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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

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  • $\begingroup$ This is only a note about symbols, not about your question. Most algebraic geometers use the symbol $A_r(X)$ to mean the Chow group of $r$-dimensional cycles modulo rational equivalence. The equivalence relation of rational equivalence is very far from being an etale equivalence relation, e.g., think of zero cycles on a K3 surface of the form $(p,q)\mapsto \underline{p}-\underline{q}$. $\endgroup$ – Jason Starr Feb 8 '18 at 23:07
  • $\begingroup$ @JasonStarr Thanks. I'm not sure I understand the last part of your comment, though. I'm not asking about any adequate equivalence relation inducing an étale equivalence relation on Chow varieties. The question is really about the following general construction about monoid objects in schemes: I included the general question at the bottom $\endgroup$ – user113393 Feb 9 '18 at 2:32

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