{"title":"Operations in connective K-theory","authors":"Alexander Merkurjev, Alexander Vishik","doi":"10.2140/ant.2023.17.1595","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1595","url":null,"abstract":"<p>We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mover accent=\"true\"><mrow><mi>ℤ</mi></mrow><mo accent=\"true\">^</mo></mover></math> case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mover accent=\"true\"><mrow><mi>ℤ</mi></mrow><mo accent=\"true\">^</mo></mover></math>-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 6","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493238","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 moment map and bigness of tangent bundles of G-varieties","authors":"Jie Liu","doi":"10.2140/ant.2023.17.1501","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1501","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> be a connected algebraic group and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> be a smooth projective <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>-variety. We prove a sufficient criterion to determine the bigness of the tangent bundle <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>T</mi><mi>X</mi></math> using the moment map <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>Φ</mi></mrow><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msubsup>\u0000<mo>:</mo> <msup><mrow><mi>T</mi></mrow><mrow><mo>∗</mo></mrow></msup><mi>X</mi>\u0000<mo>→</mo><msup><mrow>\u0000<mi mathvariant=\"fraktur\">𝔤</mi></mrow><mrow><mo>∗</mo></mrow></msup></math>. As an application, the bigness of the tangent bundles of certain quasihomogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative linear algebraic groups. Finally, we study in details the Fano manifolds <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> with Picard number <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> which is an equivariant compactification of a vector group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math>. In particular, we will determine the pseudoeffective cone of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℙ</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>∗</mo></mrow></msup><mi>X</mi><mo stretchy=\"false\">)</mo></math> and show that the image of the projectivised moment map along the boundary divisor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>D</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> is projectively equivalent to the dual variety of the variety of minimal rational tangents of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> at a general point. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 17","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493235","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":"Spectral reciprocity via integral representations","authors":"Ramon M. Nunes","doi":"10.2140/ant.2023.17.1381","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1381","url":null,"abstract":"<p>We prove a spectral reciprocity formula for automorphic forms on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> over a number field that is reminiscent of one found by Blomer and Khan. Our approach uses period representations of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions and the language of automorphic representations. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"55 3","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514514","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":"Quadratic points on intersections of two quadrics","authors":"Brendan Creutz, Bianca Viray","doi":"10.2140/ant.2023.17.1411","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1411","url":null,"abstract":"<p>We prove that a smooth complete intersection of two quadrics of dimension at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> over a number field has index dividing <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>, i.e., that it possesses a rational <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math>-cycle of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 18","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493234","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 first nontrivial strand of syzygies of projective schemes and condition ND(ℓ)","authors":"Jeaman Ahn, Kangjin Han, Sijong Kwak","doi":"10.2140/ant.2023.17.1359","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1359","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi>\u0000<mo>⊂</mo> <msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>e</mi></mrow></msup></math> be any <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mstyle mathvariant=\"bold\"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>d</mi>\u0000<mo>≥</mo> <mn>2</mn><mo>,</mo><mi>p</mi>\u0000<mo>≥</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></math>, which means that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> is <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-regular up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-th step in the minimal free resolution and the other is a new notion <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> ND</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>ℓ</mi><mo stretchy=\"false\">)</mo></math> which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained in any hypersurface of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>. </p><p> First, we introduce condition <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> ND</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>ℓ</mi><mo stretchy=\"false\">)</mo></math> and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first nontrivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mstyle mathvariant=\"bold\"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>p</mi></mrow></msub></math>, we characterize the resolution of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> to be <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-linear arithmetically Cohen–Macaulay as having property <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mstyle mathvariant=\"bold\"><mi>N</mi></mstyle></mrow><mrow><mi>d</mi><mo>,</mo><mi>e</mi></mrow></msub></math> and condition <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> ND</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>d</mi>\u0000<mo>−</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></math> at the sam","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 9","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493236","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":"A p-adic Simpson correspondence for rigid analytic varieties","authors":"Yupeng Wang","doi":"10.2140/ant.2023.17.1453","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1453","url":null,"abstract":"<p>We establish a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic Simpson correspondence in the spirit of Liu and Zhu for rigid analytic varieties <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℂ</mi></mrow><mrow><mi>p</mi></mrow></msub></math> with a liftable good reduction by constructing a new period sheaf on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><!--mstyle--><mtext mathvariant=\"normal\"> proét</mtext><!--/mstyle--></mrow></msub></math>. To do so, we use the theory of cotangent complexes described by Beilinson and Bhatt. Then we give an integral decompletion theorem and complete the proof by local calculations. Our construction is compatible with the previous works of Faltings and Liu and Zhu. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 19","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493233","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}
Cristian Lenart, Changjian Su, Kirill Zainoulline, Changlong Zhong
{"title":"Geometric properties of the Kazhdan–Lusztig Schubert basis","authors":"Cristian Lenart, Changjian Su, Kirill Zainoulline, Changlong Zhong","doi":"10.2140/ant.2023.17.435","DOIUrl":"https://doi.org/10.2140/ant.2023.17.435","url":null,"abstract":"","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"163 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092222","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}
T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, W. Yolland
{"title":"Constructions of difference sets in nonabelian 2-groups","authors":"T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, W. Yolland","doi":"10.2140/ant.2023.17.359","DOIUrl":"https://doi.org/10.2140/ant.2023.17.359","url":null,"abstract":"","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092220","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":"Tame fundamental groups of pure pairs and Abhyankar’s lemma","authors":"Javier Carvajal-Rojas, Axel Stäbler","doi":"10.2140/ant.2023.17.43","DOIUrl":"https://doi.org/10.2140/ant.2023.17.43","url":null,"abstract":"Let $(R,mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f : Y to X$ in that Galois category satisfies that $bigl(f^{-1}(P)bigr)_{text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the 'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods.","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136046273","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":"A transference principle for systems of linear equations, and applications to almost twin primes","authors":"Pierre-Yves Bienvenu, Xuancheng Shao, Joni Teräväinen","doi":"10.2140/ant.2023.17.497","DOIUrl":"https://doi.org/10.2140/ant.2023.17.497","url":null,"abstract":"","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092219","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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