A morphism a : X Y between quasi-schemes will be a right exact functor
a* : Mod(F) Mod(X) possessing a right adjoint (denoted by a*). In this way
the quasi-schemes form a category (more precisely a two-category, see Appendix
If X is a quasi-compact quasi-separated commutative scheme then Mod(X)
will be the category of quasi-coherent sheaves on X. It is proved in [32] that
this is a Grothendieck category. Rosenberg in [29] has proved a reconstruction
theorem which allows one to recover X from Mod(X) (generalizing work of Gabriel
in the noetherian case). He has also announced that the functor which assigns to
a commutative scheme its associated quasi-scheme is fully faithful if we work over
Let X be a quasi-scheme. We think of objects in Mod(X) as sheaves of right
modules on X. However to define algebras on X, it is clear that we need bimodules
on X (see [34] for the case where X is commutative). Let us for the moment define
a bimodule on X as a right exact functor from Mod(X) to itself commuting with
direct limits. Then the category of bimodules is monoidal (the tensor product being
given by composition) and hence we can define algebra objects. Let A be such an
algebra object. It is routine to define an abelian category Mod(^4) of .4-modules.
So this seems like a reasonable starting point for the theory.
However a difficulty emerges if one wants to define Rees algebras. As we have
seen, the main point is to take the sum of the In for some subbimodule / of an
invertible bimodule C. In was defined as the image of I®n £®n. Unfortunately
to take an image one needs an abelian category, and I don't see how to prove that
the above definition of a bimodule yields an abelian category, even if we drop the
requirement that bimodules should commute with direct limits. In this paper we
sidestep this difficulty by defining the category of bimodules on X as the opposite
category of the category of left exact functors from Mod(X) to itself. Since left
exact functors are determined by their values on injectives, they trivially form an
abelian category. In this way one can define Rees algebras in reasonable generality
(see Definition 3.4.13).
We will say that a quasi-scheme X is noetherian if Mod(X) is locally noetherian.
That is, if Mod(X) is generated by mod(X). As already has been indicated above, in
this paper we will study a noetherian quasi-scheme X which contains a commutative
curve Y as a divisor. To make this more precise we denote the identity functor
on Mod(X) by oX- This is an algebra on X such that Mod(ox) = Mod(X).
We will assume that ox contains an invertible subbimodule ox{—Y) such that
Mod(ox/ox(-Y)) is equivalent to Mod(F).
We also need some sort of smoothness condition on X. Since it is obviously
sufficient to impose this in a neighborhood of F , we assume that every object in
Mod(F) has finite injective dimension in Mod(X). This is the same setting as in
[36], albeit cast in a somewhat different language.
Now p £ Y defines a subbimodule mp of ox which is the analog of the maximal
ideal corresponding to p. We put / = mpox(Y). Define
v =
e / e
This is the Rees algebra associated to I. We define the blowup X of X at p as
Proj V.
Two remarks are in order here.
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