Presentations of modules when ideals need not be principal
This paper studies presentations of modules. A presentation is a surjection \(f: P\to U\), where P is a finitely generated projective \(\Lambda\)-module and U is a \(\Lambda\)-module. Generalizing elementary divisor theory, two presentations f, g of U by P are called equivalent if \(f=\beta g\alpha\) for some \(\beta\) \(\in Aut U\) and \(\alpha\) \(\in Aut P\). If \(\Lambda\) is a commutative PID, one can restate the main result of elementary divisor theory as: any two presentations of U by P are equivalent. In this article, \(\Lambda\) is semiprime and module finite over its noetherian one dimensional center R. It is shown that the set of stable equivalence classes of presentations, pres(P,U), forms an abelian group. (Under mild hypotheses, stable equivalence classes are actual equivalence classes, for example, this is always true for \(\Lambda\) commutative.) If \(\Lambda\) is a prime maximal R-order, then \(pres(P,U)=0\) and if, in addition the uniform rank of the kernel is at least two, then any two presentations of U by P are equivalent. In particular, this answers a fifty year old question of Nakayama in the special case that \(\Lambda\) is a noncommutative PID module finite over its center. (The two authors, together with \textit{C. Odenthal} [in ``Elementary divisor theorem for noncommutative PID's'', Proc. Am. Math. Soc. (1988)] have completely settled the Nakayama question.) If \(\Lambda\) is not a maximal order, then ``unique presentation'' still holds in some situations. For example, this is true when \(\Lambda\) is a finitely generated algebra over an algebraically closed field of characteristic zero and Z has finite length. To study the problem when \(\Lambda\) is not a maximal order, a Mayer- Vietoris sequence is used. From this one finds that if I is a conductor ideal from a maximal order \(\Gamma\), P is a progenerator, and ker f is faithful, then pres(P,U) is a homomorphic image of \(K_ 1(\Lambda /I\Lambda)\). If \(\Lambda\) is commutative and U of finite length, then pres(P,U) has bounded exponent. Examples are given to show that pres(P,U) can be rather large. If \(\Lambda\) is a classical \({\mathbb{Z}}\)-order, then \(| pres(P,U)| \leq n\) for \(n=n(\Lambda)\) depending only on \(\Lambda\). This is true for actual (not merely stable) equivalence classes as well if \(A={\mathbb{Q}}\Lambda\) has no simple summand \(M_ r(D)\) where D is a totally definite quaternion algebra. If A violates this ``Eichler condition'', then there is no bound on the number of presentations of U by P (as P, U vary), although there is bound for any given P, U.
finitely generated projective \(\Lambda \) - module
stable equivalence classes of presentations
noncommutative PID module finite over its center
scientific article; zbMATH DE number 4044035