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 A_1 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 A_1)

: 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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