{"title":"Soluble Lie rings of finite Morley rank","authors":"Adrien Deloro, Jules Tindzogho Ntsiri","doi":"arxiv-2409.07783","DOIUrl":"https://doi.org/arxiv-2409.07783","url":null,"abstract":"We do two things. 1. As a corollary to a stronger linearisation result\u0000(Theorem A), we prove the finite Morley rank version of the Lie-Kolchin-Malcev\u0000theorem on Lie algebras (Corollary A2). 2. We classify Lie ring actions on\u0000modules of characteristic not 2, 3 and Morley rank 2 (Theorem B).","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192650","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":"A tour of noncommutative spectral theories","authors":"Manuel Reyes","doi":"arxiv-2409.08421","DOIUrl":"https://doi.org/arxiv-2409.08421","url":null,"abstract":"This is a survey of noncommutative generalizations of the spectrum of a ring,\u0000written for the Notices of the American Mathematical Society.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247420","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":"Multiplier Hopf coquasigroup: Definition and Coactions","authors":"Tao Yang","doi":"arxiv-2409.07788","DOIUrl":"https://doi.org/arxiv-2409.07788","url":null,"abstract":"This paper uses Galois maps to give a definition of generalized multiplier\u0000Hopf coquasigroups, and give a sufficient and necessary condition for a\u0000multiplier bialgebra to be a regular multiplier Hopf coquasigroup. Then\u0000coactions and Yetter-Drinfeld quasimodules of regular multiplier Hopf\u0000coquasigroups are also considered.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"395 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192651","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":"Cleft extensions of rings and singularity categories","authors":"Panagiotis Kostas","doi":"arxiv-2409.07919","DOIUrl":"https://doi.org/arxiv-2409.07919","url":null,"abstract":"This paper provides a systematic treatment of Gorenstein homological aspects\u0000for cleft extensions of rings. In particular, we investigate Goresnteinness,\u0000Gorenstein projective modules and singularity categories in the context of\u0000cleft extensions of rings. This setting includes triangular matrix rings,\u0000trivial extension rings and tensor rings, among others. Under certain\u0000conditions, we prove singular equivalences between the algebras in a cleft\u0000extension, unifying an abundance of known results. Moreover, we compare the big\u0000singularity categories of cleft extensions of rings in the sense of Krause.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192649","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":"Injectivity of modules over trusses","authors":"Yongduo Wang, Shujuan Han, Dengke Jia, Jian He, Dejun Wu","doi":"arxiv-2409.07023","DOIUrl":"https://doi.org/arxiv-2409.07023","url":null,"abstract":"As the dual notion of projective modules over trusses, injective modules over\u0000trusses are introduced. The Schanuel Lemmas on projective and injective modules\u0000over trusses are exhibited in this paper.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192652","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":"Stable Rationality and Cyclicity","authors":"David J Saltman","doi":"arxiv-2409.07240","DOIUrl":"https://doi.org/arxiv-2409.07240","url":null,"abstract":"There are two outstanding questions about division algebras of prime degree\u0000$p$. The first is whether they are cyclic, or equivalently crossed products.\u0000The second is whether the center, $Z(F,p)$, of the generic division algebra\u0000$UD(F,p)$ is stably rational over $F$. When $F$ is characteristic 0 and\u0000contains a primitive $p$ root of one, we show that there is a connection\u0000between these two questions. Namely, we show that if $Z(F,p)$ is not stably\u0000rational then $UD(F,p)$ is not cyclic.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192680","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}
Pere Ara, Ken Goodearl, Pace P. Nielsen, Kevin C. O'Meara, Enrique Pardo, Francesc Perera
{"title":"Levels of cancellation for monoids and modules","authors":"Pere Ara, Ken Goodearl, Pace P. Nielsen, Kevin C. O'Meara, Enrique Pardo, Francesc Perera","doi":"arxiv-2409.06880","DOIUrl":"https://doi.org/arxiv-2409.06880","url":null,"abstract":"Levels of cancellativity in commutative monoids $M$, determined by stable\u0000rank values in $mathbb{Z}_{> 0} cup {infty}$ for elements of $M$, are\u0000investigated. The behavior of the stable ranks of multiples $ka$, for $k in\u0000mathbb{Z}_{> 0}$ and $a in M$, is determined. In the case of a refinement\u0000monoid $M$, the possible stable rank values in archimedean components of $M$\u0000are pinned down. Finally, stable rank in monoids built from isomorphism or\u0000other equivalence classes of modules over a ring is discussed.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192676","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":"Finite Simple Groups in the Primitive Positive Constructability Poset","authors":"Sebastian Meyer, Florian Starke","doi":"arxiv-2409.06487","DOIUrl":"https://doi.org/arxiv-2409.06487","url":null,"abstract":"We show that any clone over a finite domain that has a quasi Maltsev\u0000operation and fully symmetric operations of all arities has an incoming minion\u0000homomorphism from I, the clone of all idempotent operations on a two element\u0000set. We use this result to show that in the pp-constructability poset the lower\u0000covers of the structure with all relations that are invariant under I are the\u0000transitive tournament on three vertices and structures in one-to-one\u0000correspondence with all finite simple groups.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192677","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":"Iwasawa Theory for GU(2,1) at inert primes","authors":"Muhammad Manji","doi":"arxiv-2409.05664","DOIUrl":"https://doi.org/arxiv-2409.05664","url":null,"abstract":"Many problems of arithmetic nature rely on the computation or analysis of\u0000values of $L$-functions attached to objects from geometry. Whilst basic\u0000analytic properties of the $L$-functions can be difficult to understand, recent\u0000research programs have shown that automorphic $L$-values are susceptible to\u0000study via algebraic methods linking them to Selmer groups. Iwasawa theory,\u0000pioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an\u0000algebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this\u0000work we aim to adapt Iwasawa theory to a new context of representations of the\u0000unitary group GU(2,1) at primes inert in the respective imaginary quadratic\u0000field. This requires a novel approach using the Schneider--Venjakob regulator\u0000map, working over locally analytic distribution algebras. Subsequently, we show\u0000vanishing of some Bloch--Kato Selmer groups when a certain $p$-adic\u0000distribution is non-vanishing. These results verify cases of the Bloch--Kato\u0000conjecture for GU(2,1) at inert primes in rank 0.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192683","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":"Stability and rigidity of 3-Lie algebra morphisms","authors":"Jun Jiang, Yunhe Sheng, Geyi Sun","doi":"arxiv-2409.05041","DOIUrl":"https://doi.org/arxiv-2409.05041","url":null,"abstract":"In this paper, first we use the higher derived brackets to construct an\u0000$L_infty$-algebra, whose Maurer-Cartan elements are $3$-Lie algebra morphisms.\u0000Using the differential in the $L_infty$-algebra that govern deformations of\u0000the morphism, we give the cohomology of a $3$-Lie algebra morphism. Then we\u0000study the rigidity and stability of $3$-Lie algebra morphisms using the\u0000established cohomology theory. In particular, we show that if the first\u0000cohomology group is trivial, then the morphism is rigid; if the second\u0000cohomology group is trivial, then the morphism is stable. Finally, we study the\u0000stability of $3$-Lie subalgebras similarly.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4291 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192679","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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