We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of their algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0(L(E))$ of the Leavitt path algebra $L(E)$ coincides with Krieger's dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^*$-algebras. This result was only known for irreducible graphs.
{"title":"Graded K-theory, filtered K-theory and the\u0000classification of graph algebras","authors":"P. Ara, R. Hazrat, Huanhuan Li","doi":"10.2140/akt.2022.7.731","DOIUrl":"https://doi.org/10.2140/akt.2022.7.731","url":null,"abstract":"We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of their algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0(L(E))$ of the Leavitt path algebra $L(E)$ coincides with Krieger's dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^*$-algebras. This result was only known for irreducible graphs.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43702100","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}
Let $F$ be a field, $ell$ a prime and $D$ a central division $F$-algebra of $ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $lambda$ such that the cohomology class $(D)cup (lambda)in H^3(F,,mathbb{Q}_{ell}/Z_{ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{otimes i},,forall ige 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $ell$. When $D$ has period $ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $ell$-dimension $mathrm{cd}_{ell}(k)$ of $k$ is $le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $mathrm{cd}_{ell}(k)le 2$. In the case $ell=car(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.
设$F$是域，$ell$是素数，$D$是$ell$-幂次的中心除法$F$-代数。通过$D$的Rost核，我们指的是由元素$lamba$组成的$F^*$的子群，使得H^3（F，，mathbb）中的上同调类$（D）cup（lamba）{Q}_｛ell｝/Z_（2））$消失。1985年，Suslin推测Rost核是由$D^｛otimes i｝，，for all ige 1$的约化范数的$i$次方生成的。尽管有已知的反例，我们还是证明了Suslin猜想的一些新情况。我们假设$F$是一个henselian离散估值域，其残差域$k$的特征不同于$ell$。当$D$具有周期$ell$时，我们证明了如果$k$是$2$-局部域或上同调$ell$-维数$mathrm，Suslin猜想成立{cd}_$k$的｛ell｝（k）$是$le 2$。当周期是任意的时，当$k$本身是带有$mathrm的henselian离散估值域时，我们证明了相同的结果{cd}_｛ell｝（k）le 2$。在$ell=car（k）$的情况下，得到了温和分枝代数的一个类似物。我们猜想Suslin猜想适用于上同调维数为3的所有域。
{"title":"On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3","authors":"Yong Hu, Z. Wu","doi":"10.2140/akt.2020.5.677","DOIUrl":"https://doi.org/10.2140/akt.2020.5.677","url":null,"abstract":"Let $F$ be a field, $ell$ a prime and $D$ a central division $F$-algebra of $ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $lambda$ such that the cohomology class $(D)cup (lambda)in H^3(F,,mathbb{Q}_{ell}/Z_{ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{otimes i},,forall ige 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $ell$. When $D$ has period $ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $ell$-dimension $mathrm{cd}_{ell}(k)$ of $k$ is $le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $mathrm{cd}_{ell}(k)le 2$. In the case $ell=car(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47720696","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}
J. Bergner, A. Osorno, Viktoriya Ozornova, M. Rovelli, Claudia I. Scheimbauer
In previous work, we develop a generalized Waldhausen $S_{bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{bullet}$-constructions for exact categories and for stable and exact $(infty,1)$-categories, as well as the relative $S_{bullet}$-construction for exact functors.
{"title":"Comparison of Waldhausen constructions","authors":"J. Bergner, A. Osorno, Viktoriya Ozornova, M. Rovelli, Claudia I. Scheimbauer","doi":"10.2140/akt.2021.6.97","DOIUrl":"https://doi.org/10.2140/akt.2021.6.97","url":null,"abstract":"In previous work, we develop a generalized Waldhausen $S_{bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{bullet}$-constructions for exact categories and for stable and exact $(infty,1)$-categories, as well as the relative $S_{bullet}$-construction for exact functors.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47989965","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 devissage--type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $mathscr{C}$ when a devissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between devissage theorems of emph{single type} and of emph{multiple type} depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original devissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of devissage--type theorems and prove a general devissage theorem of single type and a general devissage theorem of multiple type.
{"title":"Dévissage for Waldhausen K-theory","authors":"G. Raptis","doi":"10.2140/akt.2022.7.467","DOIUrl":"https://doi.org/10.2140/akt.2022.7.467","url":null,"abstract":"A devissage--type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $mathscr{C}$ when a devissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between devissage theorems of emph{single type} and of emph{multiple type} depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original devissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of devissage--type theorems and prove a general devissage theorem of single type and a general devissage theorem of multiple type.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43080639","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 have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincar'e duality for infinite-dimensional manifolds. We will also discuss the assembly maps.
{"title":"An infinite-dimensional index theorem and the\u0000Higson–Kasparov–Trout algebra","authors":"Doman Takata","doi":"10.2140/akt.2022.7.1","DOIUrl":"https://doi.org/10.2140/akt.2022.7.1","url":null,"abstract":"We have been studying the index theory for some special infinite-dimensional manifolds with a \"proper cocompact\" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain relation, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincar'e duality for infinite-dimensional manifolds. We will also discuss the assembly maps.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48108163","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 study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $mathrm{K}_0(mathcal{D}^{sg}(X))$ is finite torsion, and that $mathrm{K}_1(mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.
{"title":"K-theory and the singularity category of\u0000quotient singularities","authors":"Nebojsa Pavic, E. Shinder","doi":"10.2140/akt.2021.6.381","DOIUrl":"https://doi.org/10.2140/akt.2021.6.381","url":null,"abstract":"In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $mathrm{K}_0(mathcal{D}^{sg}(X))$ is finite torsion, and that $mathrm{K}_1(mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47028185","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 use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.
{"title":"Periodic cyclic homology and derived de Rham cohomology","authors":"Benjamin Antieau","doi":"10.2140/akt.2019.4.505","DOIUrl":"https://doi.org/10.2140/akt.2019.4.505","url":null,"abstract":"We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.505","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45261340","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 lisse l-adic sheaf G on a smooth proper variety X and a lisse sheaf F on an open dense U in X, Kato and Saito conjectured a localization formula for the global l-adic epsilon factor εl(X,F ⊗ G) in terms of the global epsilon factor of F and a certain intersection number associated to det(G) and the Swan class of F . In this article, we prove an analog of this conjecture for global de Rham epsilon factors in the classical setting of DX -modules on smooth projective varieties over a field of characteristic zero.
{"title":"On a localization formula of epsilon factors via microlocal geometry","authors":"Tomoyuki Abe, D. Patel","doi":"10.2140/AKT.2018.3.461","DOIUrl":"https://doi.org/10.2140/AKT.2018.3.461","url":null,"abstract":"Given a lisse l-adic sheaf G on a smooth proper variety X and a lisse sheaf F on an open dense U in X, Kato and Saito conjectured a localization formula for the global l-adic epsilon factor εl(X,F ⊗ G) in terms of the global epsilon factor of F and a certain intersection number associated to det(G) and the Swan class of F . In this article, we prove an analog of this conjecture for global de Rham epsilon factors in the classical setting of DX -modules on smooth projective varieties over a field of characteristic zero.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2018.3.461","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45609606","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 a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $Omega(BG^wedge_p)$, when $G$ is a finite group, $BG^wedge_p$ is the $p$-completion of its classifying space, and $Omega(BG^wedge_p)$ is the loop space of $BG^wedge_p$. The main purpose of this work is to shed new light on Benson's result by extending it to a more general setting. As a special case, we show that if $mathcal{C}$ is a small category, $|mathcal{C}|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|mathcal{C}|^+_R$ is a "plus construction" for $|mathcal{C}|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_*(Omega(|mathcal{C}|^+_R);R)$ can be described as the homology of a chain complex of projective $Rmathcal{C}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson's theorem is now the case where $mathcal{C}$ is the category of a finite group $G$, $R=mathbb{F}_p$ for some prime $p$, and $|mathcal{C}|^+_R=BG^wedge_p$.
{"title":"Loop space homology of a small category","authors":"C. Broto, R. Levi, B. Oliver","doi":"10.2140/akt.2021.6.425","DOIUrl":"https://doi.org/10.2140/akt.2021.6.425","url":null,"abstract":"In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod $p$ homology of $Omega(BG^wedge_p)$, when $G$ is a finite group, $BG^wedge_p$ is the $p$-completion of its classifying space, and $Omega(BG^wedge_p)$ is the loop space of $BG^wedge_p$. The main purpose of this work is to shed new light on Benson's result by extending it to a more general setting. As a special case, we show that if $mathcal{C}$ is a small category, $|mathcal{C}|$ is the geometric realization of its nerve, $R$ is a commutative ring, and $|mathcal{C}|^+_R$ is a \"plus construction\" for $|mathcal{C}|$ in the sense of Quillen (taken with respect to $R$-homology), then $H_*(Omega(|mathcal{C}|^+_R);R)$ can be described as the homology of a chain complex of projective $Rmathcal{C}$-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson's theorem is now the case where $mathcal{C}$ is the category of a finite group $G$, $R=mathbb{F}_p$ for some prime $p$, and $|mathcal{C}|^+_R=BG^wedge_p$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44049114","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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