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 $A_r(X,j)$ 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 $A_r(X,j) := M\times_kM/R$ should be a group object in algebraic spaces, if so, separated and locally of finite type, hence a $k$-group scheme.

In other words, $A_r(X,j)$ 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?