| Sbornik: Mathematics |
|
|
|
|
|
A remark on 0-cycles as modules over algebras of finite correspondences M. Z. Rovinskyab a Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 8, DOI: https://doi.org/10.4213/sm9907 
Bibliographic databases:
    Let $k$ be a field. There are several ways and versions in which the 0-cycles on $k$-schemes of finite type can be considered as a functor. In each of these versions, we want this functor to be an object of an abelian category, and we study its structure (‘composition series’). Consider a set $S$ of smooth projective varieties over a fixed field. Let $Z_0(S)$ be the direct sum of the $\mathbb Q$-vector spaces of 0-cycles (that is, formal finite linear combinations of closed points) on varieties in $S$ with rational coefficients. We consider $Z_0(S)$ as a module over the algebra of finite correspondences. The aim of this note is to show that the rationally trivial 0-cycles form an absolutely simple submodule $Z_0^{\mathrm{rat}}(S)$ of the module $Z_0(S)$, which is contained in all nonzero submodules of $Z_0(S)$. To some extent this is analogous to the minimality of rational equivalence among all ‘adequate’ equivalence relations on algebraic cycles; cf. Proposition 8 in [12]. Assuming the Beilinson-Bloch motivic filtration conjecture, we show that the radical filtration on $Z_0(S)/Z_0^{\mathrm{rat}}(S)$ is an evident modification of the conjectural motivic filtration on Chow groups of 0-cycles. This is checked unconditionally in the case of curves. In the last section, a point of view on the 0-cycles on smooth, but not necessarily proper, varieties as a cosheaf in an appropriate topology is briefly discussed. § 1. Category algebras and nondegenerate modulesA category $\mathcal C$ is called preadditive if, for each pair of objects $X$ and $Y$, the morphism set $\mathrm{Hom}_{\mathcal C}(X,Y)$ is endowed with an abelian group structure, while the morphism composition maps
$$
\begin{equation*}
\circ_{X,Y,Z}\colon \mathrm{Hom}_{\mathcal C}(X,Y)\times\mathrm{Hom}_{\mathcal C}(Y,Z)\to \mathrm{Hom}_{\mathcal C}(X,Z) \quad\text{are bilinear}
\end{equation*}
\notag
$$
for all objects $X$, $Y$ and $Z$. For any small preadditive category $\mathcal C$, set
$$
\begin{equation*}
A_{\mathcal C}:=\bigoplus_{X,Y\in\mathcal C}\mathrm{Hom}_{\mathcal C}(X,Y).
\end{equation*}
\notag
$$
The composition pairings $\circ_{X,Y,Z}$ (and the zero pairings between the groups $\mathrm{Hom}_{\mathcal C}(W,X)$ and $\mathrm{Hom}_{\mathcal C}(Y,Z)$ for all quadruples $W$, $X$, $Y$, $Z$ such that $X\neq Y$) induce an associative ring structure on the abelian group $A_{\mathcal C}$. The ring $A_{\mathcal C}$ is unital if and only if there are only finitely many objects in $\mathcal C$. However, even if $A_{\mathcal C}$ is not unital, it is idempotented (in the sense of Definition 4 in [2]), that is, for every finite collection $B$ of elements of $A_{\mathcal C}$ there is an idempotent $e\in A_{\mathcal C}$ such that $ea=a$ for all $a\in B$, namely, the sum of identities $\mathrm{id}_X\in\mathrm{Hom}_{\mathcal C}(X,X)\subseteq A_{\mathcal C}$ for all objects $X$ in an arbitrary finite set containing the union of the supports of elements of $B$. (By definition, the support of an element $a$ is the smallest set $\mathrm{Supp}(a)$ such that $a\in\bigoplus_{X,Y\in\mathrm{Supp}(a)}\mathrm{Hom}_{\mathcal C}(X,Y)\subseteq A_{\mathcal C}$.) Recall (see, for example, [3], p. 113) that a left module $M$ over an associative ring $A$ is called nondegenerate if $AM=M$. Obviously, $A_{\mathcal C}$ is a nondegenerate left $A_{\mathcal C}$-module. Denote by $\mathrm{Mod}_{\mathcal C}$ the category of nondegenerate left $A_{\mathcal C}$-modules. Denote by $\mathcal C^{\vee}$ the category of additive functors from $\mathcal C$ to the category of abelian groups. Lemma 1 (Morita equivalence). If $\mathcal C$ is a small preadditive category, then $\mathcal C^{\vee}$ and $\mathrm{Mod}_{\mathcal C}$ are equivalent abelian categories. In particular, if two small preadditive categories $\mathcal C$ and $\mathcal C'$ are equivalent, then the categories $\mathrm{Mod}_{\mathcal C}$ and $\mathrm{Mod}_{\mathcal C'}$ are equivalent too. Proof. We send any functor $\mathcal F$ from $\mathcal C$ to the category of abelian groups to $\bigoplus_{X\in\mathcal C}\mathcal F(X)$, which is a nondegenerate $A_{\mathcal C}$-module in an obvious way.
In the opposite direction, given an $A_{\mathcal C}$-module $M$ and an object $X$, we set $\mathcal F(X):=\mathrm{id}_X(M)$. Any morphism $f\in\mathrm{Hom}_{\mathcal C}(X,X')\subseteq A_{\mathcal C}$ induces a map
$$
\begin{equation*}
\mathcal F(X)=\mathrm{id}_X(M)\xrightarrow{f}f\circ\mathrm{id}_X(M)= \mathrm{id}_{X'}\circ f\circ\mathrm{id}_X(M)\subseteq\mathrm{id}_{X'}(M)=\mathcal F(X').
\end{equation*}
\notag
$$
It is easy to see that these two functors are quasi-inverse equivalences. In particular, we obtain a chain of equivalences: $\mathrm{Mod}_{\mathcal C}\simeq\mathcal C^{\vee} \simeq(\mathcal C')^{\vee} \simeq\mathrm{Mod}_{\mathcal C'}$. The lemma is proved. The Yoneda embedding $\mathcal C\to\mathcal C^{\vee}\simeq\mathrm{Mod}_{\mathcal C}$, $X\mapsto h_X:=\mathrm{Hom}_{\mathcal C}(X,-)$, is a fully faithful functor. We are interested in the structure of the $A_{\mathcal C}$-module $h_X$ for the ‘unit’ object $X$. § 2. Algebras of finite correspondences and their modulesFix a field $k$. For each pair of smooth $k$-varieties $X$ and $Y$ we define $\mathrm{Cor}(X,Y)_{\mathbb Q}$ as the $\mathbb Q$-vector space with a basis given by the irreducible closed subsets of $X\times_kY$ whose associated integral subschemes are finite, flat and surjective over a connected component of $X$. For each triple of smooth $k$-varieties $(X,Y,Z)$ we define the bilinear pairing $\mathrm{Cor}(X,Y)_{\mathbb Q} \times\mathrm{Cor}(Y,Z)_{\mathbb Q}\xrightarrow{\circ_{X,Y,Z}}\mathrm{Cor}(X,Z)_{\mathbb Q}$ in the standard way: $(\alpha,\beta)\mapsto \mathrm{pr}_{XZ*}(\alpha\times Z\cap X\times\beta)$; see [5], Ch. 1. These pairings as compositions turn the category of smooth $k$-varieties with morphisms $\mathrm{Cor}(-,-)_{\mathbb Q}$ to an additive category, denoted by $\mathrm{SmCor}_k$. Denote the full subcategory of projective $k$-varieties by $\mathrm{SmCor}_k^{\mathrm{proj}}$. Given a set $S$ of smooth $k$-varieties, we can consider $S$ as a full subcategory of $\mathrm{SmCor}_k$. As the category $S$ is preadditive, the direct sum $A_S\!:=\!\bigoplus_{X,Y\in S}\mathrm{Cor}(X,Y)_{\mathbb Q}$ carries a ring structure. 2.1. The socle of $Z_0(S)$For each smooth variety $Y$ over $k$ let $Z_0(Y):=\mathrm{Cor}(\mathrm{Spec}(k),Y)_{\mathbb Q}$ be the $\mathbb Q$-vector space of 0-cycles on $Y$. Lemma 2. Let $X$ be a smooth quasiprojective variety over $k$, $F$ be a characteristic zero field, and $\xi\in Z_0(Y)\otimes F$ be a nonzero 0-cycle. Then there exists a correspondence $\vartheta\in\mathrm{Cor}(X,\mathbb P^1_k)\otimes F$ such that $\vartheta\xi=[0]-[\infty]\in Z_0(\mathbb P^1_k)$. Proof. Let $\xi_X=\sum_{i=1}^Nm_i[p_i]$ for nonzero $m_i\in\mathbb Q$ and closed points $p_i\in X$.
By a refinement of the projective version of the Noether normalization lemma proved in [9], Theorem 1, $X$ admits a morphism $\varphi\colon X\to\mathbb P^n_k$, where $n:=\dim X$, which maps $p_2,\dots,p_N$ into a hyperplane $H\subset\mathbb P^n_k$ and maps $p_1$ into the complement of $H$. Set $p_i':=\varphi(p_i)$ for all $i$, so that $p_2',\dots,p_N'\in H$ and $p_1'\in\mathbb P^n_k\setminus H$. This means that $\varphi_*\xi_X=\sum_{i=1}^Nm_i[p_i']\neq 0$. Let us show using induction on $N\geqslant 2$ that there exists a finite endomorphism $\psi\colon\mathbb P^n_k\to\mathbb P^n_k$ sending the points $p_2',\dots,p_N'$ to a single $k$-rational point $p\in\mathbb P^n_k$ and sending $p_1'$ to another $k$-rational point $q\in\mathbb P^n_k$, $q\neq p$. Let $W_0,\dots,W_n$ be homogeneous coordinates on $\mathbb P^n_k$ such that $H$ is given by the equation $W_0=0$, while both $p_2'$ and $p_3'$ do not lie on the hyperplane given by the equation $W_1=0$. For each $2\leqslant i\leqslant n$ set $w_i:=W_i/W_1$, and let $P_{ij}$ be the minimal polynomial of $w_i(p_j')$ over $k$. Set $d:=\max_{2\leqslant i\leqslant n}\deg(P_{i2}P_{i3})$ and $P_i\,{:=}\,P_{i2}(w_i)P_{i3}(w_i)w_i^{d-\deg(P_{i2}P_{i3})}W_1^d$. Then the map
$$
\begin{equation*}
g\colon (W_0:\cdots:W_N)\mapsto(W_0^d:W_1^d:P_1:P_2:\cdots:P_N)
\end{equation*}
\notag
$$
is a well-defined endomorphism of $\mathbb P^n_k$, $g$ preserves $H$, the point $g(p_2')=g(p_3')$ is $k$-rational, and $g$ transforms $\varphi_*\xi_X$ to $m_1[p_1'']+(m_2+m_3)[p_3'']+\sum_{i=4}^Nm_i[p_i'']$, where $p_3'',\dots,p_N''\in H$ and $p_1''\notin H$.
Then $\psi_*\varphi_*\xi_X$ is a nonzero multiple of $[p]-[q]$. Let $\Upsilon$ be an $n$-dimensional variety admitting a nonconstant morphism $h\colon\Upsilon\to\mathbb P^1_k$ (for example, $\Upsilon=\mathbb P^{n-1}\times\mathbb P^1_k$ and $h\colon\Upsilon\to\mathbb P^1_k$ is the projection). Fix a fibre $D$ of $h$, and a hyperplane $H'\subset\mathbb P^n_k$ containing $p$ but not $q$. By Theorem 1 in [9] there exists a finite morphism $\pi\colon\Upsilon\to\mathbb P^n_k$ such that $\pi(D)=H'$, so that $D$ meets $\pi^{-1}(p)$ but not $\pi^{-1}(q)$, and therefore $h_*\pi^*\psi_*\varphi_*\xi_X\neq 0$. Then $h_*({}^t\Gamma_{\pi})_*\psi_*\varphi_*\xi=h_*\pi^*\psi_*\varphi_*\xi$ is a nonzero divisor $E=\sum_{i=0}^na_i[q_i]$ on $\mathbb P^1_k$ for some $a_i\neq 0$ and pairwise distinct $q_i$. Choose a morphism $f\colon\mathbb P^1_k\to\mathbb P^1_k$ such that $f(q_0)=0$ and $f(q_i)=\infty$ for all $1\leqslant i\leqslant n$, so that $f_*h_*({}^t\Gamma_{\pi})_*\psi_*\varphi_*\xi_X=a_0([0]-[\infty])$. The lemma is proved. For each set $S$ of smooth varieties over $k$ consider $Z_0(S):=\bigoplus_{X\in S}Z_0(X)$. Then the above pairings
$$
\begin{equation*}
\circ_{\mathrm{Spec}(k),Y,Z}\colon Z_0(Y)\times\mathrm{Cor}(Y,Z)_{\mathbb Q}\to Z_0(Z),
\end{equation*}
\notag
$$
given by $(\alpha,\beta)\mapsto\mathrm{pr}_{Z*}(\alpha\times Z\cap\beta)$, induce an $A_S$-module structure on $Z_0(S)$. Define the degree of a 0-cycle $\alpha=\sum_im_iP_i$ on $X$ by $\deg(\alpha):=\sum_im_i[\varkappa(P_i):k]$, where $\varkappa(P_i)$ is the residue field of $P_i$. For each smooth variety $Y$ over $k$ let $Z_0^{\circ}(Y)$ be the subspace of 0-cycles of degree 0 on each connected component of $Y$. Obviously, $Z_0^{\circ}(S):=\bigoplus_{X\in S}Z_0^{\circ}(X)$ is an $A_S$-submodule of $Z_0(S)$. Recall that a cycle is called rationally equivalent (or rationally trivial) to zero if it is a sum of divisors of rational functions on subvarieties (see [12], § 2, and [5], Ch. 1). Theorem 1. Let $S$ be a set of smooth varieties over $k$ and $F$ be a characteristic zero field. Then 1) any proper $(A_S\otimes F)$-submodule of $Z_0(S)\otimes F$ is contained in the submodule $Z_0^{\circ}(S)\otimes F$; 2) if $S$ consists of projective varieties, then any nonzero $(A_S\otimes F)$-submodule of $Z_0(S)\otimes F$ contains the $A_S$-submodule of 0-cycles that are rationally equivalent to 0 on all the $X\in S$. Proof. It is clear that if $S'$ is the set of connected components of varieties in $S$ then $A_{S'}$ and $A_S$ are naturally isomorphic, while $Z_0(S')$ and $Z_0(S)$ coincide as $A_S$-modules. This means that we may assume that all varieties in $S$ are connected. Given any characteristic zero field $F$ and any nonzero element $\xi=(\xi_X)_{X\in S}\in Z_0(S)\otimes F$, there is $X\in S$ such that $\xi_X\neq 0$, so that $\xi':=\mathrm{id}_X\xi\neq 0$.
1. For any $Y\in S$ and any closed point $y\in Y$ the finite correspondence $[X\times_ky]\in\mathrm{Cor}(X,Y)_{\mathbb Q}$ maps $\xi_X$ to the 0-cycle $\deg(\xi_X)\cdot[y]\in Z_0(Y)\otimes F$, so if $\deg(\xi_X)\neq 0$, then $\xi'$ (and therefore $\xi$) generates the whole of the $(A_S\otimes F)$-module $Z_0(S)\otimes F$, which is equivalent to assertion 1). 2. According to assertion 1), we may assume below that $\deg(\xi_X)=0$ and, since $\xi_X\neq 0$, that $\dim X>0$. By Lemma 2 there exists a correspondence $\vartheta\in\mathrm{Cor}(X,\mathbb P^1_k)\otimes F$ such that ${\vartheta\xi_X=[0]-[\infty]\in Z_0(\mathbb P^1_k)}$. Finally, for each $Y\in S$, any 0-cycle on $Y$ rationally equivalent to 0 is a linear combination of the images of the cycle $[0]-[\infty]$ under finite correspondences $\gamma$ from $\mathbb P^1_k$ to $Y$, that is, of elements $(\gamma\circ\vartheta)_*\xi_X$ for appropriate correspondences $\gamma$. The theorem is proved. Remark 1. A module $M$ over a $\mathbb Q$-algebra $A$ is called absolutely simple if $M\otimes F$ is a simple $(A_S\otimes F)$-module for any characteristic zero field $F$. Equivalently, the $A$-module $M$ is simple and $\mathrm{End}_A(M)=\mathbb Q$. In particular, in the setting of Theorem 1, if $S$ is nonempty, then the $A$-modules $Z_0(S)/Z_0^{\circ}(S)$ and $Z_0^{\mathrm{rat}}(S)$ are absolutely simple. 2.2. Motivic $A_S$-modulesBy definition (see [12]) an equivalence relation $\sim$ between algebraic cycles is adequate if it satisfies the following conditions:
Example 1 (see [12], § 2). Apart from rational equivalence mentioned above, the following equivalence relations are adequate. Recall (see, for example, [11]) that a (homological) effective Grothendieck motive over $k$ modulo an ‘adequate’ equivalence relation $\sim$ is defined as a pair $(X,\pi)$ consisting of a smooth projective variety $X$ over $k$ and a projector $\pi$ in the algebra of self-correspondences on $X$ of dimension $\dim X$ with coefficients in $\mathbb Q$ modulo $\sim$. Morphisms between pairs $(X,\pi)$ and $(X',\pi')$ are algebraic cycles $\alpha$ on $X\times_kX'$ of dimension $\dim X$ modulo $\sim$ such that $\alpha=\pi'\circ\alpha\circ\pi$. The motives over $k$ modulo an equivalence relation $\sim$ form a pseudo-abelian category, denoted by $\mathcal{M}_{k,\mathrm{eff}}^{\sim}$. The category $\mathcal{M}_{k,\mathrm{eff}}^{\sim}$ carries a tensor structure: $(X,\pi)\otimes(X',\pi'):=(X\times_kX',\pi\times\pi')$. Denote by $\mathbb M^{\sim}\colon \mathrm{SmCor}_k^{\mathrm{proj}} \to\mathcal{M}_{k,\mathrm{eff}}^{\sim}$ the additive functor $X\mapsto(X,\Delta_X)$, where $\Delta_X$ is the class of the diagonal in $X\times_kX$. In particular,
$$
\begin{equation*}
\mathbb M^{\sim}(\mathbb P^1_k)\cong\mathbb M^{\sim}(\mathrm{Spec}(k))\oplus\mathbb L, \quad\text{where } \mathbb L=(\mathbb P^1_k,[\{q\}\times\mathbb P^1_k])
\end{equation*}
\notag
$$
for any rational point $q\in\mathbb P^1(k)$. It is easy to see that the natural map
$$
\begin{equation*}
\mathrm{Hom}_{\mathcal{M}_{k,\mathrm{eff}}^{\sim}}(U,V)\to \mathrm{Hom}_{\mathcal{M}_{k,\mathrm{eff}}^{\sim}}(U\otimes\mathbb L,V\otimes\mathbb L)
\end{equation*}
\notag
$$
is bijective for all effective motives $U$ and $V$. Denote by $\mathcal{M}_k^{\sim}$ the category of triples $(X,\pi,n)$, where $(X,\pi)$ are as above and $n$ is an integer, while $\mathrm{Hom}_{\mathcal{M}_k^{\sim}}((X,\pi,n),(X',\pi',n')) :=\mathrm{Hom}_{\mathcal{M}_{k,\mathrm{eff}}^{\sim}} ((X,\pi)\otimes\mathbb L^{\otimes(m+n-n')},(X',\pi')\otimes\mathbb L^{\otimes m})$ for any integer $m>|n'-n|$. We consider $\mathcal{M}_{k,\mathrm{eff}}^{\sim}$ as a full subcategory of $\mathcal{M}_k^{\sim}$ under the embedding $(X,\pi)\mapsto(X,\pi,0)$. For each variety $Y$ and an integer $q$ denote by $\operatorname{CH}_q(Y)$ the group of dimension $q$ cycles on $Y$ modulo rational equivalence. Theorem 2. The functor $\mathbb M^{\sim}$ is full. In other words, the natural ring homomorphism $A_S\to\bigoplus_{X,Y\in S}\operatorname{CH}_{\dim X}(X\times_kY)_{\mathbb Q}$ is surjective for any set $S$ of smooth projective varieties over $k$. This is a particular case of Theorem 7.1 in [4]. For any set $S$ of smooth projective varieties over $k$ each Grothendieck motive $N\in\mathcal{M}_k^{\sim}$ gives rise to an $A_S$-module
$$
\begin{equation*}
\mathfrak{M}^{\sim}_N(S):=\bigoplus_{X\in S}\mathrm{Hom}_{\mathcal{M}_k^{\sim}}(N,\mathbb M^{\sim}(X)).
\end{equation*}
\notag
$$
We omit symbol $\sim$ from the notation when $\sim$ denotes numerical equivalence $\sim_{\mathrm{num}}$. Proof. The $A_S$-action on $\mathfrak{M}_N(S)$ factors through an action of the algebra $A_S/{\sim_{\mathrm{num}}}$, while $A_S/{\sim_{\mathrm{rat}}} \cong\bigoplus_{X,Y\in S}\operatorname{CH}_{\dim X}(X\times_kY)_{\mathbb Q}$, so that
$$
\begin{equation*}
A_S/{\sim_{\mathrm{num}}} \cong\bigoplus_{X,Y\in S}\operatorname{CH}_{\dim X}(X\times_kY)_{\mathbb Q}/{\sim_{\mathrm{num}}}.
\end{equation*}
\notag
$$
By [7], $\mathcal{M}_k$ is an abelian semisimple category, and therefore any nondegenerate $(A_S/{\sim_{\mathrm{num}}})$-module is semisimple. In particular, so is $\mathfrak{M}_N(S)$. § 3. Loewy filtrations on $Z_0(S)$Modifying slightly the standard definition (see, for instance, [6]), a filtration of a module $M$ is called a Loewy filtration if it is finite, its successive quotients are semisimple and its length is minimal under these assumptions. Let $S$ be a set of smooth irreducible projective varieties over a field $k$. We are interested in Loewy filtrations on the $A_S$-module $Z_0(S)$. By Theorem 1 the socle (that is, the maximal semisimple submodule) of the $A_S$-module $Z_0(S)$ is $Z_0^{\mathrm{rat}}(S)$, while the radical (that is, the intersection of all maximal submodules) of the $A_S$-module $Z_0(S)$ is $Z_0^{\circ}(S)$, and $Z_0^{\mathrm{rat}}(S)$ is an essential submodule of $Z_0(S)$. The $A_S$-action on the quotient $\operatorname{CH}_0(S)_{\mathbb Q}:=Z_0(S)/Z_0^{\mathrm{rat}}(S)$ factors through an action of the quotient $A_S/{\sim_{\mathrm{rat}}}$ of $A_S$ by rational equivalence. 3.1. The case of curvesProposition 1. Let $S$ be a set of smooth projective curves over $k$. Then $Z_0^{\mathrm{rat}}(S)\subset Z_0^{\circ}(S)\subset Z_0(S)$ is the unique Loewy filtration on the $A_S$-module $Z_0(S)$. Proof. By Theorem 1 the socle of the $A_S$-module $Z_0(S)$ is simple and coincides with $Z_0^{\mathrm{rat}}(S)$, while $Z_0^{\circ}(S)$ is the unique maximal submodule of the $A_S$-module $Z_0(S)$. It remains only to check the semisimplicity of $Z_0^{\circ}(S)/Z_0^{\mathrm{rat}}(S)$.
One has $A_S/{\sim_{\mathrm{rat}}}=\bigoplus_{X,Y\in S}\mathrm{Pic}(X\times_kY)_{\mathbb Q}$. Then the subgroup
$$
\begin{equation*}
I:=\bigoplus_{X,Y\in S}\mathrm{Pic}^{\circ}(X\times_kY)_{\mathbb Q}
\end{equation*}
\notag
$$
is an ideal in $A_S/{\sim_{\mathrm{rat}}}$ such that $I^2=0$, while $(A_S/{\sim_{\mathrm{rat}}})/I=\bigoplus_{X,Y\in S}\mathrm{NS}(X\times_kY)_{\mathbb Q}$ is a semisimple algebra. Here $\mathrm{Pic}$ is the Picard group, $\mathrm{Pic}^{\circ}$ is the subgroup of algebraically trivial elements and $\mathrm{NS}:=\mathrm{Pic}/\mathrm{Pic}^{\circ}$ is the Néron-Severi group.
Then for any $(A_S/{\sim_{\mathrm{rat}}})$-module $M$ the submodule $IM$ and the quotient $M/IM$ can be considered as $(A_S/{\sim_{\mathrm{rat}}})/I$-modules, and thus they are semisimple. Applying this to the module $M=Z_0(S)/Z_0^{\mathrm{rat}}(S)$, we see that the $A_S$-module $IM=Z_0^{\circ}(S)/Z_0^{\mathrm{rat}}(S)$ is semisimple. The proposition is proved. 3.2. Consequences of the filtration conjectureAccording to the Bloch-Beilinson motivic filtration conjecture (see, for instance, [8], Conjecture 2.3, or [10], Conjecture 33), there should exist a neutral tannakian $\mathbb Q$-linear category $\mathcal{MM}_k$ (of mixed motives over $k$) containing the category $\mathcal{M}_k$ as the full subcategory of semisimple objects, covariant functors $H_i(-,\mathbb Q(j))$ (homology; $i,j\in\mathbb Z$) from the category of varieties over $k$ to $\mathcal{MM}_k$, and a functorial descending filtration $\mathcal F^{\bullet}$ on the Chow groups $\operatorname{CH}_q(X)_{\mathbb Q}$ for smooth projective $k$-varieties $X$ such that $\mathcal F^0\operatorname{CH}_q(X)_{\mathbb Q}=\operatorname{CH}_q(X)_{\mathbb Q}$ and
$$
\begin{equation*}
\operatorname{gr}^i_{\mathcal F}\operatorname{CH}_q(X)_{\mathbb Q}=\mathrm{Ext}^i_{\mathcal{MM}_k}\bigl(\mathbb Q(0),H_{2q+i}(X,\mathbb Q(-q))\bigr).
\end{equation*}
\notag
$$
As a part of the filtration conjecture, it is natural to assume Grothendieck’s ‘semisimplicity conjecture’ on the coincidence of homological $\otimes\,\mathbb Q$ and numerical equivalences, so that the motive $H_{2q+i}(X,\mathbb Q(-q))$ is semisimple by Jannsen’s theorem [7]. A simple effective motive $P\in\mathcal{M}_k$ is called primitive of weight $-i\leqslant 0$ if (i) $P\cong(X,\pi)$ for some $X$ and $\pi$ such that $\dim X=i$ and (ii) $\mathrm{Hom}_{\mathcal{M}_k}(P,\mathbb M(Y\times\mathbb P^1))=0$ for any smooth projective variety $Y$ of dimension $<i$. In particular, for $q=0$ Beilinson’s formula becomes
$$
\begin{equation*}
\begin{aligned} \, \operatorname{gr}^i_{\mathcal F}\operatorname{CH}_0(X)_{\mathbb Q} &=\mathrm{Ext}^i_{\mathcal{MM}_k}(\mathbb Q(0),H_i(X,\mathbb Q)) \\ &=\bigoplus_P\mathrm{Ext}^i_{\mathcal{MM}_k}(\mathbb Q(0),P)\otimes_{\mathrm{End}_{\mathcal{M}_k}(P)} \mathrm{Hom}_{\mathcal{MM}_k}(P,H_i(X,\mathbb Q)), \end{aligned}
\end{equation*}
\notag
$$
where $P$ runs over the isomorphism classes of simple primitive motives of weight $-i$, and we see that the spaces $\mathcal F^iCH_0(X)_{\mathbb Q}$ should be covariant functorial. For each set $S$ of smooth irreducible projective varieties over a field $k$ and each integer $i\geqslant 0$, consider
$$
\begin{equation*}
\mathcal F^i\operatorname{CH}_0(S)_{\mathbb Q}:=\bigoplus_{X\in S}\mathcal F^i\operatorname{CH}_0(X)_{\mathbb Q}.
\end{equation*}
\notag
$$
By the functoriality of $\mathcal F^{\bullet}$ this is an $A_S$-submodule of $\operatorname{CH}_0(S)_{\mathbb Q}$. The algebra $A_S$ acts on $\operatorname{gr}^i_{\mathcal F}\operatorname{CH}_0(S)_{\mathbb Q}$ via its action on the motives $H_i(X,\mathbb Q)$, so the $A_S$-action on $\operatorname{gr}^i_{\mathcal F}\operatorname{CH}_0(S)_{\mathbb Q}$ factors through an action of the quotient $A_S/{\sim_{\mathrm{num}}}$ of $A_S$, that is, of the algebra $B_S:=\bigoplus_{X,Y\in S}\operatorname{CH}_{\dim X}(X\times_kY)_{\mathbb Q}/{\sim_{\mathrm{num}}}$. As the algebra $B_S$ is semisimple, the $A_S$-module $\operatorname{gr}^i_{\mathcal F}\operatorname{CH}_0(S)_{\mathbb Q}$ is semisimple too. In particular, if the dimensions of the varieties in $S$ do not exceed $d$, then the length $\ell(S)$ of any Loewy filtration of $\operatorname{CH}_0(S)_{\mathbb Q}$ does not exceed ${d+1}$. (More precisely, $\ell(S)-1$ does not exceed the number of those $i$, $0\leqslant i\leqslant d$, for which $H_i(X,\mathbb Q)$ is not a Tate twist of an effective motive of weight $>-i$ for at least one ${X\in S}$.) It seems that the radical filtration on $\operatorname{CH}_0(S)_{\mathbb Q}$ (that is, the strictly descending sequence of iterated radicals) is the motivic one, but with the repeating terms omitted. Remark 2. Usually (see, for example, [1] or [8], Conjecture 2.3, and [10], § 5.3) one states the motivic conjectures in the contravariant setting, that is, instead of $\mathcal{M}_k$ one considers its dual category (which is, in fact, equivalent to $\mathcal{M}_k$), while the homology functors from the category of varieties over $k$ to $\mathcal{MM}_k$ are replaced by contravariant functors $H^i (-,\mathbb Q(j))$. Then the homological object is the Poincaré dual of the cohomological object $H^i(X,\mathbb Q)$, while the Beilinson formula for codimension $q$ Chow groups of smooth projective $k$-varieties $X$ can be rewritten as § 4. Correspondences on nonproper varieties?One could try to extend part 2) of Theorem 1 to collections $S$ of smooth varieties over $k$ that are not necessarily proper. However, as there are no nonconstant morphisms from projective varieties to affine ones, it seems that the structure of the $A_S$-module $Z_0(S)$ can be quite complicated. On the other hand, if the set $S$ is considered as a preadditive category, then $A_S$-modules become precosheaves with transfers (in analogy with the terminology of V. Voevodsky). To restrict the category of precosheaves one can pass to the category of cosheaves in some nontrivial Grothendieck topology where $Y\mapsto Z_0(Y)$ is a cosheaf. In [14] a Grothendieck topology was defined on the categories of schemes of finite type over noetherian bases, which is called the $h$-topology; see also [13], § 10. This topology is generated by a pretopology where coverings are finite families $(p_i\colon U_i\to X)_i$ of morphisms of finite type such that $\amalg p_i\colon\amalg U_i\to X$ is a universal topological epimorphism (that is, a subset of $X$ is open if and only if so is its preimage, and any base change has the same property). A precosheaf $\mathcal F$ of abelian groups on the category of schemes of finite type over $k$ is an $h$-cosheaf if the sequence
$$
\begin{equation*}
\mathcal F(U\times_XU)\xrightarrow{f_*\circ\mathrm{pr}_{1*}-f_*\circ\mathrm{pr}_{2*}} \mathcal F(U)\xrightarrow{f_*}\mathcal F(X)\to 0
\end{equation*}
\notag
$$
is exact for any $h$-covering $f\colon U\to X$. By an $h$-cosheaf on the category of smooth varieties over $k$ we mean the restriction of an $h$-cosheaf to the category of schemes of finite type over $k$. The following lemma is related somehow to Proposition 3.1.3 in [15], where $f$ is a Nisnevich cover. Lemma 3. If a quasi-compact morphism of schemes $Y\xrightarrow{f}X$ is surjective (on the sets of points) then Proof. Let $p$ be a closed point of $X$. Then $Y_p:=f^{-1}(p)$ is a nonempty closed subset of $Y$, so it suffices to show the existence of a closed point of $Y_p$. Suppose on the contrary that there are no closed points in $Y_p$. As $Y_p$ is quasi-compact, it can be covered by a finite collection $S$ of affine opens: $Y_p=\bigcup_{U\in S}U$. Let us construct recursively a sequence of points $q_i\in X$ and a sequence $U_1,U_2,\dots$ of elements of $S$ as follows: let $U_1$ be an arbitrary element of $S$ and $q_1$ be an arbitrary closed point of $U_1$; for $i>1$, if the closure $\overline{\{q_{i-1}\}}$ of $q_{i-1}$ is not contained in $U_1\cup\dots\cup U_{i-1}$, let (i) $q_i'$ be a point of $\overline{\{q_{i-1}\}}$ in the complement of $\bigcup_{j=1}^{i-1}U_j$, (ii) $U_i$ be an element of $S$ containing $q_i'$, (iii) $q_i$ be a closed point of $\overline{\{q_i'\}}\cap U_i$: $\overline{\{q_i\}}\cap U_i=\{q_i\}$.
Then $\overline{\{q_i\}}\cap(U_1\cup\dots\cup U_j)$ is a subset of $\{q_1,\dots,q_j\}$ for any $j\leqslant i$. As $S$ is finite, there is some $n$, $1\leqslant n\leqslant\#S$, such that $\overline{\{q_n\}}$ is contained in $U_1\cup\dots\cup U_n$. As the complement of $\bigcup_{j=1}^{n-1}U_j$ is closed, the set $\{q_n\}=\overline{\{q_n\}}\cap(X\setminus(\bigcup_{j=1}^{n-1}U_j))\subseteq U_n$ is also closed. The kernel of $f_*$ is spanned by the elements $q-q'$ for all closed points $q$ and $q'$ in $Y$ such that $f(q)=f(q')$. But $q-q'$ is the image of any closed point of $q\times_Xq'\subseteq Y\times_XY$. The lemma is proved. Remark 3. The proof of Lemma 3 can obviously be modified to show that the linear combinations of $k$-rational points on $k$-schemes of finite type ($X\mapsto\mathbb Q[X(k)]$) form a Nisnevich subcosheaf without transfers (that is, functorial with respect to morphisms of schemes, rather than with respect to finite correspondences) of the $h$-cosheaf with transfers $Z_0\colon X\mapsto Z_0(X)$. Lemma 3 suggests that, in the nonproper case, the category of $h$-cosheaves is more appropriate than the much larger category of $A_S$-modules. Then the natural guess is that the socle $\mathrm{Soc}(Z_0)$ of the $h$-cosheaf $Z_0$ is simple and consists of those 0-cycles that become rationally trivial on some smooth compactifications, while the radical filtration on $Z_0/\mathrm{Soc}(Z_0)$ is separable and coincides with the motivic one. AcknowledgementsDiscussions with Vadim Vologodsky, Sergey Gorchinskiy, Dmitry Kaledin and, especially, with Ivan Panin were very helpful for me.
Citation in format AMSBIB
\Bibitem{Rov23} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|