Sextic double solids, double covers of $mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $mathbb Q$ -factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n leq 8$ , describe models for each n explicitly, and prove that sextic double solids with $n> 3$ are birationally nonrigid.
{"title":"BIRATIONAL GEOMETRY OF SEXTIC DOUBLE SOLIDS WITH A COMPOUND SINGULARITY","authors":"ERIK PAEMURRU","doi":"10.1017/nmj.2024.17","DOIUrl":"https://doi.org/10.1017/nmj.2024.17","url":null,"abstract":"Sextic double solids, double covers of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline2.png\"/> <jats:tex-math> $mathbb P^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline3.png\"/> <jats:tex-math> $mathbb Q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline4.png\"/> <jats:tex-math> $A_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> singularity. We prove a sharp bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline5.png\"/> <jats:tex-math> $n leq 8$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, describe models for each <jats:italic>n</jats:italic> explicitly, and prove that sextic double solids with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000175_inline6.png\"/> <jats:tex-math> $n> 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are birationally nonrigid.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"207 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using the technique of Gabor analysis, we characterize the boundedness of $e^{iDelta }: W^{p_1,q_1}_mrightarrow W^{p_2,q_2}$ with modulation and translation operators, where and m is a v-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.
{"title":"SCHRÖDINGER PROPAGATOR ON WIENER AMALGAM SPACES IN THE FULL RANGE","authors":"GUOPING ZHAO, WEICHAO GUO","doi":"10.1017/nmj.2024.14","DOIUrl":"https://doi.org/10.1017/nmj.2024.14","url":null,"abstract":"Using the technique of Gabor analysis, we characterize the boundedness of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S002776302400014X_inline1.png\"/> <jats:tex-math> $e^{iDelta }: W^{p_1,q_1}_mrightarrow W^{p_2,q_2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with modulation and translation operators, where <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S002776302400014X_inline2.png\"/> and <jats:italic>m</jats:italic> is a <jats:italic>v</jats:italic>-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.
{"title":"CONSTANCY OF THE HILBERT–SAMUEL FUNCTION","authors":"VINCENT COSSART, OLIVIER PILTANT, BERND SCHOBER","doi":"10.1017/nmj.2024.13","DOIUrl":"https://doi.org/10.1017/nmj.2024.13","url":null,"abstract":"We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.
{"title":"WHEN IS THE SILTING-DISCRETENESS INHERITED?","authors":"TAKUMA AIHARA, TAKAHIRO HONMA","doi":"10.1017/nmj.2024.8","DOIUrl":"https://doi.org/10.1017/nmj.2024.8","url":null,"abstract":"<p>We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.</p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"59 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We invoke the Bernstein–Gel $'$ fand–Gel $'$ fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case.
{"title":"SUBCOMPLEXES OF CERTAIN FREE RESOLUTIONS","authors":"MAYA BANKS, ALEKSANDRA SOBIESKA","doi":"10.1017/nmj.2024.7","DOIUrl":"https://doi.org/10.1017/nmj.2024.7","url":null,"abstract":"We invoke the Bernstein–Gel<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000072_inline1.png\" /> <jats:tex-math> $'$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>fand–Gel<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000072_inline2.png\" /> <jats:tex-math> $'$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.
{"title":"TILTING COMPLEXES AND CODIMENSION FUNCTIONS OVER COMMUTATIVE NOETHERIAN RINGS","authors":"MICHAL HRBEK, TSUTOMU NAKAMURA, JAN ŠŤOVÍČEK","doi":"10.1017/nmj.2024.1","DOIUrl":"https://doi.org/10.1017/nmj.2024.1","url":null,"abstract":"In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"69 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 prove the nonvanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field <jats:italic>k</jats:italic> of characteristic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline2.png" /> <jats:tex-math> $p> 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. More precisely, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline3.png" /> <jats:tex-math> $(X,B)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a projective log canonical threefold pair over <jats:italic>k</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline4.png" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is pseudo-effective, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline5.png" /> <jats:tex-math> $kappa (K_{X}+B)geq 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline6.png" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nef and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline7.png" /> <jats:tex-math> $kappa (K_{X}+B)geq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline8.png" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is semi-ample. As applications, we show that the log canonical rings of projective log canonical threefold pairs over <jats:italic>k</jats:italic> are finitely generated and the abundance holds when the nef dimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline9.png" /> <jats:tex-math> $n(K_{X}+B)leq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> or when the Albanese map <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763024000035_inline10.png" /> <jats:tex-math> $a_{X}:Xto mathrm {Alb}(X)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nontrivial. Moreo
在本文中，我们证明了在特征为 $p> 3$ 的代数闭域 k 上的 log canonical threefold 对的丰度的不消失性和一些特例。更准确地说，我们证明了如果 $(X,B)$ 是 k 上的投影对数典型三折对，并且 $K_{X}+B$ 是伪有效的，那么 $kappa (K_{X}+B)geq 0$ ，如果 $K_{X}+B$ 是新有效的，并且 $kappa (K_{X}+B)geq 1$ ，那么 $K_{X}+B$ 是半范例。作为应用，我们证明了在 k 上的投影对数对数对数三重环是有限生成的，并且当 nef 维度 $n(K_{X}+B)leq 2$ 或 Albanese 映射 $a_{X}:Xto mathrm {Alb}(X)$ 是非微观时，丰度成立。此外，我们还证明了 k 上 klt 三重对的丰度意味着 k 上 log canonical 三重对的丰度。
{"title":"NOTE ON THE THREE-DIMENSIONAL LOG CANONICAL ABUNDANCE IN CHARACTERISTIC","authors":"ZHENG XU","doi":"10.1017/nmj.2024.3","DOIUrl":"https://doi.org/10.1017/nmj.2024.3","url":null,"abstract":"In this paper, we prove the nonvanishing and some special cases of the abundance for log canonical threefold pairs over an algebraically closed field <jats:italic>k</jats:italic> of characteristic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline2.png\" /> <jats:tex-math> $p> 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. More precisely, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline3.png\" /> <jats:tex-math> $(X,B)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a projective log canonical threefold pair over <jats:italic>k</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline4.png\" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is pseudo-effective, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline5.png\" /> <jats:tex-math> $kappa (K_{X}+B)geq 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline6.png\" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nef and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline7.png\" /> <jats:tex-math> $kappa (K_{X}+B)geq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline8.png\" /> <jats:tex-math> $K_{X}+B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is semi-ample. As applications, we show that the log canonical rings of projective log canonical threefold pairs over <jats:italic>k</jats:italic> are finitely generated and the abundance holds when the nef dimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline9.png\" /> <jats:tex-math> $n(K_{X}+B)leq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> or when the Albanese map <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763024000035_inline10.png\" /> <jats:tex-math> $a_{X}:Xto mathrm {Alb}(X)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nontrivial. Moreo","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"49 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.
{"title":"COUNTING GEOMETRIC BRANCHES VIA THE FROBENIUS MAP AND F-NILPOTENT SINGULARITIES","authors":"HAILONG DAO, KYLE MADDOX, VAIBHAV PANDEY","doi":"10.1017/nmj.2024.4","DOIUrl":"https://doi.org/10.1017/nmj.2024.4","url":null,"abstract":"We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes <jats:italic>F</jats:italic>-nilpotent curves. Further, we show that a reduced, local <jats:italic>F</jats:italic>-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to <jats:italic>F</jats:italic>-nilpotent affine semigroup rings.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"134 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that perfectoidization can be (almost) calculated by using p-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidizations and uniform completions, as well as the p-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “p-root closure” used by Roberts in mixed characteristic commutative algebra and a more recent concept of “perfectoidization” introduced by Bhatt and Scholze in their theory of prismatic cohomology.
我们证明，在某些情况下，包括在半完形情况下，完形化（几乎）可以用 p 根封闭来计算。为此，我们重点研究了完形化和均匀完形的普遍性，以及积分完形环的 p 根封闭性质。通过这一计算，我们建立了罗伯茨在混合特征交换代数中使用的经典闭合运算 "p 根闭合 "与巴特和肖尔茨在棱柱同调理论中引入的最新概念 "完形化 "之间的联系。
{"title":"A CALCULATION OF THE PERFECTOIDIZATION OF SEMIPERFECTOID RINGS","authors":"RYO ISHIZUKA","doi":"10.1017/nmj.2024.2","DOIUrl":"https://doi.org/10.1017/nmj.2024.2","url":null,"abstract":"We show that perfectoidization can be (almost) calculated by using <jats:italic>p</jats:italic>-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidizations and uniform completions, as well as the <jats:italic>p</jats:italic>-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “<jats:italic>p</jats:italic>-root closure” used by Roberts in mixed characteristic commutative algebra and a more recent concept of “perfectoidization” introduced by Bhatt and Scholze in their theory of prismatic cohomology.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"99 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott–Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.
{"title":"A HOMOMORPHISM BETWEEN BOTT–SAMELSON BIMODULES","authors":"NORIYUKI ABE","doi":"10.1017/nmj.2023.38","DOIUrl":"https://doi.org/10.1017/nmj.2023.38","url":null,"abstract":"<p>In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott–Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.</p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"53 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139515375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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