# (Weil divisors : Cartier divisors) = (p-Cycles : ? )

Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense.

The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by codimention $1$ irreducible subvarieties.

The sheaf of Cartier divisors is $\mathrm{Cart}_X:=\mathcal{K}_X^{\times}/\mathcal{O}_X^{\times}$.

The group of Cartier divisors is $\mathrm{Cart}(X):=\Gamma(X,\mathrm{Cart}_X)$.

The group $Z^p(X)$ of $p$-cycles is the free abelian group generated by irreducible subvarieties of codimension $p$, so in particular $\mathrm{Div}(X)=Z^1(X)$. So the notion of $p$-cycle is a direct generalization of the notion of Weil divisor.

My question:

Is there an analogous notion of group of "Cartier $p$-cycles" $\mathrm{Cart}^p(X)$? If yes, is there a sheaf $\mathrm{Cart}^p_X$ such that (naturally in $X$) we have $\mathrm{Cart}^p(X)=\Gamma(X,\mathrm{Cart}^p_X)$?

• One naive attempt would be to say that Cartier divisors are "locally principal." To get a codimension p object we'd need something like "locally the zero set of p things" which we could take to mean, lci or something. I have no idea how to make a sheaf that describes that, though. – Matt Nov 1 '12 at 17:50