Dan Abramovich, Michael Temkin, Jarosław Włodarczyk
{"title":"Functorial embedded resolution via weighted blowings up","authors":"Dan Abramovich, Michael Temkin, Jarosław Włodarczyk","doi":"10.2140/ant.2024.18.1557","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1557","url":null,"abstract":"<p>We provide a simple procedure for resolving, in characteristic 0, singularities of a variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> embedded in a smooth variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Y</mi> </math> by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. </p><p> A similar result was discovered independently by McQuillan (2020). </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exceptional characters and prime numbers in sparse sets","authors":"Jori Merikoski","doi":"10.2140/ant.2024.18.1305","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1305","url":null,"abstract":"<p>We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>b</mi></mrow><mrow><mn>8</mn></mrow></msup></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Combining Igusa’s conjectures on exponential sums and monodromy with semicontinuity of the minimal exponent","authors":"Raf Cluckers, Kien Huu Nguyen","doi":"10.2140/ant.2024.18.1275","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1275","url":null,"abstract":"<p>We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math> variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"59 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Maximal subgroups of exceptional groups and Quillen’s dimension","authors":"Kevin I. Piterman","doi":"10.2140/ant.2024.18.1375","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1375","url":null,"abstract":"<p>Given a finite group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> and a prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> be the poset of nontrivial elementary abelian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-subgroups of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>. The group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> satisfies the Quillen dimension property at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> has nonzero homology in the maximal possible degree, which is the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> minus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>. For example, D. Quillen showed that solvable groups with trivial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-core satisfy this property, and later, M. Aschbacher and S. D. Smith provided a list of all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-extensions of simple groups that may fail this property if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> is odd. In particular, a group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> with this property satisfies Quillen’s conjecture: <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> has trivial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-core and the poset <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> is not contractible. </p><p> In this article, we focus on the prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>=</mo> <mn>2</mn></math> and prove that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>-extensions of finite simple groups of exceptional Lie type in odd characteristic satisfy the Quillen dimension property, wit","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra
{"title":"Serre weights for three-dimensional wildly ramified Galois representations","authors":"Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra","doi":"10.2140/ant.2024.18.1221","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1221","url":null,"abstract":"<p>We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> Galois representations under a genericity condition when the field is unramified at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. This removes the assumption made previously that the representation be tamely ramified at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. We also prove a version of Breuil’s lattice conjecture and a mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> multiplicity one result for the cohomology of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>U</mi><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math>-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023). </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonathan Boretsky, Christopher Eur, Lauren Williams
{"title":"Polyhedral and tropical geometry of flag positroids","authors":"Jonathan Boretsky, Christopher Eur, Lauren Williams","doi":"10.2140/ant.2024.18.1333","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1333","url":null,"abstract":"<p>A <span>flag positroid </span>of ranks <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>r</mi></mstyle>\u0000<mo>:</mo><mo>=</mo>\u0000<mo stretchy=\"false\">(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub>\u0000<mo><</mo>\u0000<mo>⋯</mo>\u0000<mo><</mo> <msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mi>n</mi><mo stretchy=\"false\">]</mo></math> is a flag matroid that can be realized by a real <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub>\u0000<mo>×</mo>\u0000<mi>n</mi></math> matrix <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> such that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub>\u0000<mo>×</mo> <msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> minors of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> involving rows <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> are nonnegative for all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn>\u0000<mo>≤</mo>\u0000<mi>i</mi>\u0000<mo>≤</mo>\u0000<mi>k</mi></math>. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>r</mi></mstyle>\u0000<mo>:</mo><mo>=</mo>\u0000<mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>a</mi>\u0000<mo>+</mo> <mn>1</mn><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo></math> is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi> TrFl</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mstyle mathvariant=\"bold-italic\"><mi>r</mi></mstyle><mo>,</mo><mi>n</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></math> equals the nonnegative flag Dressian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi> FlDr</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mstyle mathvariant=\"bold-italic\"><mi>r</mi></mstyle><mo>,</mo><mi>n</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></math>, and that the points <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>μ</mi>\u0000<mo>=</mo>\u0000<mo stretchy=\"false\">(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>μ</mi></mrow>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the p-adic interpolation of unitary Friedberg–Jacquet periods","authors":"Andrew Graham","doi":"10.2140/ant.2024.18.1117","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1117","url":null,"abstract":"<p>We establish functoriality of higher Coleman theory for certain unitary Shimura varieties and use this to construct a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic analytic function interpolating unitary Friedberg–Jacquet periods. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"19 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Refined height pairing","authors":"Bruno Kahn","doi":"10.2140/ant.2024.18.1039","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1039","url":null,"abstract":"<p>For a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-dimensional regular proper variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> over the function field of a smooth variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> over a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> and for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>i</mi>\u0000<mo>≥</mo> <mn>0</mn></math>, we define a subgroup <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> and construct a “refined height pairing” </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<msup><mrow><mi>CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup>\u0000<mo>×</mo><msup><mrow><mi> CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup>\u0000<mo>→</mo><msup><mrow><mi> CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>B</mi><mo stretchy=\"false\">)</mo>\u0000</math>\u0000</div>\u0000<p> in the category of abelian groups up to isogeny. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>i</mi>\u0000<mo>=</mo> <mn>1</mn><mo>,</mo><mi>d</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup></math> is the group of cycles numerically equivalent to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math>. This pairing relates to pairings defined by P. Schneider and A. Beilinson if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> is a curve, to a refined height defined by ","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"70 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Enumeration of conjugacy classes in affine groups","authors":"Jason Fulman, Robert M. Guralnick","doi":"10.2140/ant.2024.18.1189","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1189","url":null,"abstract":"<p>We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the noncoprime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mi>V</mi>\u0000<mo stretchy=\"false\">)</mo></math> problem of Brauer. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Balmer spectra and Drinfeld centers","authors":"Kent B. Vashaw","doi":"10.2140/ant.2024.18.1081","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1081","url":null,"abstract":"<p>The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>C</mi></mstyle></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>C</mi></mstyle></math> extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson–Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"58 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140818076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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