Simple modules and non-commutative quadrics (Q2847693)

The article considers noncommutative quadrics, that is, noncommutative \(k\)-algebras in two variables, divided out by a polynomial of degree two: \(A=k\langle x,y\rangle/(f),\) \(k\) algebraically closed of characteristic \(0\). There is a canonical epimorphism \(\kappa:k\langle x,y\rangle\rightarrow k[x,y]\) where \(k[x,y]\) is the ordinary polynomial ring. The author uses the notation \(\kappa(f)=f_0\), and then note that a one-dimensional representation of \(A\) can be considered as a point on the commutative curve \(f_0\). If one adds an element from the commutator ideal \(([x,y])\) to \(f\), one gets the same commutative curve \(f_0\), and exactly the same one-dimensional representations.NEWLINENEWLINEFor a commutative ring \(R\), \(\text{Ext}^1_R(S,T)=0\) for two non-isomorphic simple modules \(S\) and \(T\). This is not the case in the noncommutative situation.The author uses information about these dimensions to classify the quadrics.NEWLINENEWLINEFor any two points \(P\;,Q\) on a noncommutative curve \(I=(f)\), there is a set of partial derivatives such that a Jacobian matrix \(J(I;P)(Q)\) can be defined. This is used to prove that for \(A=k\langle x_1,\dots,x_m\rangle/I\) and \(\phi_P\) and \(\psi_Q\) two simple one-dimensional representations with \(P\neq Q\), we have NEWLINE\[NEWLINE\dim_k\text{Ext}^1_A(k(P),k(Q))=m-1-\text{rk} J(I;P)(Q).NEWLINE\]NEWLINENEWLINENEWLINEFirst of all, it is proved that the commutative algebras corresponding to noncommutative quadrics are the six non-isomorphic algebras \(k[x,y]/(f)\) with (1) \(f=xy-1\), (2) \(f=xy\) (3) \(f=x\), (4) \(f=x^2\) (5) \(f=x(x-1)\), (or \(f=x(x+1))\), (6) \(f=1\) or \(f=0\). With this classification ready, the author goes through and classifies all noncommutative quadrics corresponding to each commutative case. This is done in various ways, by smart tricks and logical force. There is no emphasizing of the main result, and one has to go directly into the text to try to get the results. Except from that, the article is very interesting, it is in a more or less uninvestigated area, and it gives a lot of different techniques to use in the various situations that occur in noncommutative algebraic geometry.