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 $\{C{H}^p(\mathcal{X}){\}}_{p\in \mathbb{Z}}$. We also introduce Chow groups $\{{\mathcal{C}\mathcal{H}}^p(\mathcal{G}){\}}_{p\in \mathbb{Z}}$ for a presheaf with transfers $\mathcal{G}$ on $Sm/K$. Then, we show that we have natural isomorphisms of Chow groups $$C{H}^p(\mathcal{X})\cong {\mathcal{C}\mathcal{H}}^p(Cor(\mathcal{X}))\forall p\in \mathbb{Z}$$ 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 $C{H}^p(\mathcal{X})$ 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 $C{H}^p(\mathcal{X})\cong {\mathcal{C}\mathcal{H}}^p(Cor(\mathcal{X}))$ is actually a filtered isomorphism.

Several changes