Francisco Miraglia

Modules in the category of sheaves over quantales

In this paper we develop the elementary theory of modules in the category Sh(Q) of sheaves over right-sided idempotent quantales. The main ingredient is the construction of a (first-order) logic sound for Sh(Q). As an application we prove that in Sh(Q), a finitely generated projective module is free (Theorem 7.2), a result that is relevant to the study of representation of non-commutative C∗-algebras.

Non-Commutative Topology and Quantales

The relationship between q-spaces (c.f. [9]) and quantum spaces (c.f. [5]) is studied, proving that both models coincide in the case of Spec A, the spectrum of a non-commutative C*-algebra A. It is shown that a sober T1 quantum space is a classical topological space. This difficulty is circumvented through a new definition of point in a quantale. With this new definition, it is proved that Lid A has enough points. A notion of orthogonality in quantum spaces is introduced, which permits us to express the usual topological properties of separation. The notion of stalks of sheaves over quantales is introduced, and some results in categorial model theory are obtained.

Lattice-ordered reduced special groups

Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups (RSG) which are a lattice under their natural representation partial order (for motivation see Open Problem 1, Introduction); we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular finite RSGs and RSGs of finite chain length are lattice ordered. We prove that the lattice property fails for the RSGs of function fields of real algebraic varieties over a uniquely ordered field dense in its real closure, unless their stability index is 1 (Section 3). We show that Open Problem 1 (a strong local-global principle) has a positive answer for the RSG of the field Q(X) (Theorem 4.1). In the final section we explore the meaning of Open Problem 1 for formally real fields, in terms of their orders and real valuations; we introduce (and employ) the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups.

The Profinite Hull of Special Groups and Local-Global Principles

We introduce the Profinite Hull functor of special groups, showing that it gives rise to a new (and strong) local-global principle, the subform reflection property. We also indicate applications of this principle to the abstract algebraic theory of quadratic forms.