We study (relative) $mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $cal K$-Mittag-Leffler modules.
{"title":"Mittag-Leffler modules and definable\u0000 subcategories","authors":"P. Rothmaler","doi":"10.1090/CONM/730/14716","DOIUrl":"https://doi.org/10.1090/CONM/730/14716","url":null,"abstract":"We study (relative) $mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $cal K$-Mittag-Leffler modules.","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"224 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116211868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Multisorted modules and their model\u0000 theory","authors":"M. Prest","doi":"10.1090/CONM/730/14714","DOIUrl":"https://doi.org/10.1090/CONM/730/14714","url":null,"abstract":"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.","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123673046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.
{"title":"Derived categories for Grothendieck\u0000 categories of enriched functors","authors":"G. Garkusha, Darren J. R. Jones","doi":"10.1090/CONM/730/14708","DOIUrl":"https://doi.org/10.1090/CONM/730/14708","url":null,"abstract":"The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122410979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that a finite dimensional algebra is $tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $tau$-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms $Alongrightarrow B$ with ${rm Tor}_1^A(B,B)=0$. It follows that a finite dimensional algebra is $tau$-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.
我们证明了一个有限维代数是$tau$-倾斜有限的当且仅当它不允许大的淤积模。此外，我们还证明了对于一个$tau$-倾斜有限代数$ a $，在基支持$tau$-倾斜(即有限维淤积)模的同构类与${rm Tor}_1^ a (B,B)=0$的环上胚的等价类$ a 长列B$之间存在双射。由此得出，有限维代数是$ $倾斜有限的当且仅当只有有限多个等价类的环泛胚。
{"title":"A characterisation of 𝜏-tilting finite\u0000 algebras","authors":"Lidia Angeleri Hugel, F. Marks, Jorge Vit'oria","doi":"10.1090/CONM/730/14711","DOIUrl":"https://doi.org/10.1090/CONM/730/14711","url":null,"abstract":"We prove that a finite dimensional algebra is $tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $tau$-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms $Alongrightarrow B$ with ${rm Tor}_1^A(B,B)=0$. It follows that a finite dimensional algebra is $tau$-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114622272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the notion of intrinsic entropy for endomorphisms of Abelian groups to endomorphisms of modules over an Archimedean non-discrete valuation domain $R$, using the natural non-discrete length function introduced by Northcott and Reufel for such a category of modules. We prove that this notion of entropy is a length function for the category of $R[X]$-modules, it satisfies (a suitably adapted version of) the Intrinsic Algebraic Yuzvinski Formula and that it is essentially the unique invariant for $Mod(R[X])$ with these properties.
{"title":"Intrinsic valuation entropy","authors":"L. Salce, Simone Virili","doi":"10.1090/CONM/730/14717","DOIUrl":"https://doi.org/10.1090/CONM/730/14717","url":null,"abstract":"We extend the notion of intrinsic entropy for endomorphisms of Abelian groups to endomorphisms of modules over an Archimedean non-discrete valuation domain $R$, using the natural non-discrete length function introduced by Northcott and Reufel for such a category of modules. We prove that this notion of entropy is a length function for the category of $R[X]$-modules, it satisfies (a suitably adapted version of) the Intrinsic Algebraic Yuzvinski Formula and that it is essentially the unique invariant for $Mod(R[X])$ with these properties.","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114414064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
{"title":"Left determined morphisms and free\u0000 realisations","authors":"L. Gregory","doi":"10.1090/conm/730/14709","DOIUrl":"https://doi.org/10.1090/conm/730/14709","url":null,"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","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115470411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Describing models of Th(ℤ) in adelic\u0000 terms","authors":"A. Macintyre","doi":"10.1090/CONM/730/14712","DOIUrl":"https://doi.org/10.1090/CONM/730/14712","url":null,"abstract":"","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129781035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pure projective modules over non-singular\u0000 serial rings","authors":"P. Př́ıhoda","doi":"10.1090/CONM/730/14715","DOIUrl":"https://doi.org/10.1090/CONM/730/14715","url":null,"abstract":"","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124142406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Decidability and modules over Bézout\u0000 domains","authors":"C. Toffalori","doi":"10.1090/conm/730/14718","DOIUrl":"https://doi.org/10.1090/conm/730/14718","url":null,"abstract":"","PeriodicalId":318971,"journal":{"name":"Model Theory of Modules, Algebras and\n Categories","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129876922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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