1,640 research outputs found
Chow groups of ind-schemes and extensions of Saito's filtration
Let be a field of characteristic zero and let be the category of
smooth and separated schemes over . For an ind-scheme (and more
generally for any presheaf of sets on ), we define its Chow groups
. We also introduce Chow groups
for a presheaf with transfers
on . Then, we show that we have natural isomorphisms of Chow
groups where is the
presheaf with transfers that associates to any the collection of
finite correspondences from to . Additionally, when , we show that Saito's filtration on the Chow groups of a smooth projective
scheme can be extended to the Chow groups and more
generally, to the Chow groups of an arbitrary presheaf of sets on . Similarly, there exists an extension of Saito's filtration to the Chow
groups of a presheaf with transfers on . Finally, when the
ind-scheme is ind-proper, we show that the isomorphism
is actually a filtered
isomorphism.Comment: Exposition improve
Classifying subcategories and the spectrum of a locally noetherian category
Let be a locally noetherian Grothendieck category. In this
paper, we study subcategories of using subsets of the spectrum
. Along the way, we also develop results in local
algebra with respect to the category 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 be a commutative monoid object in an
abelian symmetric monoidal category satisfying certain
conditions and let . If the subobjects of
satisfy a certain compactness property, we say that is Noetherian. We study
the localisation of with respect to any and define the
quotient of with respect to any ideal . 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 .
Our notion of a scheme over a symmetric monoidal category is that of To\"en and Vaqui\'e.Comment: Some proofs modified, some references adde
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