Chow groups of ind-schemes and extensions of Saito's filtration

Abstract

Let $K$ be a field of characteristic zero and let $Sm/K$ be the category of smooth and separated schemes over $K$. For an ind-scheme $\mathcal X$ (and more generally for any presheaf of sets on $Sm/K$), we define its Chow groups $\{CH^p(\mathcal X)\}_{p\in \mathbb Z}$. We also introduce Chow groups $\{\mathcal{CH}^p(\mathcal G)\}_{p\in \mathbb Z}$ for a presheaf with transfers $\mathcal G$ on $Sm/K$. Then, we show that we have natural isomorphisms of Chow groups $$ CH^p(\mathcal X)\cong \mathcal{CH}^p(Cor(\mathcal X))\qquad\forall\textrm{ }p \in \mathbb Z$$ where $Cor(\mathcal X)$ is the presheaf with transfers that associates to any $Y\in Sm/K$ the collection of finite correspondences from $Y$ to $\mathcal X$. Additionally, when $K=\mathbb C$, we show that Saito's filtration on the Chow groups of a smooth projective scheme can be extended to the Chow groups $CH^p(\mathcal X)$ and more generally, to the Chow groups of an arbitrary presheaf of sets on $Sm/\mathbb C$. Similarly, there exists an extension of Saito's filtration to the Chow groups of a presheaf with transfers on $Sm/\mathbb C$. Finally, when the ind-scheme $\mathcal X$ is ind-proper, we show that the isomorphism $CH^p(\mathcal X)\cong \mathcal{CH}^p(Cor(\mathcal X))$ is actually a filtered isomorphism.