We do two things. 1. As a corollary to a stronger linearisation result (Theorem A), we prove the finite Morley rank version of the Lie-Kolchin-Malcev theorem on Lie algebras (Corollary A2). 2. We classify Lie ring actions on modules of characteristic not 2, 3 and Morley rank 2 (Theorem B).
{"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}
This is a survey of noncommutative generalizations of the spectrum of a ring, written for the Notices of the American Mathematical Society.
这是为美国数学学会公告撰写的关于环谱的非交换概论的调查报告。
{"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}
This paper uses Galois maps to give a definition of generalized multiplier Hopf coquasigroups, and give a sufficient and necessary condition for a multiplier bialgebra to be a regular multiplier Hopf coquasigroup. Then coactions and Yetter-Drinfeld quasimodules of regular multiplier Hopf coquasigroups are also considered.
{"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}
This paper provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings. In particular, we investigate Goresnteinness, Gorenstein projective modules and singularity categories in the context of cleft extensions of rings. This setting includes triangular matrix rings, trivial extension rings and tensor rings, among others. Under certain conditions, we prove singular equivalences between the algebras in a cleft extension, unifying an abundance of known results. Moreover, we compare the big singularity categories of cleft extensions of rings in the sense of Krause.
{"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}
As the dual notion of projective modules over trusses, injective modules over trusses are introduced. The Schanuel Lemmas on projective and injective modules over trusses are exhibited in this paper.
{"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}
There are two outstanding questions about division algebras of prime degree $p$. The first is whether they are cyclic, or equivalently crossed products. The second is whether the center, $Z(F,p)$, of the generic division algebra $UD(F,p)$ is stably rational over $F$. When $F$ is characteristic 0 and contains a primitive $p$ root of one, we show that there is a connection between these two questions. Namely, we show that if $Z(F,p)$ is not stably rational then $UD(F,p)$ is not cyclic.
{"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
Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $mathbb{Z}_{> 0} cup {infty}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k in mathbb{Z}_{> 0}$ and $a in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.
研究了交换单元$M$中的可取消性等级，它是由$M$元素在$mathbb{Z}_{> 0} cup {infty}$中的稳定等级值决定的。在 $k inmathbb{Z}_{> 0}$ 和 $a in M$ 的情况下，确定了倍数 $ka$ 的稳定等级的行为。在细化单元 $M$ 的情况下，确定了 $M$ 的阿基米德成分中可能的稳定秩值。最后，讨论了由环上模块的同构等价类建立的单元的稳定秩。
{"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}
We show that any clone over a finite domain that has a quasi Maltsev operation and fully symmetric operations of all arities has an incoming minion homomorphism from I, the clone of all idempotent operations on a two element set. We use this result to show that in the pp-constructability poset the lower covers of the structure with all relations that are invariant under I are the transitive tournament on three vertices and structures in one-to-one correspondence with all finite simple groups.
{"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}
Many problems of arithmetic nature rely on the computation or analysis of values of $L$-functions attached to objects from geometry. Whilst basic analytic properties of the $L$-functions can be difficult to understand, recent research programs have shown that automorphic $L$-values are susceptible to study via algebraic methods linking them to Selmer groups. Iwasawa theory, pioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an algebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this work we aim to adapt Iwasawa theory to a new context of representations of the unitary group GU(2,1) at primes inert in the respective imaginary quadratic field. This requires a novel approach using the Schneider--Venjakob regulator map, working over locally analytic distribution algebras. Subsequently, we show vanishing of some Bloch--Kato Selmer groups when a certain $p$-adic distribution is non-vanishing. These results verify cases of the Bloch--Kato conjecture for GU(2,1) at inert primes in rank 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}
In this paper, first we use the higher derived brackets to construct an $L_infty$-algebra, whose Maurer-Cartan elements are $3$-Lie algebra morphisms. Using the differential in the $L_infty$-algebra that govern deformations of the morphism, we give the cohomology of a $3$-Lie algebra morphism. Then we study the rigidity and stability of $3$-Lie algebra morphisms using the established cohomology theory. In particular, we show that if the first cohomology group is trivial, then the morphism is rigid; if the second cohomology group is trivial, then the morphism is stable. Finally, we study the stability of $3$-Lie subalgebras similarly.
{"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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