Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

A stable range for quadratic forms over commutative rings

A commutative ring A has quadratic stable range 1 (qsr(A) = 1) if each primitive binary quadratic form over A represents a unit. It is shown that qsr(A) = 1 implies that every primitive quadratic form over A represents a unit, A has stable range 1 and finitely generated constant rank projectives over A are free. A classification of quadratic forms is provided over Bezout domains with characteristic other than 2, quadratic stable range 1, and a strong approximation property for a certain subset of their maximum spectrum. These domains include rings of holomorphic functions on connected noncompact Riemann surfaces. Examples of localizations of rings of algebraic integers are provided to show that the classical concept of stable range does not behave well in either direction under finite integral extensions and that qsr(A) = 1 does not descend from such extensions.     

Strongly stable rank and applications to matrix completion

We perform an in-depth study of strongly stable ranks of modules over a commutative ring. Here we define the strongly stable rank of a module to be the supremum of the stable ranks of its finitely generated submodules. As an application, we give non-Noetherian generalizations of known facts about outer products and matrix completions over PIRs and Dedekind domains. We construct Noetherian and non-Noetherian domains of arbitrary strongly stable rank. We also consider strongly n-generated ideals, and we characterize the rings in which every ideal is strongly 2-generated and the domains in which every ideal is strongly 3-generated.     

Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring $L_s$ in s indeterminates. Given a group homomorphism $p:{\mathbb{Z}}^s?{\mathbb{Z}}^t$ we let $p_{!}(C)=C{?}_{L_s}{L}_t$ denote the resulting induced complex over the Laurent polynomial ring $L_t$ in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that $b_k(p_{!}(C))>b_k(C)$, have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.