Let $X$ be a quasi-projective variety over a perfect field $k$. Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety parametrizing effective proper cycles on $X$ of codimension $r\ge 0$.
Is there some variant of the Chow construction, so as to provide a quasi-projective variety $\text{Cyc}_r(X,j)$ parametrizing (proper?) cycles on $X$? Namely, removing the effectivity condition?
A naive attempt is, calling $M := \text{Chow}_r(X,j)$, to take the quotient of $M\times_kM$ by the equivalence relation $R$ defined by
$$R := (M\times_k M\times_k M\times_k M)\times_{\mu, M\times_kM, \Delta} M$$
where the map $\mu:M\times_k M\times_k M\times_k M\to M\times_kM$ sends $(a,b,c,d)$ to $(a + d, b+c)$ and the map $\Delta : M\to M\times_kM$ is the diagonal.
It seems $R$ is an étale equivalence relation, and then $\text{Cyc}_r(X,j) := M\times_kM/R$ should be a group object in algebraic spaces, if so, separated and locally of finite type, hence a $k$-group scheme.
In other words, $\text{Cyc}_r(X,j)$ is meant to be a naive "scheme theoretic group completion", using the fact that $M$ is a monoid object in $k$-schemes under usual addition of effective cycles, and that it is cancellative.
Is this too naive a try? Is there a construction that actually works?
EDIT: a general question about monoid objects in algebraic spaces.
Let $S$ be a scheme, $M\to S$ a (commutative) cancellative monoid object in algebraic spaces (ie. for any $S$-scheme $T$, the set $M(T)$ is a commutative monoid with zero, functorially in $T$, and it is cancellative).
We define $M^{\rm gp}$ the fppf quotient sheaf of $M\times_SM$ by the following equivalence relation:
$$R := (M\times_SM\times_SM\times_SM)\times_{\mu, M\times_SM,\Delta_{M/S}}M$$
with:
- $\mu : M\times_SM\times_SM\times_SM\to M\times_SM$ the map defined functorially on $T$-sections by sending $(a_T, b_T, c_T, d_T)$ to $(a_T+d_T, b_T+c_T)$.
- $\Delta_{M/S} : M\to M\times_SM$ is the diagonal.
- $s,t : R\to M\times_SM$ are defined to be the pullback along $\mu$ of $\text{pr}_1\circ\Delta_{M/S}$ and $\text{pr}_2\circ\Delta_{M/S}$.
Is $M^{\rm gp}$ a (commutative) group object in algebraic spaces? (ie. is it an algebraic space?)
Remark.
If so, the naive idea is then to consider the algebraic space $\text{Cyc}_r(X,j) := \text{Chow}_r(X,j)^{\rm gp}$, which would be, in addition, locally of finite type and separated over $k$, hence a $k$-group scheme.
EDIT: see the more general question Algebraic-space theoretic group completions