{"title":"K0-stability over monoid algebras","authors":"Husney Parvez Sarwar","doi":"10.2140/akt.2021.6.629","DOIUrl":"https://doi.org/10.2140/akt.2021.6.629","url":null,"abstract":"","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41966972","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 construct the homomorphism of presheaves ${mathrm{K}}^mathrm{MW}_* to {pi}^{*,*}$ over an arbitrary base scheme $S$, where $mathrm{K}^mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $unKMW_nsimeq underline{pi}^{n,n}_s$, $nin mathbb Z$, over a filed $k$ originally proved in cite{M02} and cite{M-A1Top}.
{"title":"The naive Milnor–Witt K-theory relations in the\u0000stable motivic homotopy groups over a base","authors":"A. Druzhinin","doi":"10.2140/akt.2021.6.651","DOIUrl":"https://doi.org/10.2140/akt.2021.6.651","url":null,"abstract":"We construct the homomorphism of presheaves ${mathrm{K}}^mathrm{MW}_* to {pi}^{*,*}$ over an arbitrary base scheme $S$, where $mathrm{K}^mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. \u0000Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $unKMW_nsimeq underline{pi}^{n,n}_s$, $nin mathbb Z$, over a filed $k$ originally proved in cite{M02} and cite{M-A1Top}.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"60 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88469458","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 calculate $mathrm K_*(k[x]/x^e;mathbf Z_p)$ by evaluating the syntomic cohomology $mathbf Z_p(i)(k[x]/x^e)$ introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case $e=2$ and $p>2$. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for $e=2$. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.
{"title":"Floor, ceiling, slopes, and K-theory","authors":"Yuri J. F. Sulyma","doi":"10.2140/akt.2023.8.331","DOIUrl":"https://doi.org/10.2140/akt.2023.8.331","url":null,"abstract":"We calculate $mathrm K_*(k[x]/x^e;mathbf Z_p)$ by evaluating the syntomic cohomology $mathbf Z_p(i)(k[x]/x^e)$ introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case $e=2$ and $p>2$. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for $e=2$. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49470328","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 $1 to Gamma to tilde{G} to G to 1$ be a central extension by an abelian finite group. In this paper, we compute the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$. We then introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle $mathcal{A}$. We define and compute the index of such families using the cohomological index formula for families of $SU(N)$-transversally elliptic operators. More precisely, a family $A$ of projective operators can be pulled back in a family $tilde{A}$ of $SU(N)$-transversally elliptic operators on the $PU(N)$-principal bundle of trivialisations of $mathcal{A}$. Through the distributional index of $tilde{A}$, we can define an index for the family $A$ of projective operators and using the index formula in equivariant cohomology for families of $SU(N)$-transversally elliptic operators, we derive an explicit cohomological index formula in de Rham cohomology. Once this is done, we define and compute the index of families of projective Dirac operators. As a second application of our computation of the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$, we consider the special case of a family of $Spin(2n)$-transversally elliptic Dirac operators over the bundle of oriented orthonormal frames of an oriented fibration and we relate its distributional index with the index of the corresponding family of projective Dirac operators.
{"title":"The index of families of projective operators","authors":"Alexandre Baldare","doi":"10.2140/akt.2023.8.285","DOIUrl":"https://doi.org/10.2140/akt.2023.8.285","url":null,"abstract":"Let $1 to Gamma to tilde{G} to G to 1$ be a central extension by an abelian finite group. In this paper, we compute the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$. We then introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle $mathcal{A}$. We define and compute the index of such families using the cohomological index formula for families of $SU(N)$-transversally elliptic operators. More precisely, a family $A$ of projective operators can be pulled back in a family $tilde{A}$ of $SU(N)$-transversally elliptic operators on the $PU(N)$-principal bundle of trivialisations of $mathcal{A}$. Through the distributional index of $tilde{A}$, we can define an index for the family $A$ of projective operators and using the index formula in equivariant cohomology for families of $SU(N)$-transversally elliptic operators, we derive an explicit cohomological index formula in de Rham cohomology. Once this is done, we define and compute the index of families of projective Dirac operators. As a second application of our computation of the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$, we consider the special case of a family of $Spin(2n)$-transversally elliptic Dirac operators over the bundle of oriented orthonormal frames of an oriented fibration and we relate its distributional index with the index of the corresponding family of projective Dirac operators.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46869877","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 BSTRACT . We show that Mandell’s inverse K -theory functor is a categorically- enriched non-symmetric multifunctor. In particular, it preserves algebraic structures parametrized by non-symmetric operads. As applications, we describe how ring categories arise as the images of inverse K -theory.
{"title":"Multifunctorial inverse K-theory","authors":"Niles Johnson, Donald Yau","doi":"10.2140/akt.2022.7.507","DOIUrl":"https://doi.org/10.2140/akt.2022.7.507","url":null,"abstract":"A BSTRACT . We show that Mandell’s inverse K -theory functor is a categorically- enriched non-symmetric multifunctor. In particular, it preserves algebraic structures parametrized by non-symmetric operads. As applications, we describe how ring categories arise as the images of inverse K -theory.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45698034","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 initiate the study of real C ∗ -algebras associated to higher-rank graphs Λ, with a focus on their K -theory. Following Kasparov and Evans, we identify a spectral sequence which computes the CR K -theory of C ∗ R (Λ , γ ) for any involution γ on Λ, and show that the E 2 page of this spectral sequence can be straightforwardly computed from the combinatorial data of the k -graph Λ and the involution γ . We provide a complete description of K CR ( C ∗ R (Λ , γ )) for several examples of higher-rank graphs Λ with involution.
{"title":"K-theory for real k-graph C∗-algebras","authors":"Jeffrey L. Boersema, E. Gillaspy","doi":"10.2140/akt.2022.7.395","DOIUrl":"https://doi.org/10.2140/akt.2022.7.395","url":null,"abstract":". We initiate the study of real C ∗ -algebras associated to higher-rank graphs Λ, with a focus on their K -theory. Following Kasparov and Evans, we identify a spectral sequence which computes the CR K -theory of C ∗ R (Λ , γ ) for any involution γ on Λ, and show that the E 2 page of this spectral sequence can be straightforwardly computed from the combinatorial data of the k -graph Λ and the involution γ . We provide a complete description of K CR ( C ∗ R (Λ , γ )) for several examples of higher-rank graphs Λ with involution.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43160028","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 study multilinear symbols on fields taking values in reciprocity sheaves. We prove that any such symbol satisfying natural axioms automatically has Steinberg-type relations, which is a manifestation of the geometry of modulus pairs lying behind.
{"title":"Steinberg symbols and reciprocity sheaves","authors":"Junnosuke Koizumi","doi":"10.2140/akt.2022.7.695","DOIUrl":"https://doi.org/10.2140/akt.2022.7.695","url":null,"abstract":"We study multilinear symbols on fields taking values in reciprocity sheaves. We prove that any such symbol satisfying natural axioms automatically has Steinberg-type relations, which is a manifestation of the geometry of modulus pairs lying behind.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42385985","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 the sheaf of $mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $mathbb A^1$-invariant quotient (obtained by iterating the $mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $mathbb A^1$-connected space on which the iterations of the naive $mathbb A^1$-connected components construction do not stabilize before the $n$th stage.
{"title":"Remarks on iterations of the 𝔸1-chain connected\u0000components construction","authors":"Chetan T. Balwe, B. Rani, Anand Sawant","doi":"10.2140/akt.2022.7.385","DOIUrl":"https://doi.org/10.2140/akt.2022.7.385","url":null,"abstract":"We show that the sheaf of $mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $mathbb A^1$-invariant quotient (obtained by iterating the $mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $mathbb A^1$-connected space on which the iterations of the naive $mathbb A^1$-connected components construction do not stabilize before the $n$th stage.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46760080","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 the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.
{"title":"Deformation theory of perfect complexes and traces","authors":"Max Lieblich, Martin Olsson","doi":"10.2140/akt.2022.7.651","DOIUrl":"https://doi.org/10.2140/akt.2022.7.651","url":null,"abstract":"We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46657523","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 study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes.
{"title":"K-theory of admissible Zariski–Riemann\u0000spaces","authors":"Christian Dahlhausen","doi":"10.2140/akt.2023.8.1","DOIUrl":"https://doi.org/10.2140/akt.2023.8.1","url":null,"abstract":"We study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41903620","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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