Over the real numbers with $Z/2-$coefficients, we compute the $C_2$-equivariant Borel motivic cohomology ring, the Bredon motivic cohomology groups and prove that the Bredon motivic cohomology ring of the real numbers is a proper subring in the $RO(C_2times C_2)$-graded Bredon cohomology ring of a point. This generalizes Voevodsky's computation of the motivic cohomology ring of the real numbers to the $C_2$-equivariant setting. These computations are extended afterwards to any real closed field.
{"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":null,"pages":null},"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}
We prove symplectic versions of Suslin's famous $n!$-theorem for algebras over quadratically closed perfect fields of characteristic $neq 2$ and for algebras over finite fields of characteristic $neq 2$.
{"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":null,"pages":null},"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}
Two important invariants of directed graphs, namely magnitude homology and path homology, have recently been shown to be intimately connected: there is a 'magnitude-path spectral sequence' or 'MPSS' in which magnitude homology appears as the first page, and in which path homology appears as an axis of the second page. In this paper we study the homological and computational properties of the spectral sequence, and in particular of the full second page, which we now call 'bigraded path homology'. We demonstrate that every page of the MPSS deserves to be regarded as a homology theory in its own right, satisfying excision and Kunneth theorems (along with a homotopy invariance property already established by Asao), and that magnitude homology and bigraded path homology also satisfy Mayer-Vietoris theorems. We construct a homotopy theory of graphs (in the form of a cofibration category structure) in which weak equivalences are the maps inducing isomorphisms on bigraded path homology, strictly refining an existing structure based on ordinary path homology. And we provide complete computations of the MPSS for two important families of graphs - the directed and bi-directed cycles - which demonstrate the power of both the MPSS, and bigraded path homology in particular, to distinguish graphs that ordinary path homology cannot.
{"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":null,"pages":null},"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}
Gersten's injectivity conjecture for a functor $F$ of ``motivic type'', predicts that given a semilocal, ``non-singular'', integral domain $R$ with a fraction field $K$, the restriction morphism induces an injection of $F(R)$ inside $F(K)$. We prove two new cases of this conjecture for smooth algebras over valuation rings. Namely, we show that the higher algebraic $K$-groups of a semilocal, integral domain that is an essentially smooth algebra over an equicharacteristic valuation ring inject inside the same of its fraction field. Secondly, we show that Gersten's injectivity is true for smooth algebras over, possibly of mixed-characteristic, valuation rings in the case of torsors under tori and also in the case of the Brauer group.
{"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":null,"pages":null},"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}
Kaique Matias de Andrade Roberto, Hugo Luiz mariano
We build on previous work on multirings (cite{roberto2021quadratic}) that provides generalizations of the available abstract quadratic forms theories (special groups and real semigroups) to the context of multirings (cite{marshall2006real}, cite{ribeiro2016functorial}). Here we raise one step in this generalization, introducing the concept of pre-special hyperfields and expand a fundamental tool in quadratic forms theory to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory (cite{milnor1970algebraick}) and Special Groups K-theory, developed by Dickmann-Miraglia (cite{dickmann2006algebraic}). We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in cite{dickmann1998quadratic} in order to provide a solution of Marshall's Signature Conjecture.
我们在先前关于多重irings的工作（cite{roberto2021quadratic}）基础上，将现有的抽象二次型理论（特殊群和实半群）推广到多重irings的语境中（cite{marshall2006real}, cite{ribeiro2016functorial}）。在此，我们将这一泛化提升了一步，引入了前特殊超场的概念，并将二次型理论中的一个基本工具扩展到了更一般的多值环境中：K理论。我们引入并发展了双曲超场的K理论，它同时概括了米尔诺的K理论（cite{milnor1970algebraick}）和迪克曼-米拉利亚（Dickmann-Miraglia）发展的特殊群K理论（cite{dickmann2006algebraic}）。我们发展了这个广义 K 理论的一些性质，它可以被看作是一个自由归纳分级环，这个概念是在《迪克曼 1998 四元组》中引入的，目的是为马歇尔签名猜想提供一个解决方案。
{"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":null,"pages":null},"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}
A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. We answer this question positively by constructing an enriched multifunctor K from the G-categorically enriched multicategory of O-pseudoalgebras to the symmetric monoidal category of orthogonal G-spectra, for a compact Lie group G and a 1-connected pseudo-commutative G-categorical operad O. As the main application of its enriched multifunctoriality, K preserves all equivariant algebraic structures parametrized by multicategories enriched in either G-spaces or G-categories. For example, for a finite group G and the G-Barratt-Eccles operad, K transports equivariant E-infinity algebras, in the sense of Guillou-May or Blumberg-Hill, of genuine symmetric monoidal G-categories to equivariant E-infinity algebras of orthogonal G-spectra.
等变代数 K 理论的一个核心问题是，从真正的对称一元 G 范畴到正交 G 范畴，是否存在一个保留等变代数结构的等变 K 理论机。我们正面回答了这个问题，即针对一个紧凑的李群G和一个1连接的伪交换G范畴O，构造了一个从G范畴丰富的O伪基多范畴到正交G谱的对称一元范畴的丰富多矢量K。例如，对于有限群Gand的G-Barratt-Eccles操作数，K将真正对称单环G-类的等变E-无穷代数（Guillou-May或Blumberg-Hill意义上的等变E-无穷代数）转移到正交G-谱的等变E-无穷代数。
{"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":null,"pages":null},"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}
We show that algebraizability of the functors $R^1pi_*mathcal{K}^M_{2,X}$ and $R^2pi_*mathcal{K}^M_{2,X}$ is a stable birational invariant for smooth and proper varieties $pi:Xrightarrow k$ defined over an algebraic extension $k$ of $mathbb{Q}$. The same is true for the 'etale sheafifications of these functors as well. To get these results we introduce a notion of relative $K$-homology for schemes of finite type over a finite dimensional, Noetherian, excellent base scheme over a field. We include this material in an appendix.
{"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":null,"pages":null},"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}
A $3$-fold and a $5$-fold quadratic Pfister forms are canonically associated to every symplectic involution on a central simple algebra of degree $8$ over a field of characteristic $2$. The same construction on central simple algebras of degree $4$ associates to every unitary involution a $2$-fold and a $4$-fold Pfister quadratic forms, and to every orthogonal involution a $1$-fold and a $3$-fold quasi-Pfister forms. These forms hold structural information on the algebra with involution.
{"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":null,"pages":null},"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}
Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fr'echet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories [T,A] for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of [T,A] whose objects are given by the functors G with G(X)=0 for all objects X not in S. We prove that F is a torsion-free Birkhoff subcategory of [T,A]. This allows us to study (higher) central extensions from categorical Galois theory in [T,A] with respect to F and generalized Hopf formulae for homology.
给定一个有限范畴 T，我们考虑函数范畴 [T,A]，其中，Ac 可以是任何准阿贝尔范畴。准阿贝尔范畴的例子有任何无性范畴，也有非完全加法范畴，如无扭（-free）无性群、拓扑无性群、局部紧凑无性群、巴拿赫空间和 Fr'echetspaces 的范畴。对于 T 的一个完整子类 S，我们定义 F 为 [T,A] 的完整子类，其对象由函数 G 给出，对于不在 S 中的所有对象 X，函数 G(X)=0 。这使我们能够研究[T,A]中相对于 F 的分类伽罗瓦理论的（高）中心扩展以及同调的广义霍普夫公式。
{"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":null,"pages":null},"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}
In this article, we produce Grothendieck-Riemann-Roch formulas for cohomology theories that are not oriented in the classical sense. We then specialize to the case of cohomology theories that admit a so-called symplectic orientation and show how to compute the relevant Todd classes in that situation. At the end of the article, we illustrate our methods on the Borel character linking Hermitian K-theory and rational MW-motivic cohomology.
在这篇文章中，我们为非经典意义上定向的同调理论提出了格罗恩迪克-黎曼-罗赫公式。然后，我们专门讨论了允许所谓交映定向的同调理论的情况，并展示了如何计算这种情况下的相关托德类。在文章的最后，我们说明了我们在连接赫米蒂 K 理论和有理 MW 动机同调的伯勒尔特性上的方法。
{"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":null,"pages":null},"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}
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