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

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.Comment: Exposition improve

Classifying subcategories and the spectrum of a locally noetherian category

Let $\mathcal A$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal A$ using subsets of the spectrum $\mathfrak Spec(\mathcal A)$. Along the way, we also develop results in local algebra with respect to the category $\mathcal A$ that we believe to be of independent interest.Comment: 40 pages, some new results adde

Noetherian Schemes over abelian symmetric monoidal categories

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let $A$ be a commutative monoid object in an abelian symmetric monoidal category $(\mathbf C,\otimes,1)$ satisfying certain conditions and let $\mathcal E(A)=Hom_{A-Mod}(A,A)$. If the subobjects of $A$ satisfy a certain compactness property, we say that $A$ is Noetherian. We study the localisation of $A$ with respect to any $s\in \mathcal E(A)$ and define the quotient $A/\mathscr I$ of $A$ with respect to any ideal $\mathscr I\subseteq \mathcal E(A)$. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc) for schemes over $(\mathbf C,\otimes,1)$ . Our notion of a scheme over a symmetric monoidal category $(\mathbf C,\otimes,1)$ is that of To\"en and Vaqui\'e.Comment: Some proofs modified, some references adde