightarrow \mathbf{C} \hookrightarrow \Omega^\bullet$ induce a canonical morphism
(given by the difference of the two inclusions).
For $p \in \mathbf{Z}$, the Deligne complex of weight $p$, denoted $A(p)_\D$, is defined as the mapping cone of the above morphism shifted by -1, hence fitting in the distinguished triangle
The Deligne cohomology is just the hypercohomology of this complex. That is, consider the right derived functor $R\Gamma(-, A(p)_D)$ of the functor of global sections on $An$ with values in $D^+(A-mod)$.
The Deligne cohomology of $X \in An$ of weight $p$ and degree $q$ with coefficients in $A$ is
There is a canonical long exact sequence
This follows by applying the cohomological functor $R\Gamma(X, -)$ to the above distinguished triangle.
For $p \le 0$ there is a canonical quasi-isomorphism
For $p \gt 0$ there are canonical quasi-isomorphisms
There exists a canonical morphism in $D^+(An)$
inducing the structure of a graded commutative ring on
for all $X \in An$.
Beilinson gives an explicit formula using the usual explicit model for the mapping cone. He also remarks shortly that the product can be defined by observing that the obvious multiplicative structures on $\Omega^\bullet$, $\Omega^{\ge *}$, $A(*)$, turn each into a monoid object in the symmetric monoidal category of cochain complexes (of abelian sheaves), that is, into dg-algebras (of complexes of sheaves). Consider then the homotopy pullback of the diagram $A(*) \to \Omega \leftarrow \Omega^{\ge *}$; Beilinson claims that the underlying complex of this dg-algebra has in each degree $p$ the Deligne complex $A(p)_D$ of weight $p$. This point is expanded on in Hopkins-Quick.
Here we will simply give a formula for the quasi-isomorphic complex $(*)$ of Lemma 1 (assuming $i,j \gt 0$), which we will denote for the moment by $A(i)_E$. For $X \in An$, take $x \in \Gamma(X, A(i)_E)$, $y \in \Gamma(X, A(j)_E)$, and define
We omit the various verifications, that this defines a morphism of complexes, is associative, commutative, etc. One gets a monoid object in the category of cochain complexes of abelian sheaves. It only remains to see that $R\Gamma(X, -)$ preserves monoids, so that one gets the structure of a graded commutative ring on the hypercohomology groups $H^*_D(X, A(*))$.
Beilinson uses this cup product for $i=j=1$ to recover the Bloch regulator?.
(Bloch). For each algebraic curve $X$ over $\mathbf{R}$, there is a canonical functorial homomorphism
from the second algebraic K-theory group to the first Betti cohomology group with coefficients in $\mathbf{C}^*$.
By Lemma 1, there are quasi-isomorphisms
induced by the exponential map, and
induced by $x \mapsto \exp(x/2\pi i)$. It follows that the cup product
corresponds to a canonical homomorphism
According to Deligne, the RHS classifies isomorphism classes of line bundles with holomorphic connection?. Since $\dim(X) = 1$, all connections are integrable and this group is identified with $H^1_B(X, \mathbf{C}^*)$.
Now by Matsumoto’s theorem, giving a presentation of the $K_2$ group of a field by two generators and certain relations, one has
On the other hand, one has the Steinberg identity $t \cup (1-t) = 0$ for $t \in \mathcal{O}^*(X)$. It follows that the homomorphism above factors through $K_2(\mathcal{O}(X)) = K_2(\eta)$ for $\eta$ a generic point. To extend it to all of $X$, one uses the commutative diagram of localization exact sequences
The first row comes from the Gersten-Quillen resolution? for K-theory.
If $S$ and $T$ are toposes and $u^* : S \rightleftarrows T : u_*$ is a geometric morphism, consider the Artin gluing, i.e. the topos $(id_S/u_*) = (u^*/id_T)$ whose objects are morphisms $u^*(F) \to G$ for $F \in S$, $G \in T$, and morphisms are commutative diagrams. Write $(S, T)$ for this topos.
The functor of global sections on $(S, T)$ is the left exact functor
defined by
for each sheaf $F \in (S, T)$ given by $u^*(F_S) \to F_T$ with $F_S \in S$ and $F_T \in T$. Let $R\Gamma(S, T, -) : D^+(Ab(S, T)) \to D^+(Ab)$ denote its right derived functor.
The rest of this section is just defining a monoidal product on $Ch^+(Ab((S, T)))$, and explaining that a monoid in $D^+(Ab((S,T)))$ will induce a monoid in $D^+(Ab)$, i.e. a dg-algebra, after taking cohomology.
Let $\Pi$ denote the category of pairs $(X, \overline{X})$ with $\overline{X} \in An$ smooth and $j : X \hookrightarrow \overline{X}$ an open subspace such that the complement $\overline{X} - X$ is a normal crossing divisor?. The open immersion $j$ induces a functor $\Ouv(X) \to \Ouv(\overline{X})$ on the petit sites by mapping an open subspace $U \subset X$ to $U \subset X \subset \overline{X}$. This induces a geometric morphism $j^* : \overline{X}^{\sim} \rightleftarrows X^{\sim} : j_*$ on the petit toposes. Following the discussion of the previous section, we make the
The topos of the pair $(X, \overline{X})$ is defined to be the Artin gluing $(j^*/id)$, and will be denoted $(X, \overline{X})^{\sim}$. Hence a sheaf on the pair $(X, \overline{X})$ is a sheaf $F$ on $X$, a sheaf $G$ on $\overline{X}$, and a connecting morphism $j^*(G) \to F$.
Let $\Omega^\bullet_{X,\overline{X}} \in \Ch^+(Ab((X, \overline{X})^{\sim}))$ denote the de Rham complex of holomorphic forms on $C$ with logarithmic singularities? along $\overline{X} - X$. That is, $\Omega^\bullet_{X,\overline{X}}$ is an object of $Ch^+(Ab(\overline{X}^\sim))$ and we view it as a complex on $(X, \overline{X})^\sim$ by taking the part on $X^\sim$ to be trivial. Let $\Omega^{\ge p}_{X, \overline{X}}$ denote the stupid filtration.
Now we define a complex of abelian sheaves $A(p)_D$ in $Ch^+(Ab((X, \overline{X})^{\sim}))$ as follows.
The Beilinson-Deligne complex with logarithmic singularities of weight $p$ of the pair $(X, \overline{X})$ is a complex of abelian sheaves on $(X, \overline{X})^\sim$, denoted $A(p)_D \in Ch^+(Ab((X, \overline{X})^\sim))$ and defined as follows. In degree $n \in \mathbf{Z}$, take the sheaf
in $X^\sim$ (where the mapping cone is taken in $Ch^+(Ab(X^\sim))$), and the sheaf
in $\overline{X}^\sim$, together with the connecting morphism induced by the inclusion $j^*(\Omega^{\ge p}) \hookrightarrow \Omega^\bullet_X$.
The Beilinson-Deligne cohomology with logarithmic singularities of the pair $(X, \overline{X})$ in weight $p$ and degree $q$ and with coefficients in $A$, is the hypercohomology
One defines a cup product on these complexes in the same way as above, and gets a graded commutative ring structure on Beilinson-Deligne cohomology with logarithmic singularities.
Let $\Pi' \subset \Pi$ denote the full subcategory spanned by pairs $(X, \overline{X})$ for which $\overline{X}$ is a smooth projective algebraic variety. Let $V = V_\mathbf{R}$ denote the category of smooth quasi-projective schemes over $\mathbf{R}$. By the GAGA principle, we have a functor $\sigma : \Pi' \to V$ which sends a pair $(X, \overline{X})$ to $X$. Conversely given $X \in V$, by Hironaka? there exists a pair $(X, \overline{X}) \in \Pi'$ (a compactification).
Let $X \in V$ be a smooth quasi-projective algebraic variety over $\mathbf{R}$. Let $(X, \overline{X}) \in \Pi'$ be a compactification and define the Beilinson-Deligne cohomology of $X$ as
and
One shows that these definitions are independent of the chosen compactification. By the above, one gets a cup product also on these cohomology groups.
Next Beilinson shows that $H_D$ can be defined as cohomologies of certain complexes of Zariski sheaves. He notes that this is not necessary for the remainder of the paper, so we omit this here.
There is a canonical morphism
for each $X \in V$. This induces a canonical homomorphism
from the Picard group of invertible sheaves?.
For an invertible sheaf? $\mathcal{L}$ on $X$, its first Chern class is defined to be the image of the class of $\mathcal{L}$ under the above homomorphism.
One can show that for $A = \mathbf{Z}$, this homomorphism is injective, and further surjective if $X$ is compact.
Next Beilinson shows the projective bundle formula? for Beilinson-Deligne cohomology.
(Projective bundle formula). Let $E$ be an vector bundle of rank $r$ on $X$, let $\pi : \mathbf{P}(E) \to X$ be the associated projective bundle, and $\mathcal{O}(1)$ the tautological sheaf? on $\mathbf{P}(E)$. The homomorphism
is invertible.
By definition we have the distinguished triangle
One checks that it is compatible with the cup product. Since the morphism $A(i)_D \to A(i)$ sends first Chern classes in Deligne cohomology to first Chern classes in Betti cohomology, the projective bundle formula?s for Betti cohomology and de Rham cohomology show that the map in question induces an isomorphism on the leftmost and rightmost members of the triangle. Hence the result follows.
After the projective bundle theorem, one can define Chern classes of vector bundles following Grothendieck. In particular one gets the Chern character
For each $i$, there exists a unique assignment to a vector bundle $E$ over $X \in \mathcal{V}$ a class
that is functorial with respect to inverse images and for which $A(i)_D \to A(i)$ sends $c_i(E)$ to the usual Chern class in Betti cohomology.
We omit the proof and just recall the construction due to Grothendieck of the Chern classes. Let $r = rk(E)$ and write $P = \mathbf{P}(E)$. By the projective bundle formula one has
There exist $\gamma_i$ such that
with $\gamma_i \in H_D^{2i}(X, \mathbf{Z}(i))$ and $\gamma_0 = 1$. We define $c_i(E) = \gamma_i$.
In this paragraph, our goal is to define the homology theory dual to Deligne cohomology, for schemes over $\mathbf{R}$. To do this, we first define functorial complexes on $\Pi_*$. Then they extend, more or less formally, to the category of finite type schemes over $\mathbf{R}$ and proper maps. Then we establish Poincare duality.
Let $X \in An$ be smooth. Let
denote the complex of $C^\infty$ forms (resp. with distribution coefficients on $\mathcal{A}^{-p,-q}_{X}$). This is the totalization of the double complex $\mathcal{A}^{*,*}_X$ (resp. $\mathcal{D}^{*,*}_X$) formed by sheaves of $(p,q)$-forms (resp. with distribution coefficients). Let $\mathcal{A}^{\ge *}_{X}$ and $\mathcal{D}^{\ge *}_{X}$ denote the respective induced filtrations.
Let
denote the complex of $C^\infty$ singular chains with coefficents in the constant sheaf $A(p)$.
Let
denote the complex of global sections with compact support. (Here we view $\mathcal{D}_X$ as a sheaf on $X$.)
Let $\Pi_* \subset \Pi$ denote the subcategory with the same objects and only morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ which satisfy $f(\overline{X} - X) \subset \overline{Y} - Y$.
Let $(X, \overline{X}) \in \Pi_*$. Define the complexes
and
with the induced filtrations $\mathcal{A}^{\ge *}_{(X, \overline{X})}$ and $\mathcal{D}^{\ge *}_{(X, \overline{X})}$.
Define the complex of relative $C^\infty$ singular chains on $(X, \overline{X})$ as the quotient chain complex
Let
denote the complex of sections with compact support, with the induced filtration $\mathcal{D}^{\ge *}(X, \overline{X})$.
Finally, define the complex $C'_D(X, \overline{X}, A(p))$ (this is the complex that will give us the Deligne homology groups $H'_D^*(X, \overline{X}, A(p))$).
For $(X, \overline{X}) \in \Pi_*$, define the complex
This is functorial on $\Pi_*$.
Let $Sch_*$ denote the category of finite type schemes over $\mathbf{R}$ and proper maps. Let $V_* \subset Sch_*$ denote the subcategory of smooth quasi-projective schemes. Let $\Pi'_* = \Pi' \cap \Pi_*$ denote the category of pairs $(X, \overline{X})$ with $\overline{X}$ smooth projective, $X \subset \overline{X}$ open with $\overline{X} - X$ a normal crossing divisor, and with morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ such that $f(\overline{X} - X) \subset \overline{Y} - Y$.
The functor $C'^\bullet_D(-, A(p))$ on the category $\Pi'_*$ extends uniquely to a functor on $Sch_*$. In particular one gets has a distinguished triangle in $D^+(A-mod)$
where $H'_{dR}$ is de Rham homology?, $F^i$ is the Hodge filtration, $H'_B$ is the Borel-Moore homology with coefficients in the constant sheaf $A(p)$, and $H'_D(X, A(p))$ is the Deligne homology?, defined by the complex $C'_D(X, A(p))$.
(Poincare duality). Let $X$ be a smooth scheme of dimension $n$. There is a canonical isomorphism
Let $(X, \overline{X}) \in \Pi'_*$ be a compactification of $X$. Consider the presheaf on $\overline{X}$ of complexes of abelian groups, defined by
Take its associated sheaf, and consider it as a complex of abelian sheaves, $\overline{C}'_{X, \overline{X}}(A(p))$. First of all note that
where $j : X \hookrightarrow \overline{X}$ is the open immersion. Note that $j^* \overline{C}'_{X, \overline{X}}(A(p))$ is a flasque resolution? of the sheaf $A(p+n)[2n]$ on $X$. Since the embedding
is a filtered quasi-isomorphism, and the associated graded objects $gr^p \mathcal{D}_{X, \overline{X}}$ are soft sheaves?, one gets
Let $X$ be a scheme and $Y \in Z_n(X)$ an irreducible subscheme of dimension $n$. Note that the canonical homomorphism
is invertible. Let $cl_D(Y)$ denote the element of $H'_D^{-2n}(Y, A(-n))$ corresponding to the unit $1 \in A$. Hence one gets a homomorphism
given by $cl_D[Y] = i_*(\cl_D(Y))$. If $X$ is smooth, by Poincare duality this corresponds to a homomorphism
on the group of algebraic cyles? of codimension $n$.
If $X$ is smooth and compact, then for each $Y \in Z_n(X)$, if $cl_B(Y) \in H'_B^{-2n}(X, \mathbf{Z}(-n))$ is equal to 0, then $cl_D(Y)$ coincides with the Abel-Jacobi-Griffiths periods of the cycle $Y$.
The distinguished triangle defining $\mathbf{Z}(n)_D$ induces, after passing to the associated long exact sequence, a short exact sequence
where $\mathcal{I}^n$ is the $n$th intermediate Jacobian of Griffiths?, defined as
and $\Hdn^n(X)$ is the group of integral Hodge cycles of type $(n, n)$.
Using the usual explicit model for the mapping cone, $cl_D(Y)$ is the homology class of the cycle
where $i : Y \hookrightarrow X$ denotes the closed immersion, and $cl_F(Y) \in F^{-n}\mathcal{D}^{-2n}(X)$ is a distribution defined by integration over $Y$. Since $cl_B(Y) = 0$, we can choose $s \in C'^{-2n-1}(X, \mathbf{Z}(-n))$ such that $d(s) = i_*(cl_B(Y))$. By subtracting from $cl_D(Y)$ the boundary $(0, s, 0)$, we see that $cl_D(Y) = (cl_F(Y), 0, s)$. But the latter is precisely the definition of the periods of the cycle $Y$.
…
Alexander BeilinsonHigher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036-2070, (mathnet (Russian), DOI) (reviewed in Esnault-Viehweg 88)
Hélène Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)
Michael Hopkins, Gereon Quick, Hodge filtered complex bordism, arXiv.
Last revised on October 28, 2014 at 19:52:11. See the history of this page for a list of all contributions to it.