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Lectures on Chow-Witt groups 关于Chow-Witt小组的讲座
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2019-11-19 DOI: 10.1090/conm/745/15023
J. Fasel
{"title":"Lectures on Chow-Witt groups","authors":"J. Fasel","doi":"10.1090/conm/745/15023","DOIUrl":"https://doi.org/10.1090/conm/745/15023","url":null,"abstract":"In these lectures, we provide a toolkit to work with Chow-Witt groups, and more generally with the homology and cohomology of the Rost-Schmid complex associated to Milnor-Witt $K$-theory.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128870604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
SL-oriented cohomology theories 面向sl的上同理论
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2019-01-06 DOI: 10.1090/conm/745/15020
A. Ananyevskiy
{"title":"SL-oriented cohomology theories","authors":"A. Ananyevskiy","doi":"10.1090/conm/745/15020","DOIUrl":"https://doi.org/10.1090/conm/745/15020","url":null,"abstract":"We show that a representable motivic cohomology theory admits a unique normalized SL^c-orientation if the zeroth cohomology presheaf is a Zariski sheaf. We also construct Thom isomorphisms in SL-oriented cohomology for SL^c-bundles and obtain new results on the eta-torsion characteristic classes, in particular, we prove that the Euler class of an oriented bundle admitting a (possibly non-orientable) odd rank subbundle is annihilated by the Hopf element.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121574396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
The homotopy Leray spectral sequence 同伦Leray谱序列
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2018-12-22 DOI: 10.1090/conm/745/15021
A. Asok, F. D'eglise, J. Nagel
{"title":"The homotopy Leray spectral sequence","authors":"A. Asok, F. D'eglise, J. Nagel","doi":"10.1090/conm/745/15021","DOIUrl":"https://doi.org/10.1090/conm/745/15021","url":null,"abstract":"In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126380317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Examples of wild ramification in an enriched Riemann–Hurwitz formula 丰富黎曼-赫维茨公式中野分枝的例子
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2018-12-08 DOI: 10.1090/conm/745/15022
Candace Bethea, J. Kass, K. Wickelgren
{"title":"Examples of wild ramification in an enriched\u0000 Riemann–Hurwitz formula","authors":"Candace Bethea, J. Kass, K. Wickelgren","doi":"10.1090/conm/745/15022","DOIUrl":"https://doi.org/10.1090/conm/745/15022","url":null,"abstract":"M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"56 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123443720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Chow-Witt rings of split quadrics 分裂二次曲面的Chow-Witt环
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2018-11-30 DOI: 10.1090/conm/745/15024
J. Hornbostel, H. Xie, Marcus Zibrowius
{"title":"Chow-Witt rings of split quadrics","authors":"J. Hornbostel, H. Xie, Marcus Zibrowius","doi":"10.1090/conm/745/15024","DOIUrl":"https://doi.org/10.1090/conm/745/15024","url":null,"abstract":"We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116217667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Oriented Schubert calculus in Chow–Witt rings of Grassmannians 格拉斯曼人的Chow-Witt环中的定向Schubert微积分
Motivic Homotopy Theory and Refined Enumerative Geometry Pub Date : 2018-08-22 DOI: 10.1090/conm/745/15027
Matthias Wendt
{"title":"Oriented Schubert calculus in Chow–Witt rings\u0000 of Grassmannians","authors":"Matthias Wendt","doi":"10.1090/conm/745/15027","DOIUrl":"https://doi.org/10.1090/conm/745/15027","url":null,"abstract":"We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and twisted Witt groups. In the resulting arithmetic refinements of Schubert calculus, the multiplicity of a solution subspace is a quadratic form encoding additional orientation information. We also discuss a couple of applications, such as a Chow-Witt version of the signed count of balanced subspaces of Feh'er and Matszangosz.","PeriodicalId":246899,"journal":{"name":"Motivic Homotopy Theory and Refined\n Enumerative Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130583203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 19
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