We give a construction of a generalized Vaserstein symbol associated to any finitely generated projective module of rank $2$ over a commutative ring with unit.
{"title":"The generalized Vaserstein symbol revisited","authors":"Tariq Syed","doi":"arxiv-2408.10164","DOIUrl":"https://doi.org/arxiv-2408.10164","url":null,"abstract":"We give a construction of a generalized Vaserstein symbol associated to any\u0000finitely generated projective module of rank $2$ over a commutative ring with\u0000unit.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226400","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 D'{e}vissage theorems for Hermitian $K$ Theory (or $GW$ theory), analogous to Quillen's D'{e}vissage theorem for $K$-theory. For abelian categories ${mathscr A}:=({mathscr A}, ^{vee}, varpi)$ with duality, and appropriate abelian subcategories ${mathscr B}subseteq {mathscr A}$, we prove D'{e}vissage theorems for ${bf GW}$ spaces, $G{mathcal W}$-spectra and ${mathbb G}W$ bispectra. As a consequence, for regular local rings $(R, m, kappa)$ with $1/2in R$, we compute the ${BG}W$ groups ${mathbb G}W^{[n]}_k(spec{R})~forall k, nin {mathbb Z}$, where $n$ represent the translation.
{"title":"Dévissage Hermitian Theory","authors":"Satya Mandal","doi":"arxiv-2408.09633","DOIUrl":"https://doi.org/arxiv-2408.09633","url":null,"abstract":"We prove D'{e}vissage theorems for Hermitian $K$ Theory (or $GW$ theory),\u0000analogous to Quillen's D'{e}vissage theorem for $K$-theory. For abelian\u0000categories ${mathscr A}:=({mathscr A}, ^{vee}, varpi)$ with duality, and\u0000appropriate abelian subcategories ${mathscr B}subseteq {mathscr A}$, we\u0000prove D'{e}vissage theorems for ${bf GW}$ spaces, $G{mathcal W}$-spectra and\u0000${mathbb G}W$ bispectra. As a consequence, for regular local rings $(R, m,\u0000kappa)$ with $1/2in R$, we compute the ${BG}W$ groups ${mathbb\u0000G}W^{[n]}_k(spec{R})~forall k, nin {mathbb Z}$, where $n$ represent the\u0000translation.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206316","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 compute the Chow ring of a quasi-split geometrically almost simple algebraic group assuming the coefficients to be a field. This extends the classical computation for split groups done by Kac to the non-split quasi-split case. For the proof we introduce and study equivariant conormed Chow rings, which are well adapted to the study of quasi-split groups and their homogeneous varieties.
{"title":"Chow rings of quasi-split geometrically almost simple algebraic groups","authors":"Alexey Ananyevskiy, Nikita Geldhauser","doi":"arxiv-2408.09390","DOIUrl":"https://doi.org/arxiv-2408.09390","url":null,"abstract":"We compute the Chow ring of a quasi-split geometrically almost simple\u0000algebraic group assuming the coefficients to be a field. This extends the\u0000classical computation for split groups done by Kac to the non-split quasi-split\u0000case. For the proof we introduce and study equivariant conormed Chow rings,\u0000which are well adapted to the study of quasi-split groups and their homogeneous\u0000varieties.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206317","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}
Noe Barcenas, Luis Eduardo Garcia-Hernandez, Raphael Reinauer
We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by computing the connective real k-homology of the classifying space with the Adams spectral sequence and two types of detection theorems for the kernel of the alpha invariant: one based on eta-invariants, closely following work of Botvinnik-Gilkey-Stolz, and a second one based on homological methods. Along the way, we determine differentials of the Adams spectral sequence for classifying spaces involved in the computation, and we study the cap structure of the Adams spectral sequence for sub-hopf algebras of the Steenrod algebra relevant to the computation of connective real and complex k-homology.
{"title":"The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4","authors":"Noe Barcenas, Luis Eduardo Garcia-Hernandez, Raphael Reinauer","doi":"arxiv-2408.07895","DOIUrl":"https://doi.org/arxiv-2408.07895","url":null,"abstract":"We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by\u0000computing the connective real k-homology of the classifying space with the\u0000Adams spectral sequence and two types of detection theorems for the kernel of\u0000the alpha invariant: one based on eta-invariants, closely following work of\u0000Botvinnik-Gilkey-Stolz, and a second one based on homological methods. Along\u0000the way, we determine differentials of the Adams spectral sequence for\u0000classifying spaces involved in the computation, and we study the cap structure\u0000of the Adams spectral sequence for sub-hopf algebras of the Steenrod algebra\u0000relevant to the computation of connective real and complex k-homology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206333","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}
Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, Robin J. Sroka
In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. We apply this to various families of groups, such as interval exchange groups and Brin--Thompson groups, recovering results of Szymik--Wahl, Li, and Tanner, and obtaining new results as well.
在任何具有合理覆盖概念的范畴中，每个对象都有一个剪刀自动形群。我们证明，在温和的条件下，这个群的同调与对象无关，可以用扎哈雷维奇定义的剪刀同调 K 理论谱来表示。因此，我们既获得了扎哈雷维奇高阶剪刀同构 K 理论的群论解释，也获得了计算剪刀同构群同调的方法。我们将其应用于不同的群族，如间隔交换群和布林-汤普森群，恢复了希米克-华尔、李和坦纳的结果，同时也获得了新的结果。
{"title":"Scissors automorphism groups and their homology","authors":"Alexander Kupers, Ezekiel Lemann, Cary Malkiewich, Jeremy Miller, Robin J. Sroka","doi":"arxiv-2408.08081","DOIUrl":"https://doi.org/arxiv-2408.08081","url":null,"abstract":"In any category with a reasonable notion of cover, each object has a group of\u0000scissors automorphisms. We prove that under mild conditions, the homology of\u0000this group is independent of the object, and can be expressed in terms of the\u0000scissors congruence K-theory spectrum defined by Zakharevich. We therefore\u0000obtain both a group-theoretic interpretation of Zakharevich's higher scissors\u0000congruence K-theory, as well as a method to compute the homology of scissors\u0000automorphism groups. We apply this to various families of groups, such as\u0000interval exchange groups and Brin--Thompson groups, recovering results of\u0000Szymik--Wahl, Li, and Tanner, and obtaining new results as well.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206318","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 paper, we construct a Real equivariant version of the B"okstedt spectral sequence which takes inputs in the theory of Real Hochschild homology developed by Angelini-Knoll, Gerhardt, and Hill and converges to the equivariant homology of Real topological Hochschild homology, $text{THR}$. We also show that when $A$ is a commutative $C_2$-ring spectrum, $text{THR}(A)$ has the structure of an $A$-Hopf algebroid in the $C_2$-equivariant stable homotopy category.
{"title":"Computational tools for Real topological Hochschild homology","authors":"Chloe Lewis","doi":"arxiv-2408.07188","DOIUrl":"https://doi.org/arxiv-2408.07188","url":null,"abstract":"In this paper, we construct a Real equivariant version of the B\"okstedt\u0000spectral sequence which takes inputs in the theory of Real Hochschild homology\u0000developed by Angelini-Knoll, Gerhardt, and Hill and converges to the\u0000equivariant homology of Real topological Hochschild homology, $text{THR}$. We\u0000also show that when $A$ is a commutative $C_2$-ring spectrum, $text{THR}(A)$\u0000has the structure of an $A$-Hopf algebroid in the $C_2$-equivariant stable\u0000homotopy category.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226419","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 paper, we take advantage of a reinterpretation of differential modules admitting a flag structure as a special class of perturbations of complexes. We are thus able to leverage the machinery of homological perturbation theory to prove strong statements on the homological theory of differential modules admitting additional auxiliary gradings and having infinite homological dimension. One of the main takeaways of our results is that the category of differential modules is much more similar than expected to the category of chain complexes, and from the K-theoretic perspective such objects are largely indistinguishable. This intuition is made precise through the construction of so-called anchored resolutions, which are a distinguished class of projective flag resolutions that possess remarkably well-behaved uniqueness properties in the (flag-preserving) homotopy category. We apply this theory to prove an analogue of the Total Rank Conjecture for differential modules admitting a ZZ/2-grading in a large number of cases.
在本文中，我们利用了将允许旗结构的微分模重新解释为一类特殊的复数扰动的方法。因此，我们能够利用同调扰动理论的机制，证明容许额外辅助等级并具有无限同调维度的微分模的同调理论的强声明。我们结果的主要启示之一是，微分模范畴与链复数范畴的相似程度远超预期，而且从 K 理论的角度来看，这类对象基本上是不可区分的。通过构建所谓的锚定决议，这一直觉变得更加精确了，锚定决议是射影旗决议的一个杰出类别，在（保旗）同调范畴中具有非常良好的唯一性。我们应用这一理论证明了在大量情况下允许 ZZ/2 等级的微分模块的总等级猜想。
{"title":"Flagged Perturbations and Anchored Resolutions","authors":"Keller VandeBogert","doi":"arxiv-2408.02749","DOIUrl":"https://doi.org/arxiv-2408.02749","url":null,"abstract":"In this paper, we take advantage of a reinterpretation of differential\u0000modules admitting a flag structure as a special class of perturbations of\u0000complexes. We are thus able to leverage the machinery of homological\u0000perturbation theory to prove strong statements on the homological theory of\u0000differential modules admitting additional auxiliary gradings and having\u0000infinite homological dimension. One of the main takeaways of our results is\u0000that the category of differential modules is much more similar than expected to\u0000the category of chain complexes, and from the K-theoretic perspective such\u0000objects are largely indistinguishable. This intuition is made precise through\u0000the construction of so-called anchored resolutions, which are a distinguished\u0000class of projective flag resolutions that possess remarkably well-behaved\u0000uniqueness properties in the (flag-preserving) homotopy category. We apply this\u0000theory to prove an analogue of the Total Rank Conjecture for differential\u0000modules admitting a ZZ/2-grading in a large number of cases.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938311","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 two (hereditary) complete cotorsion pairs $(mathcal{X}_1,mathcal{Y}_1)$ and $(mathcal{X}_2,mathcal{Y}_2)$ in an exact category with $mathcal{X}_1subseteq mathcal{Y}_2$, we prove that $left({rm Smd}langle mathcal{X}_1,mathcal{X}_2 rangle,mathcal{Y}_1cap mathcal{Y}_2right)$ is also a (hereditary) complete cotorsion pair, where ${rm Smd}langle mathcal{X}_1,mathcal{X}_2 rangle$ is the class of direct summands of extension of $mathcal{X}_1$ and $mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^perpmathcal{GI}^{leqslant n},mathcal{GI}^{leqslant n})$, where $mathcal{GI}^{leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions.
{"title":"Intersection of complete cotorsion pairs","authors":"Qikai Wang, Haiyan Zhu","doi":"arxiv-2408.01922","DOIUrl":"https://doi.org/arxiv-2408.01922","url":null,"abstract":"Given two (hereditary) complete cotorsion pairs\u0000$(mathcal{X}_1,mathcal{Y}_1)$ and $(mathcal{X}_2,mathcal{Y}_2)$ in an exact\u0000category with $mathcal{X}_1subseteq mathcal{Y}_2$, we prove that $left({rm\u0000Smd}langle mathcal{X}_1,mathcal{X}_2 rangle,mathcal{Y}_1cap\u0000mathcal{Y}_2right)$ is also a (hereditary) complete cotorsion pair, where\u0000${rm Smd}langle mathcal{X}_1,mathcal{X}_2 rangle$ is the class of direct\u0000summands of extension of $mathcal{X}_1$ and $mathcal{X}_2$. As an\u0000application, we construct complete cotorsion pairs, such as\u0000$(^perpmathcal{GI}^{leqslant n},mathcal{GI}^{leqslant n})$, where\u0000$mathcal{GI}^{leqslant n}$ is the class of modules of Gorenstein injective\u0000dimension at most $n$. And we also characterize the left orthogonal class of\u0000exact complexes of injective modules and the classes of modules with finite\u0000Gorenstein projective, Gorenstein flat, and PGF dimensions.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969033","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}
As part of a program to develop $K$-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas $overline{mathfrak{A}}_a = sum_b Q_b^a(beta)overline{mathfrak{P}}_b$ and $overline{mathfrak{Q}}_a = sum_b M_b^a(beta)overline{mathfrak{F}}_b,$ where each of $overline{mathfrak{A}}_a$, $overline{mathfrak{P}}_a$, $overline{mathfrak{Q}}_a$ and $overline{mathfrak{F}}_a$ is a family of polynomials that forms a basis for $mathbb{Z}[x_1,dots,x_n][beta]$ indexed by weak compositions $a,$ and $Q_b^a(beta)$ and $M_b^a(beta)$ are monomials in $beta$ for each pair $(a,b)$ of weak compositions. The polynomials $overline{mathfrak{A}}_a$ are the Lascoux atoms, $overline{mathfrak{P}}_a$ are the kaons, $overline{mathfrak{Q}}_a$ are the quasiLascoux polynomials, and $overline{mathfrak{F}}_a$ are the glide polynomials; these are respectively the $K$-analogues of the Demazure atoms $mathfrak{A}_a$, the fundamental particles $mathfrak{P}_a$, the quasikey polynomials $mathfrak{Q}_a$, and the fundamental slide polynomials $mathfrak{F}_a$. Monical, Pechenik, and Searles conjectured that for any fixed $a,$ $sum_b Q_b^a(-1), sum_b M_b^a(-1) in {0,1},$ where $b$ ranges over all weak compositions. We prove this conjecture using a sign-reversing involution.
{"title":"Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles","authors":"Laura Pierson","doi":"arxiv-2408.01390","DOIUrl":"https://doi.org/arxiv-2408.01390","url":null,"abstract":"As part of a program to develop $K$-theoretic analogues of combinatorially\u0000important polynomials, Monical, Pechenik, and Searles (2021) proved two\u0000expansion formulas $overline{mathfrak{A}}_a = sum_b\u0000Q_b^a(beta)overline{mathfrak{P}}_b$ and $overline{mathfrak{Q}}_a = sum_b\u0000M_b^a(beta)overline{mathfrak{F}}_b,$ where each of\u0000$overline{mathfrak{A}}_a$, $overline{mathfrak{P}}_a$,\u0000$overline{mathfrak{Q}}_a$ and $overline{mathfrak{F}}_a$ is a family of\u0000polynomials that forms a basis for $mathbb{Z}[x_1,dots,x_n][beta]$ indexed\u0000by weak compositions $a,$ and $Q_b^a(beta)$ and $M_b^a(beta)$ are monomials\u0000in $beta$ for each pair $(a,b)$ of weak compositions. The polynomials\u0000$overline{mathfrak{A}}_a$ are the Lascoux atoms, $overline{mathfrak{P}}_a$\u0000are the kaons, $overline{mathfrak{Q}}_a$ are the quasiLascoux polynomials,\u0000and $overline{mathfrak{F}}_a$ are the glide polynomials; these are\u0000respectively the $K$-analogues of the Demazure atoms $mathfrak{A}_a$, the\u0000fundamental particles $mathfrak{P}_a$, the quasikey polynomials\u0000$mathfrak{Q}_a$, and the fundamental slide polynomials $mathfrak{F}_a$.\u0000Monical, Pechenik, and Searles conjectured that for any fixed $a,$ $sum_b\u0000Q_b^a(-1), sum_b M_b^a(-1) in {0,1},$ where $b$ ranges over all weak\u0000compositions. We prove this conjecture using a sign-reversing involution.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938390","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 define a motivic Greenlees spectral sequence by characterising an associated $t$-structure. We then examine a motivic version of topological Hochschild homology for the motivic cohomology spectrum modulo a prime number $p$. Finally, we use the motivic Greenlees spectral sequence to determine the homotopy ring of a related spectrum, given that the base field is algebraically closed with a characteristic that is coprime to $p$.
我们通过描述相关的 $t$ 结构来定义动机格林列斯谱序列。然后，我们研究了动机同调谱 modulo a prime number$p$的拓扑霍赫希尔德同调的动机版本。最后，我们利用动机格林列斯谱序列来确定相关谱的同调环，条件是基域是代数封闭的，其特征与$p$共乘。
{"title":"A motivic Greenlees spectral sequence towards motivic Hochschild homology","authors":"Federico Ernesto Mocchetti","doi":"arxiv-2408.00338","DOIUrl":"https://doi.org/arxiv-2408.00338","url":null,"abstract":"We define a motivic Greenlees spectral sequence by characterising an\u0000associated $t$-structure. We then examine a motivic version of topological\u0000Hochschild homology for the motivic cohomology spectrum modulo a prime number\u0000$p$. Finally, we use the motivic Greenlees spectral sequence to determine the\u0000homotopy ring of a related spectrum, given that the base field is algebraically\u0000closed with a characteristic that is coprime to $p$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885218","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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