Modules as exact functors

We can define a module to be an exact functor on a small abelian category. This is explained and shown to be equivalent to the usual definition but it does offer a different perspective, inspired by the notions from model theory of imaginary sort and interpretation. A number of examples are worked through

Multisorted modules and their model theory

Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted extension by imaginaries. The model theory of multisorted modules works just as for the usual, 1-sorted modules. A number of examples are presented, some in considerable detail

Where are the vrais dévots and are they véritables gens de bien? Eloquent slippage in the Tartuffe controversy

The famous controversy provoked by Molière’s Tartuffe (1664–1669) is usually read in terms of vrais and faux dévots and thought to turn on the question of sincerity versus hypocrisy. Here the vrai-faux dichotomy is challenged and a third term introduced in the form of the véritable homme de bien of Molière’s Preface to the published edition of the play. In the slippage between a vrai dévot and a véritable homme de bien (considered by most critics to be synonymous), I argue, lies a value-judgment and the suggestion of an alternative, more secular worldview that persisted even in the 1669 version of the play. The scandal of Tartuffe thus lies less with the threat of religious hypocrisy and more with the possibility that true morality could be found outside the Church.PostprintPeer reviewe

Ringel's conjecture for domestic string algebras

We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.Comment: minor correction

Modules with irrational slope over tubular algebras

Let $A$ be a tubular algebra and let $r$ be a positive irrational. Let ${\mathcal D}_r$ be the definable subcategory of $A$-modules of slope $r$. Then the width of the lattice of pp formulas for ${\mathcal D}_r$ is $\infty$. It follows that if $A$ is countable then there is a superdecomposable pure-injective module of slope $r$.Comment: minor corrections/improvements to argument

Reconstructing projective schemes from Serre subcategories

Given a positively graded commutative coherent ring A which is finitely generated as an A_0-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y\subseteq Proj A of the form Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all i\in\Omega is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) --> (Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr A.Comment: some minor corrections mad

Torsion classes of finite type and spectra

Given a commutative ring R (respectively a positively graded commutative ring A=\ps_{j\geq 0}A_j which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y\subset Spec R (respectively Y\subset Proj A) of the form Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\Y_i) quasi-compact and open for all i\in\Omega, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R,O_R)-->(Spec(Mod R),O_{Mod R}) and (Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R}) and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices L_{tor}(Mod R) and L_{tor}(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes perf(R) and the torsion classes of finite type in Mod R is established