{"title":"Operations in connective K-theory","authors":"Alexander Merkurjev, Alexander Vishik","doi":"10.2140/ant.2023.17.1595","DOIUrl":null,"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":0.9000,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2023.17.1595","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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 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 -coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.
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ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms.
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