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Bredon motivic cohomology of the real numbers 实数的布雷顿动机同调
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-10 DOI: arxiv-2404.06697
Bill Deng, Mircea Voineagu
{"title":"Bredon motivic cohomology of the real numbers","authors":"Bill Deng, Mircea Voineagu","doi":"arxiv-2404.06697","DOIUrl":"https://doi.org/arxiv-2404.06697","url":null,"abstract":"Over the real numbers with $Z/2-$coefficients, we compute the\u0000$C_2$-equivariant Borel motivic cohomology ring, the Bredon motivic cohomology\u0000groups and prove that the Bredon motivic cohomology ring of the real numbers is\u0000a proper subring in the $RO(C_2times C_2)$-graded Bredon cohomology ring of a\u0000point. This generalizes Voevodsky's computation of the motivic cohomology ring of\u0000the real numbers to the $C_2$-equivariant setting. These computations are\u0000extended afterwards to any real closed field.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564812","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: 0
A symplectic version of Suslin's $n!$-theorem 苏斯林$n！$定理的折射版本
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-10 DOI: arxiv-2404.07077
Tariq Syed
{"title":"A symplectic version of Suslin's $n!$-theorem","authors":"Tariq Syed","doi":"arxiv-2404.07077","DOIUrl":"https://doi.org/arxiv-2404.07077","url":null,"abstract":"We prove symplectic versions of Suslin's famous $n!$-theorem for algebras\u0000over quadratically closed perfect fields of characteristic $neq 2$ and for\u0000algebras over finite fields of characteristic $neq 2$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565090","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: 0
Bigraded path homology and the magnitude-path spectral sequence 大梯度路径同源性和幅值路径谱序列
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-10 DOI: arxiv-2404.06689
Richard Hepworth, Emily Roff
{"title":"Bigraded path homology and the magnitude-path spectral sequence","authors":"Richard Hepworth, Emily Roff","doi":"arxiv-2404.06689","DOIUrl":"https://doi.org/arxiv-2404.06689","url":null,"abstract":"Two important invariants of directed graphs, namely magnitude homology and\u0000path homology, have recently been shown to be intimately connected: there is a\u0000'magnitude-path spectral sequence' or 'MPSS' in which magnitude homology\u0000appears as the first page, and in which path homology appears as an axis of the\u0000second page. In this paper we study the homological and computational\u0000properties of the spectral sequence, and in particular of the full second page,\u0000which we now call 'bigraded path homology'. We demonstrate that every page of\u0000the MPSS deserves to be regarded as a homology theory in its own right,\u0000satisfying excision and Kunneth theorems (along with a homotopy invariance\u0000property already established by Asao), and that magnitude homology and bigraded\u0000path homology also satisfy Mayer-Vietoris theorems. We construct a homotopy\u0000theory of graphs (in the form of a cofibration category structure) in which\u0000weak equivalences are the maps inducing isomorphisms on bigraded path homology,\u0000strictly refining an existing structure based on ordinary path homology. And we\u0000provide complete computations of the MPSS for two important families of graphs\u0000- the directed and bi-directed cycles - which demonstrate the power of both the\u0000MPSS, and bigraded path homology in particular, to distinguish graphs that\u0000ordinary path homology cannot.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565048","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: 0
Gersten's Injectivity for Smooth Algebras over Valuation Rings 估价环上光滑代数的格尔斯滕注入性
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-09 DOI: arxiv-2404.06655
Arnab Kundu
{"title":"Gersten's Injectivity for Smooth Algebras over Valuation Rings","authors":"Arnab Kundu","doi":"arxiv-2404.06655","DOIUrl":"https://doi.org/arxiv-2404.06655","url":null,"abstract":"Gersten's injectivity conjecture for a functor $F$ of ``motivic type'',\u0000predicts that given a semilocal, ``non-singular'', integral domain $R$ with a\u0000fraction field $K$, the restriction morphism induces an injection of $F(R)$\u0000inside $F(K)$. We prove two new cases of this conjecture for smooth algebras\u0000over valuation rings. Namely, we show that the higher algebraic $K$-groups of a\u0000semilocal, integral domain that is an essentially smooth algebra over an\u0000equicharacteristic valuation ring inject inside the same of its fraction field.\u0000Secondly, we show that Gersten's injectivity is true for smooth algebras over,\u0000possibly of mixed-characteristic, valuation rings in the case of torsors under\u0000tori and also in the case of the Brauer group.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564801","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: 0
K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories 抽象二次型理论中的 K 理论和自由归纳分级环
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-04 DOI: arxiv-2404.05750
Kaique Matias de Andrade Roberto, Hugo Luiz mariano
{"title":"K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories","authors":"Kaique Matias de Andrade Roberto, Hugo Luiz mariano","doi":"arxiv-2404.05750","DOIUrl":"https://doi.org/arxiv-2404.05750","url":null,"abstract":"We build on previous work on multirings (cite{roberto2021quadratic}) that\u0000provides generalizations of the available abstract quadratic forms theories\u0000(special groups and real semigroups) to the context of multirings\u0000(cite{marshall2006real}, cite{ribeiro2016functorial}). Here we raise one step\u0000in this generalization, introducing the concept of pre-special hyperfields and\u0000expand a fundamental tool in quadratic forms theory to the more general\u0000multivalued setting: the K-theory. We introduce and develop the K-theory of\u0000hyperbolic hyperfields that generalize simultaneously Milnor's K-theory\u0000(cite{milnor1970algebraick}) and Special Groups K-theory, developed by\u0000Dickmann-Miraglia (cite{dickmann2006algebraic}). We develop some properties of\u0000this generalized K-theory, that can be seen as a free inductive graded ring, a\u0000concept introduced in cite{dickmann1998quadratic} in order to provide a\u0000solution of Marshall's Signature Conjecture.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"111 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564761","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: 0
Multifunctorial Equivariant Algebraic K-Theory 多扇面等变代数 K 理论
arXiv - MATH - K-Theory and Homology Pub Date : 2024-04-03 DOI: arxiv-2404.02794
Donald Yau
{"title":"Multifunctorial Equivariant Algebraic K-Theory","authors":"Donald Yau","doi":"arxiv-2404.02794","DOIUrl":"https://doi.org/arxiv-2404.02794","url":null,"abstract":"A central question in equivariant algebraic K-theory asks whether there\u0000exists an equivariant K-theory machine from genuine symmetric monoidal\u0000G-categories to orthogonal G-spectra that preserves equivariant algebraic\u0000structures. We answer this question positively by constructing an enriched\u0000multifunctor K from the G-categorically enriched multicategory of\u0000O-pseudoalgebras to the symmetric monoidal category of orthogonal G-spectra,\u0000for a compact Lie group G and a 1-connected pseudo-commutative G-categorical\u0000operad O. As the main application of its enriched multifunctoriality, K\u0000preserves all equivariant algebraic structures parametrized by multicategories\u0000enriched in either G-spaces or G-categories. For example, for a finite group G\u0000and the G-Barratt-Eccles operad, K transports equivariant E-infinity algebras,\u0000in the sense of Guillou-May or Blumberg-Hill, of genuine symmetric monoidal\u0000G-categories to equivariant E-infinity algebras of orthogonal G-spectra.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564749","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: 0
On the algebraizability of formal deformations in $K$-cohomology 论形式变形在 $K$-cohomology 中的可代数性
arXiv - MATH - K-Theory and Homology Pub Date : 2024-03-27 DOI: arxiv-2403.19008
Eoin Mackall
{"title":"On the algebraizability of formal deformations in $K$-cohomology","authors":"Eoin Mackall","doi":"arxiv-2403.19008","DOIUrl":"https://doi.org/arxiv-2403.19008","url":null,"abstract":"We show that algebraizability of the functors $R^1pi_*mathcal{K}^M_{2,X}$\u0000and $R^2pi_*mathcal{K}^M_{2,X}$ is a stable birational invariant for smooth\u0000and proper varieties $pi:Xrightarrow k$ defined over an algebraic extension\u0000$k$ of $mathbb{Q}$. The same is true for the 'etale sheafifications of these\u0000functors as well. To get these results we introduce a notion of relative $K$-homology for\u0000schemes of finite type over a finite dimensional, Noetherian, excellent base\u0000scheme over a field. We include this material in an appendix.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"130 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323642","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: 0
Invariants de Witt des involutions de bas degré en caractéristique 2 特征 2 中低度卷积的维特不变式
arXiv - MATH - K-Theory and Homology Pub Date : 2024-03-22 DOI: arxiv-2403.15561
Jean-Pierre Tignol
{"title":"Invariants de Witt des involutions de bas degré en caractéristique 2","authors":"Jean-Pierre Tignol","doi":"arxiv-2403.15561","DOIUrl":"https://doi.org/arxiv-2403.15561","url":null,"abstract":"A $3$-fold and a $5$-fold quadratic Pfister forms are canonically associated\u0000to every symplectic involution on a central simple algebra of degree $8$ over a\u0000field of characteristic $2$. The same construction on central simple algebras\u0000of degree $4$ associates to every unitary involution a $2$-fold and a $4$-fold\u0000Pfister quadratic forms, and to every orthogonal involution a $1$-fold and a\u0000$3$-fold quasi-Pfister forms. These forms hold structural information on the\u0000algebra with involution.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302453","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: 0
Galois theory and homology in quasi-abelian functor categories 准阿贝尔函数范畴中的伽罗瓦理论和同源性
arXiv - MATH - K-Theory and Homology Pub Date : 2024-03-19 DOI: arxiv-2403.12750
Nadja Egner
{"title":"Galois theory and homology in quasi-abelian functor categories","authors":"Nadja Egner","doi":"arxiv-2403.12750","DOIUrl":"https://doi.org/arxiv-2403.12750","url":null,"abstract":"Given a finite category T, we consider the functor category [T,A], where A\u0000can in particular be any quasi-abelian category. Examples of quasi-abelian\u0000categories are given by any abelian category but also by non-exact additive\u0000categories as the categories of torsion(-free) abelian groups, topological\u0000abelian groups, locally compact abelian groups, Banach spaces and Fr'echet\u0000spaces. In this situation, the categories of various internal categorical\u0000structures in A, such as the categories of internal n-fold groupoids, are\u0000equivalent to functor categories [T,A] for a suitable category T. For a replete\u0000full subcategory S of T, we define F to be the full subcategory of [T,A] whose\u0000objects are given by the functors G with G(X)=0 for all objects X not in S. We\u0000prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to\u0000study (higher) central extensions from categorical Galois theory in [T,A] with\u0000respect to F and generalized Hopf formulae for homology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166175","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: 0
Quadratic Riemann-Roch formulas 二次黎曼-罗赫公式
arXiv - MATH - K-Theory and Homology Pub Date : 2024-03-14 DOI: arxiv-2403.09266
Frédéric Déglise, Jean Fasel
{"title":"Quadratic Riemann-Roch formulas","authors":"Frédéric Déglise, Jean Fasel","doi":"arxiv-2403.09266","DOIUrl":"https://doi.org/arxiv-2403.09266","url":null,"abstract":"In this article, we produce Grothendieck-Riemann-Roch formulas for cohomology\u0000theories that are not oriented in the classical sense. We then specialize to\u0000the case of cohomology theories that admit a so-called symplectic orientation\u0000and show how to compute the relevant Todd classes in that situation. At the end\u0000of the article, we illustrate our methods on the Borel character linking\u0000Hermitian K-theory and rational MW-motivic cohomology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150613","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: 0
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