{"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":"Geometric rigidity of simple modules for algebraic groups","authors":"Michael Bate, David I. Stewart","doi":"arxiv-2409.05221","DOIUrl":"https://doi.org/arxiv-2409.05221","url":null,"abstract":"Let k be a field, let G be a smooth affine k-group and V a finite-dimensional\u0000G-module. We say V is emph{rigid} if the socle series and radical series\u0000coincide for the action of G on each indecomposable summand of V; say V is\u0000emph{geometrically rigid} (resp.~emph{absolutely rigid}) if V is rigid after\u0000base change of G and V to bar k (resp.~any field extension of k). We show that\u0000all simple G-modules are geometrically rigid, though not in general absolutely\u0000rigid. More precisley, we show that if V is a simple G-module, then there is a\u0000finite purely inseparable extension k_V/k naturally attached to V such that\u0000V_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G\u0000turns on an investigation of algebras of the form Kotimes_k E where K and E\u0000are field extensions of k; we give an example of such an algebra which is not\u0000rigid as a module over itself. We establish the existence of the purely\u0000inseparable field extension k_V/k through an analogous version for artinian\u0000algebras. In the second half of the paper we apply recent results on the structure and\u0000representation theory of pseudo-reductive groups to gives a concrete\u0000description of k_V. Namely, we combine the main structure theorem of the\u0000Conrad--Prasad classification of pseudo-reductive G together with our previous\u0000high weight theory. For V a simple G-module, we calculate the minimal field of\u0000definition of the geometric Jacobson radical of End_G(V) in terms of the high\u0000weight of G and the Conrad--Prasad classification data; this gives a concrete\u0000construction of the field k_V as a subextension of the minimal field of\u0000definition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone\u0000the dimension formula for G we had previously established; we also use it to\u0000give a description of End_G(V) which includes a dimension formula.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192574","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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