Duality and noncommutative planes
The article considers extensions of simple modules over an associative ring \(A\). The main result is that if \(\mathfrak m\) and \(\mathfrak n\) are two two-sided ideals which have artinian factors, then \(\text{Ext}^1_A(A/\mathfrak m,A/\mathfrak n)\neq 0\) as left \(A\)-modules, if and only if the same is true with \(\mathfrak m, \mathfrak n\) as right \(A\)-modules. This implies that that algebras of the form \(k\langle x,y\rangle /(f)\) with \(f\in([x,y])\), which are the noncommutative models of the plane, are noetherian only in the case that \(f=([x,y])\). The main result can be considered as a sort of duality between simple right modules and simple left modules over a noncommutative ring \(A\) with unit, and the duality is used to give several examples of non-noetherian noncommutative curves. The proof of the main result is clear, simple, and algebraic. It is built on classical results on noncommutative rings. The most impressing thing about the article, is the easy explanation of the geometric meaning of noncommutative algebras: This includes the definition of noncommutative \(k\)-th partial derivatives and the corresponding Jacobi matrix. A nice geometric result in this direction is that for \(A=\langle x_1,\dots,x_m\rangle/I\) a \(k\)-algebra, and \(\phi_P\) and \(\phi_Q\) two \(1\)-dimensional representations of \(A\) corresponding to different points \(P\) and \(Q\), then \[ \dim_k\text{Ext}^1_A(P,Q)=m-1-\text{rk}J(I;P)(Q) \] where \(J(I;P)\) is the Jacobi matrix. A one-dimensional representation of an algebra \(A=k\langle x,y\rangle/(f)\) can be considered a point on the corresponding commutative curve \(f_0\), and \(f_0\) can be zero without \(f\) being so. Adding an element from the commutator ideal ([x,y]) to \(f\), we get the same commutative curve \(f_0\), but the \(\text{Ext}^1\)-groups usually change. The author gives the definition of noncommutative algebraic sets, and prove that they are generalizations of the commutative ones. All in all a very nice article explaining the basis of noncommutative algebraic geometry in a strict, but geometric way.
scientific article; zbMATH DE number 6371712