right .A-modules, modulo the modules with right bounded grading (the "projective
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
© m
© - - -
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
y). Hence the analog of
the exceptional curve
D/mD = R/m ©
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
) . . .T n - 1 (ra)
n. Put L = D/r~1(m)D. One now verifies that dimL
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
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