Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, Janet Vassilev
{"title":"Differential operators, retracts, and toric face rings","authors":"Christine Berkesch, C-Y. Jean Chan, Patricia Klein, Laura Felicia Matusevich, Janet Page, Janet Vassilev","doi":"10.2140/ant.2023.17.1959","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1959","url":null,"abstract":"<p>We give explicit descriptions of rings of differential operators of toric face rings in characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math>. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators. </p><p> Our main technique is to realize the k-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any k-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> char</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>k</mi><mo stretchy=\"false\">)</mo></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 8","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493577","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":"Axiomatizing the existential theory of 𝔽q((t))","authors":"Sylvy Anscombe, Philip Dittmann, Arno Fehm","doi":"10.2140/ant.2023.17.2013","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2013","url":null,"abstract":"<p>We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of — and therefore an algorithm to decide — the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 9","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493576","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":"The diagonal coinvariant ring of a complex reflection group","authors":"Stephen Griffeth","doi":"10.2140/ant.2023.17.2033","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2033","url":null,"abstract":"<p>For an irreducible complex reflection group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math> of rank <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> containing <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> reflections, we put <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>=</mo> <mn>2</mn><mi>N</mi><mo>∕</mo><mi>n</mi></math> and construct a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math>-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math>. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal coinvariant ring, and that further corrections to this estimate of the dimension of the diagonal coinvariant ring by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> should be orders of magnitude smaller. A crucial ingredient in the construction is the existence of a dot action of a certain product of symmetric groups (the Namikawa–Weyl group) acting on the parameter space of the rational Cherednik algebra and leaving invariant both the finite Hecke algebra and the spherical subalgebra; this fact is a consequence of ideas of Berest and Chalykh on the relationship between the Cherednik algebra and quasiinvariants. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 48","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514515","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":"Separation of periods of quartic surfaces","authors":"Pierre Lairez, Emre Can Sertöz","doi":"10.2140/ant.2023.17.1753","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1753","url":null,"abstract":"<p>We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 14","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493569","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":"Special cycles on the basic locus of unitary Shimura varieties at ramified primes","authors":"Yousheng Shi","doi":"10.2140/ant.2023.17.1681","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1681","url":null,"abstract":"<p>We study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogs on the corresponding Rapoport–Zink spaces. We study the support and compute the dimension of these cycles. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 15","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493571","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 fake linear cycles inside Fermat varieties","authors":"Jorge Duque Franco, Roberto Villaflor Loyola","doi":"10.2140/ant.2023.17.1847","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1847","url":null,"abstract":"<p>We introduce a new class of Hodge cycles with nonreduced associated Hodge loci; we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math>, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension, contrary to a conjecture of Movasati. They provide examples of algebraic cycles not generated by their periods in the sense of Movasati and Sertöz (2021). To study them we compute their Galois action in cohomology and their second-order invariant of the IVHS. We conclude that for any degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo> <mn>2</mn>\u0000<mo>+</mo> <mfrac><mrow><mn>6</mn></mrow>\u0000<mrow><mi>n</mi></mrow></mfrac></math>, the minimal codimension component of the Hodge locus passing through the Fermat variety is the one parametrizing hypersurfaces containing linear subvarieties of dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mi>n</mi></mrow>\u0000<mrow><mn>2</mn></mrow></mfrac> </math>, extending results of Green, Voisin, Otwinowska and the Villaflor Loyola. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 26","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493244","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":"Global dimension of real-exponent polynomial rings","authors":"Nathan Geist, Ezra Miller","doi":"10.2140/ant.2023.17.1779","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1779","url":null,"abstract":"<p>The ring <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> of real-exponent polynomials in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> variables over any field has global dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>+</mo> <mn>1</mn></math> and flat dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>. In particular, the residue field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits-->\u0000<mo>=</mo>\u0000<mi>R</mi><mo>∕</mo><mi mathvariant=\"fraktur\">𝔪</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> modulo its maximal graded ideal <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"fraktur\">𝔪</mi></math> has flat dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> via a Koszul-like resolution. Projective and flat resolutions of all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math>-modules are constructed from this resolution of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></math>. The same results hold when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> is replaced by the monoid algebra for the positive cone of any subgroup of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> satisfying a mild density condition. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493567","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":"Differences between perfect powers: prime power gaps","authors":"Michael A. Bennett, Samir Siksek","doi":"10.2140/ant.2023.17.1789","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1789","url":null,"abstract":"<p>We develop machinery to explicitly determine, in many instances, when the difference <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>−</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math> is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equations, and the primitive divisor theorem of Bilu, Hanrot and Voutier) and classical algebraic number theory, with results derived from the modularity of Galois representations attached to Frey–Hellegoaurch elliptic curves. By way of example, we completely solve the equation </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup>\u0000<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo>\u0000</math>\u0000</div>\u0000<p> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn>\u0000<mo>≤</mo>\u0000<mi>q</mi>\u0000<mo><</mo> <mn>1</mn><mn>0</mn><mn>0</mn></math> is prime, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>α</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> are integers with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>3</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> gcd</mi><mo> <!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>1</mn></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 25","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493566","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":"Unipotent ℓ-blocks for simply connected p-adic groups","authors":"Thomas Lanard","doi":"10.2140/ant.2023.17.1533","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1533","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> be a nonarchimedean local field and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>-points of a connected simply connected reductive group over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>. We study the unipotent <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-blocks of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>, for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math>. To that end, we introduce the notion of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-Harish-Chandra theory. The <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-blocks are then constructed using these <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>-series, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> the order of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>, and consistent systems of idempotents on the Bruhat–Tits building of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>. We also describe the stable <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-block decomposition of the depth zero category of an unramified classical group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493240","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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