Left determined morphisms and free realisations

Abstract

We investigate the connection between Prest’s notion of the free realisation of a pp formula and Auslander’s notion of determiners of functor and morphisms. The aim of this note is to explain the connections between Auslander’s notion of morphisms and subfunctors determined by objects introduced in [Aus78] and Prest’s notion of free realisations of pp formulae introduced in [Pre88]. The concept of determiners of morphisms and subfunctors were largely ignored until recently. In the last 5-10 years, effort has been made to understand them (see for instance [Rin13], [Rin12], [Kra13]). On the other hand, the algebraic study of model theory of modules is unimaginable without the concept of free realisations of a pp formulae. In 2.4 we explicitly describe the connection between determiners of functors defined by pp formulae and free realisations of pp formulae. This will give another proof, 2.5, of the existence of left determiners of morphisms between finitely presented modules for artin algebras. We then use determiners and free realisations to show that if M ∈ mod-R and R is an artin algebra, then the lattice homomorphism ppR → ppR(M) which sends φ ∈ ppR to φ(M) ∈ ppR(M) has both a left and a right adjoint, both of which we explicitly describe. Finally, in section 3, we will show that pushing the ideas from section 2 slightly harder actually gives a proof of the existence of minimal left determiners of morphisms between finitely presented modules for artin algebras. Acknowledgements: The content of this note was developed while attending Mike Prest’s research group seminars while I was his postdoc in Manchester. I would like to thank him for introducing me to morphisms determined by objects and encouraging me to publish these results. I would Date: January 11, 2018. 2010 Mathematics Subject Classification. Primary 03C60, Secondary 16G10. The content of the paper was created while the author was a postdoc at the University of Manchester and prepared for publication while the author was a postdoc a the University of Camerino. The author acknowledges the support of EPSRC through Grant