Noncommutative graded domains with quadratic growth
The Gelfand-Kirillov dimension GK(--) is an extremely useful tool for the study of algebras over a field \(k\), since information about the growth of a \(k\)-algebra \(A\) often has far-reaching consequences for its structure. Thus, for example, if \(A\) is a domain (i.e. a ring without zero divisors) with \(\text{GK}(A)<\infty\), then \(A\) is left and right Ore. Or, by an important result of \textit{L. Small} and \textit{R. B. Warfield} jun. [J. Algebra 91, 386-389 (1984; Zbl 0545.16011)], if \(A\) is affine and prime with \(\text{GK}(A)=1\), then \(A\) is finite as a module over its center which turns out to be a finitely generated domain of dimension 1. Furthermore, if in this case \(A\) is a domain and \(k\) is algebraically closed, then \(A\) is commutative. Since no \(k\)-algebra can have Gelfand-Kirillov dimension strictly between 1 and 2, it is natural to investigate domains \(A\) with \(\text{GK}(A)=2\). In full generality this still seems to be intractable, so the authors restrict themselves to graded domains of Gelfand-Kirillov dimension two and provide a fairly complete classification of these. Let \(A=\bigoplus_{n\geq 0}A_n\) be a finitely graded domain (i.e. \(A\) is finitely generated and \(A_0\) is a finite dimensional vector space over \(k\)) with \(2\leq\text{GK}(A)\leq{11\over 5}\). Then the nonzero homogeneous elements of \(A\) form an Ore set, yielding the so-called graded quotient ring of \(A\), which is known to be a skew Laurent polynomial ring \(D[z,z^{-1};\sigma]\), where \(D\) is a division algebra, \(z\) is a homogeneous element of positive degree and \(\sigma\) is an automorphism of \(D\). The first major result of the paper under review is that \(D\) is finitely generated as a module over it center \(K\), and that \(K\) is a finitely generated field extension of \(k\) of transcendence degree one. It turns out that \(\text{GK}(A)=2\), which is proved by establishing \(\text{GK}(D)=1\) and then invoking the result of Small and Warfield mentioned above. That \(11\over 5\) appears in the above inequality is due to the technical limitations of the proof that \(\text{GK}(D)=1\), which is based on an intricate analysis of the relationship between the Hilbert series of \(A\) and the Ore condition. Since this number seems to have no obvious intrinsic significance, the authors conjecture that a graded domain cannot have Gelfand-Kirillov dimension strictly between 2 and 3. Assuming \(k\) to be algebraically closed yields \(D=K\), and it follows from algebraic geometry that \(K\) is the function field of a smooth projective curve, motivating a description of \(A\) in terms of geometric data. It is proved that if \(A\) is finitely generated by elements of degree 1 then, up to a finite dimensional vector space, \(A\) is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curve. In this case, \(A\) is always noetherian. Furthermore, if the order of the automorphism \(\sigma\) is infinite then \(A\) is primitive, and if \(\sigma\) has finite order then \(A\) is finitely generated as a module over its center. If \(A\) is not generated in degree one, then it is still noetherian and primitive when \(\sigma\) has infinite order, and a satisfactory description of the structure of \(A\) in geometric terms is obtained for this case as well. However, if the order of \(\sigma\) is finite, then \(A\) is less well-behaved. Although a PI-ring, it need not be noetherian.
graded domains of Gelfand-Kirillov dimension two
function fields of smooth projective curves
scientific article; zbMATH DE number 861963
scientific article