Pub Date : 2022-02-18DOI: 10.2140/pjm.2022.320.391
Chao Qian, Zizhou Tang, Wenjiao Yan
Associated with a symmetric Clifford system ${P_0, P_1,cdots, P_{m}}$ on $mathbb{R}^{2l}$, there is a canonical vector bundle $eta$ over $S^{l-1}$. For $m=4$ and $8$, we construct explicitly its characteristic map, and determine completely when the sphere bundle $S(eta)$ associated to $eta$ admits a cross-section. These generalize the results in cite{St51} and cite{Ja58}. As an application, we establish new harmonic representatives of certain elements in homotopy groups of spheres (cf. cite{PT97} cite{PT98}). By a suitable choice of Clifford system, we construct a metric of non-negative curvature on $S(eta)$ which is diffeomorphic to the inhomogeneous focal submanifold $M_+$ of OT-FKM type isoparametric hypersurfaces with $m=3$.
{"title":"Clifford systems, harmonic maps and metrics with nonnegative curvature","authors":"Chao Qian, Zizhou Tang, Wenjiao Yan","doi":"10.2140/pjm.2022.320.391","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.391","url":null,"abstract":"Associated with a symmetric Clifford system ${P_0, P_1,cdots, P_{m}}$ on $mathbb{R}^{2l}$, there is a canonical vector bundle $eta$ over $S^{l-1}$. For $m=4$ and $8$, we construct explicitly its characteristic map, and determine completely when the sphere bundle $S(eta)$ associated to $eta$ admits a cross-section. These generalize the results in cite{St51} and cite{Ja58}. As an application, we establish new harmonic representatives of certain elements in homotopy groups of spheres (cf. cite{PT97} cite{PT98}). By a suitable choice of Clifford system, we construct a metric of non-negative curvature on $S(eta)$ which is diffeomorphic to the inhomogeneous focal submanifold $M_+$ of OT-FKM type isoparametric hypersurfaces with $m=3$.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42125351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we propose a reduction scheme for polydifferential operators phrased in terms of $L_infty$-morphisms. The desired reduction $L_infty$-morphism has been obtained by applying an explicit version of the homotopy transfer theorem. Finally, we prove that the reduced star product induced by this reduction $L_infty$-morphism and the reduced star product obtained via the formal Koszul complex are equivalent.
{"title":"The strong homotopy structure of BRST reduction","authors":"C. Esposito, Andreas Kraft, Jonas Schnitzer","doi":"10.2140/pjm.2023.325.47","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.47","url":null,"abstract":"In this paper we propose a reduction scheme for polydifferential operators phrased in terms of $L_infty$-morphisms. The desired reduction $L_infty$-morphism has been obtained by applying an explicit version of the homotopy transfer theorem. Finally, we prove that the reduced star product induced by this reduction $L_infty$-morphism and the reduced star product obtained via the formal Koszul complex are equivalent.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42638365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-04DOI: 10.2140/pjm.2022.319.189
E. Nart
. Let ( K, v ) be a valued field and let ( K h , v h ) be the henselization determined by the choice of an extension of v to an algebraic closure of K . Consider an embedding v ( K ∗ ) ֒ → Λ of the value group into a divisible ordered abelian group. Let T ( K, Λ), T ( K h , Λ) be the trees formed by all Λ-valued extensions of v , v h to the polynomial rings K [ x ], K h [ x ], respectively. We show that the natural restriction mapping T ( K h , Λ) → T ( K, Λ) is an isomorphism of posets. As a consequence, the restriction mapping T v h → T v is an isomorphism of posets too, where T v , T v h are the trees whose nodes are the equivalence classes of valuations on K [ x ], K h [ x ] whose restriction to K , K h are equivalent to v , v h , respectively.
． 设(K, v)是一个值域，设(K h, v h)是由选择v的扩展到K的代数闭包所决定的自化。考虑将值群的v (K∗)→Λ嵌入到可整除的有序阿贝尔群中。设T (K， Λ)， T (K, h， Λ)分别为v, v h对多项式环K [x]， K h [x]的所有Λ-valued次扩展所形成的树。我们证明了自然约束映射T (K h， Λ)→T (K， Λ)是偏序集的同构。因此，约束映射T v h→T v也是偏序集的同构，其中T v, T v h是树，其节点是K [x]， K h [x]上赋值的等价类，其对K, K h的约束分别等价于v, v h。
{"title":"Rigidity of valuative trees under henselization","authors":"E. Nart","doi":"10.2140/pjm.2022.319.189","DOIUrl":"https://doi.org/10.2140/pjm.2022.319.189","url":null,"abstract":". Let ( K, v ) be a valued field and let ( K h , v h ) be the henselization determined by the choice of an extension of v to an algebraic closure of K . Consider an embedding v ( K ∗ ) ֒ → Λ of the value group into a divisible ordered abelian group. Let T ( K, Λ), T ( K h , Λ) be the trees formed by all Λ-valued extensions of v , v h to the polynomial rings K [ x ], K h [ x ], respectively. We show that the natural restriction mapping T ( K h , Λ) → T ( K, Λ) is an isomorphism of posets. As a consequence, the restriction mapping T v h → T v is an isomorphism of posets too, where T v , T v h are the trees whose nodes are the equivalence classes of valuations on K [ x ], K h [ x ] whose restriction to K , K h are equivalent to v , v h , respectively.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43470199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-03DOI: 10.2140/pjm.2023.324.111
Urs Fuchs, J. Purcell, J. Stewart
It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.
{"title":"Constructing knots with specified geometric limits","authors":"Urs Fuchs, J. Purcell, J. Stewart","doi":"10.2140/pjm.2023.324.111","DOIUrl":"https://doi.org/10.2140/pjm.2023.324.111","url":null,"abstract":"It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46713424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.2140/pjm.2022.320.223
N. Fakhruddin, Chandrashekhar B. Khare, Stefan Patrikis
We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and trianguline at all primes of $F$ dividing $p$. We also prove variants of this result for representations valued in connected reductive groups.
{"title":"Trianguline lifts of global mod p Galois\u0000representations","authors":"N. Fakhruddin, Chandrashekhar B. Khare, Stefan Patrikis","doi":"10.2140/pjm.2022.320.223","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.223","url":null,"abstract":"We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and trianguline at all primes of $F$ dividing $p$. We also prove variants of this result for representations valued in connected reductive groups.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42322380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-24DOI: 10.2140/pjm.2022.318.433
G. Chang, Victor Fadinger, Daniel Windisch
. Let D be an integral domain and Γ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group G . In this paper, we show that if char( D ) = 0 (resp., char( D ) = p > 0), then D [Γ] is a weakly Krull domain if and only if D is a weakly Krull UMT-domain, Γ is a weakly Krull UMT-monoid, and G is of type (0 , 0 , 0 ,... ) (resp., type (0 , 0 , 0 ,... ) except p ). Moreover, we give arithmetical applications of this result. Our results show that there is also a class of weakly Krull domains, which are not Krull but have full system of sets of lengths.
{"title":"Semigroup rings as weakly Krull domains","authors":"G. Chang, Victor Fadinger, Daniel Windisch","doi":"10.2140/pjm.2022.318.433","DOIUrl":"https://doi.org/10.2140/pjm.2022.318.433","url":null,"abstract":". Let D be an integral domain and Γ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group G . In this paper, we show that if char( D ) = 0 (resp., char( D ) = p > 0), then D [Γ] is a weakly Krull domain if and only if D is a weakly Krull UMT-domain, Γ is a weakly Krull UMT-monoid, and G is of type (0 , 0 , 0 ,... ) (resp., type (0 , 0 , 0 ,... ) except p ). Moreover, we give arithmetical applications of this result. Our results show that there is also a class of weakly Krull domains, which are not Krull but have full system of sets of lengths.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45873098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-18DOI: 10.2140/pjm.2022.320.425
A. Singh, Rupam Barman
For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$.
{"title":"Proofs of some conjectures of Keith and Zanello\u0000on t-regular partition","authors":"A. Singh, Rupam Barman","doi":"10.2140/pjm.2022.320.425","DOIUrl":"https://doi.org/10.2140/pjm.2022.320.425","url":null,"abstract":"For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44558159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-07DOI: 10.2140/pjm.2023.323.307
Cleto B. Miranda-Neto, D. S. Queiroz, Thyago S. Souza
In this paper we further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem of when the operations of taking the Hom functor and horizontal linkage preserve the Ulrich property. One of the applications is a new characterization of quadratic hypersurface rings. Moreover, in the Gorenstein case, we deduce that applying linkage to sufficiently high syzygy modules of Ulrich ideals yields Ulrich modules. Finally, we explore connections to the theory of modules with minimal multiplicity, and as a byproduct we determine the Chern number of an Ulrich module as well as the Castelnuovo-Mumford regularity of its Rees module.
{"title":"On the theory of generalized Ulrich modules","authors":"Cleto B. Miranda-Neto, D. S. Queiroz, Thyago S. Souza","doi":"10.2140/pjm.2023.323.307","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.307","url":null,"abstract":"In this paper we further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem of when the operations of taking the Hom functor and horizontal linkage preserve the Ulrich property. One of the applications is a new characterization of quadratic hypersurface rings. Moreover, in the Gorenstein case, we deduce that applying linkage to sufficiently high syzygy modules of Ulrich ideals yields Ulrich modules. Finally, we explore connections to the theory of modules with minimal multiplicity, and as a byproduct we determine the Chern number of an Ulrich module as well as the Castelnuovo-Mumford regularity of its Rees module.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47118755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This paper re-develops Nevanlinna theory for meromorphic functions on C in the viewpoint of holomorphic forms. According to our observation, Nevanlinna’s functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna’s functions using a holomorphic form S . With the new settings, an analogue of Nevanlinna theory on the S -exhausted Riemann surfaces is obtained, which is viewed as a generalization of the classical Nevanlinna theory for C and D .
{"title":"Nevanlinna theory via holomorphic forms","authors":"Xianjing Dong, Shuangshuang Yang","doi":"10.2140/pjm.2022.319.55","DOIUrl":"https://doi.org/10.2140/pjm.2022.319.55","url":null,"abstract":". This paper re-develops Nevanlinna theory for meromorphic functions on C in the viewpoint of holomorphic forms. According to our observation, Nevanlinna’s functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna’s functions using a holomorphic form S . With the new settings, an analogue of Nevanlinna theory on the S -exhausted Riemann surfaces is obtained, which is viewed as a generalization of the classical Nevanlinna theory for C and D .","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42420315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We classify all prime thick subcategories in the derived category of coherent sheaves on elliptic curves, and determine the Serre invariant locus of Matsui spectrum of derived category of coherent sheaves on any smooth projective curves.
{"title":"Prime thick subcategories on elliptic curves","authors":"Yukikazu Hirano, Genki Ouchi","doi":"10.2140/pjm.2022.318.69","DOIUrl":"https://doi.org/10.2140/pjm.2022.318.69","url":null,"abstract":". We classify all prime thick subcategories in the derived category of coherent sheaves on elliptic curves, and determine the Serre invariant locus of Matsui spectrum of derived category of coherent sheaves on any smooth projective curves.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44596554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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