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Search Results: 1 - 10 of 2211 matches for " Mike Prest "
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Spectra of small abelian categories
Mike Prest
Mathematics , 2012,
Abstract: We investigate the Ziegler and Zariski topologies on the lattice of Serre subcategories of a small abelian category.
Categories of imaginaries for definable additive categories
Mike Prest
Mathematics , 2012,
Abstract: We develop some aspects of the model theory of additive structures, with particular emphasis on the abelian category of pp-imaginaries.
Abelian categories and definable additive categories
Mike Prest
Mathematics , 2012,
Abstract: We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".
Reconstructing projective schemes from Serre subcategories
Grigory Garkusha,Mike Prest
Mathematics , 2006,
Abstract: 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.
Ringel's conjecture for domestic string algebras
Gena Puninski,Mike Prest
Mathematics , 2014,
Abstract: We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.
Modules with irrational slope over tubular algebras
Richard Harland,Mike Prest
Mathematics , 2013, DOI: 10.1112/plms/pdu068
Abstract: 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 $r$.
Representation embeddings, interpretation functors and controlled wild algebras
Lorna Gregory,Mike Prest
Mathematics , 2014,
Abstract: We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.
Torsion classes of finite type and spectra
Grigory Garkusha,Mike Prest
Mathematics , 2006,
Abstract: 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.
The Gabriel-Roiter filtration of the Ziegler spectrum
Henning Krause,Mike Prest
Mathematics , 2011,
Abstract: Inclusion preserving maps from modules over an Artin algebra to complete partially ordered sets are studied. This yields a filtration of the Ziegler spectrum which is indexed by all Gabriel-Roiter measures. Another application is a compactness result for the set of subcategories of finitely presented modules that are closed under submodules.
Almost dual pairs and definable classes of modules
Akeel Ramadan Mehdi,Mike Prest
Mathematics , 2013,
Abstract: Holm (H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543-560) considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his results by placing them in the context of elementary duality on definable subcategories
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