2 1. INTRODUCTION

right .A-modules, modulo the modules with right bounded grading (the "projective

case").

Let us first consider the affine case. Assume that R is a finitely generated

/c-algebra and let C be the commutator ideal. C is the natural analog of the zero

divisor of a Poisson bracket. Now Spec R/C is a commutative affine scheme and a

/c-point in Spec R/C corresponds to a maximal ideal m i n i ? with R/m — k. Hence

a natural idea is to define the blowup of Spec R at p as Proj D where D is the Rees

algebra associated to m,

L = i ? © r a © r a

2

© m

3

© - - -

It is easily seen however that this definition is faulty. Consider the following example

[4] R = k(x, y)/{yx — xy — y), m = (x,y). Then

mn

=

(xn,

y). Hence the analog of

the exceptional curve

D/mD = R/m ©

ra/ra2

©

ra2/m3©

is isomorphic to k[x]. Thus Proj D/m is a point, whereas intuitively we would

expect it to be one-dimensional in some sense.

It turns out however that in this example one can use a certain twisting of the

Rees algebra which yields a reasonably behaved blowing up. This is based on the

observation that the commutator ideal C = (y) is an invertible i?-bimodule. Let J

be its inverse and put I = mJ. Then we define In as the image of I®n in J®n and

we define the modified Rees algebra D as

£ = # © / © J2 © J 3 © . . .

The blowup of Specif at p is now defined as Proj D for this new D. We refer

the reader to [4] for a detailed workout of this example. However we will indicate

how one finds the analog of the exceptional curve. Let r be the automorphism

of R given by a ^ y~lay. For M an i?-bimodule let Mr be the bimodule whose

right inaction is twisted by tau (i.e. m • r = mr(r)). Then J = RT and hence

jn _

m r

(

m

) . . .T n - 1 (ra)

r

n. Put L = D/r~1(m)D. One now verifies that dimL

n

=

u + 1. So L plays the role of the exceptional curve. Note however that L is a right

D-module but not a left module. In retrospect this was to be expected since, as we

have said in the first paragraph, if we blow up a Poisson surface, then the extended

Poisson bracket (if it exists) will in general not vanish on the exceptional curve.

This example indicates the way to go for rings whose commutator ideal is

invertible. The latter hypothesis is not unreasonable since if we look at the case of

a Poisson surface then we see that we expect a non-commutative smooth surface

to contain a commutative curve. Additional motivation comes from considering

the local rings k((x,y))/((/) with fi a (non-commutative) formal power series whose

lowest degree term is a non-degenerate quadratic tensor in x,y. These local rings

are the non-commutative analogs of complete two dimensional regular local rings

and one verifies that their commutator ideal is indeed invertible (see e.g. [36]).

There is one important hitch however. The commutator ideal is not invariant

under Morita equivalence! This indicates that it is important to develop the the-

ory in a more category-theoretical frame work. This will make it possible to talk

about non-commutative schemes containing a commutative curve, without refering

to rings or ideals at all.

To stress this point even more let us consider the case of graded rings. In [7]

Artin and Schelter introduced so-called regular rings. These are basically graded