1.2. CONSTRUCTION 3 rings which have the Hilbert series of three dimensional polynomial rings, together with a few other reasonable properties. They were classified in [7, 8, 9] and also, with different methods, in [13]. Let A be such a regular ring. We view X = Proj A as a quantum P 2 . Since on P 2 the anti-canonical sheaf has degree three, the zero divisor of a Poisson bracket will be a (possibly singular and non-reduced) elliptic curve. Therefore we would also expect X Proj A to contain an elliptic curve in some reasonable sense. It turns out that this is indeed true! It was shown in [7, 10, 8] that A contains a normal element g in degree three such that Proj A/gA is equivalent to the category of quasi-coherent sheaves over an elliptic curve Y. Thus if we actually identify Y with Proj A/gA then Y ^-» X. Now let p G Y. The previous discussion suggests that it should be possible to blow up p. However it is not clear how to proceed. Under the inclusion 7 ^ I , p corresponds to a so-called point module [9] over A. This is by definition a graded right module, which is generated in degree zero and which is one-dimensional in every degree. However such a point module is only a right module and hence it cannot be used to construct a Rees algebra. Our solution is to construct the Rees algebra directly over Proj A. To do this we have to invoke the theory of monads [24]. Since we only consider monads satisfying a lot of additional hypotheses we prefer to call our monads algebras. This is at variance with the use of "algebra" in the theory of categories [24] but in our context it seems reasonable. For us an algebra over an abelian category C is in principle an algebra object in the monoidal category of right exact functors from C to itself. There are however some technical problems with this so we end up using a less intuitive definition (see below). The importance of monads in non-commutative algebraic geometry was noticed by various people, in particular by Rosenberg. See for example [28, 23]. In the last chapter of his book Rosenberg actually defines a blow up of an arbitrary "closed" subcategory of an abelian category. While this definition is also in terms of monads, it is as far as I can see, somewhat different from ours. To see this let us again consider the affine case. Then Rosenberg's construction is in terms of the functor M i—• Mm, which is not right exact. If we replace this functor by M H- M S^R m then one would get the Proj of the ordinary Rees algebra of R, which (depending on what one wants to achieve) might not be the right answer (as we have shown above). 1.2. Constructio n Following [28] we introduce the notion of a quasi-scheme. For us this will be a Grothendieck category (that is : an abelian category with a generator and exact direct limits). However we tend to think of quasi-schemes as geometric objects, so we denote them by roman capitals X , Y, .. . . If we really refer to the category represented by a quasi-scheme X then we write Mod(X). Note that in fact X = Mod(X), but it is very useful to nevertheless make this notational distinction since it allows us to introduce other notations in a consistent way. For example we will denote the noetherian objects in Mod(X) by mod(X). Furthermore we can absorb additional structure into the symbol X (such as a morphism to a base quasi-scheme) which is not related to Mod(X). This would be awkward without the two different notations X and Mod(X).
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