{"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}
{"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}
{"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}
{"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}