After yesterday’s post I had to explain today what point-modules and line-modules are and that one can really describe them as points in a (commutative) variety. Seemingly, the present focus on categorical methods scares possibly interested students away and none of them seems to know that this non-commutative projective algebraic geometry once dealt with very concrete examples.
Let us fix the setting : A will be a quadratic algebra, that is, A is a positively graded algebra, part of degree zero the basefield k, generated by its homogeneous part A1 of degree one (which we take to be of k-dimension n 1) and with all defining relations quadratic in these generators. Take m k-independent linear terms (that is, elements of A1) : l1,…,lm and consider the graded left A-module

L
= A/(Al1 + ... + Alm)

Clearly, the Hilbert series of this module (that is, the formal power series in t with coefficient of t^a the k-dimension of the homogeneous part of L of degree a) starts off with

Hilb(L,t) = 1  + (n+1-m) t  + ...

and we call L a linear d-dimensional module if the Hilbert series is the power series expansion of

1/(1-t)^{d +1} = 1  + (d+1)t   +(d
+1)(d +2)/2 t^2   ... 

In particular, if d=0 (that is, m=n) then L is said to be a point-module and if d=1 (that is, m=n-1) then L is said to be a line-module. To a d-dimensional linear module L one can associate a d-dimensional linear subspace of ordinary (that is, commutative) projective n-space P^n. To do this, identify

P^n
= P(A 1^*)

the projective space of the n 1 dimensional space of linear functions on the homogeneous part of degree one. Then each of the linear elements li determines a hyperplane V(li) in P^n and the intersection of the m hyperplanes V(l1),…,V(lm) is the wanted subspace. In particular, to a point-module corresponds a point in P^n and to a line-module a line in P^n. So, where is the non-commutativity of A hidden? Well, if P is a point-module

P
= P0  + P1 +  P2   +... 

(with all components P_a one dimensional) then the twisted module

P' = P1 +  P2  + P3  + ...

is again a point-module and the map P–>P’ defines an automorphism on the point variety. In low dimensions, it is often possible to reconstruct A from the point-variety and automorphism. In higher dimensions, one has to consider also the higher dimensional linear modules.
When I explained all this (far clumsier as it was a long time since I worked with this) I was asked for an elementary text on all this. ‘Why hasn’t anybody written a book on all this?’ Well, Paul Smith wrote such a book so have a look at his homepage. But then, it turned out that the version one can download from one of his course pages is a more recent and a lot more categorical version. There is no more sight of a useful book on non-commutative projective spaces and their linear modules which might give starting students an interesting way to learn some non-commutative algebra and the beginnings of algebraic geometry (commutative and non-commutative). So, hopefully Paul still has the old version around and will make it available… The only webpage on this I could find in short time are the slides of a talk by Michaela Vancliff.

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Artin, geometry, non-commutative, teaching

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Posted in geometry
Written on Sun, 06 June 2004 at 11:29 am
Tags: Artin, geometry, non-commutative, teaching
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