We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement, and new proof, of R. Thomason's formula.
{"title":"Virtual excess intersection theory","authors":"Adeel A. Khan","doi":"10.2140/akt.2021.6.559","DOIUrl":"https://doi.org/10.2140/akt.2021.6.559","url":null,"abstract":"We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement, and new proof, of R. Thomason's formula.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43363295","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 give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.
{"title":"C2-equivariant stable homotopy from real\u0000motivic stable homotopy","authors":"M. Behrens, J. Shah","doi":"10.2140/AKT.2020.5.411","DOIUrl":"https://doi.org/10.2140/AKT.2020.5.411","url":null,"abstract":"We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2020.5.411","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44987041","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}
The 8-periodic theory that comes from the KO-theory of the mod 2 Moore space is the same as the real first Morava K-theory obtained from the homotopy fixed points of the Z/(2) action on the first Morava K-theory. The first Morava K-theory, K(1), is just mod 2 KU-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum. There are a lot of maps into and out of these spaces and the spaces for KO- theory, KU-theory and the first Morava K-theory. For every one of these 98 maps (counting suspensions) there is a spectral sequence. We describe all 98 maps and spectral sequences. 48 of these maps involve our new spaces and 56 of the spectral sequences do. In addition, the maps on homotopy are all written down.
{"title":"The Omega spectrum for mod 2\u0000KO-theory","authors":"W. Wilson","doi":"10.2140/akt.2020.5.357","DOIUrl":"https://doi.org/10.2140/akt.2020.5.357","url":null,"abstract":"The 8-periodic theory that comes from the KO-theory of the mod 2 Moore space is the same as the real first Morava K-theory obtained from the homotopy fixed points of the Z/(2) action on the first Morava K-theory. The first Morava K-theory, K(1), is just mod 2 KU-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum. There are a lot of maps into and out of these spaces and the spaces for KO- theory, KU-theory and the first Morava K-theory. For every one of these 98 maps (counting suspensions) there is a spectral sequence. We describe all 98 maps and spectral sequences. 48 of these maps involve our new spaces and 56 of the spectral sequences do. In addition, the maps on homotopy are all written down.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2020.5.357","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41773561","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 that for a quasi-regular semiperfectoid $mathbb{Z}_p^{rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;mathbb{Z}_p)$ of $R$ to the topological cyclic homology $mathrm{TC}(R;mathbb{Z}_p)$ of $R$ identifies on $pi_2$ with a $q$-deformation of the logarithm.
{"title":"The p-completed cyclotomic trace in degree\u00002","authors":"J. Anschutz, A. C. Bras","doi":"10.2140/akt.2020.5.539","DOIUrl":"https://doi.org/10.2140/akt.2020.5.539","url":null,"abstract":"We prove that for a quasi-regular semiperfectoid $mathbb{Z}_p^{rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;mathbb{Z}_p)$ of $R$ to the topological cyclic homology $mathrm{TC}(R;mathbb{Z}_p)$ of $R$ identifies on $pi_2$ with a $q$-deformation of the logarithm.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2020.5.539","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42897447","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 a cycle class map from the higher Chow groups of 0-cycles to the relative $K$-theory of a modulus pair. We show that this induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and relative $K$-theory of truncated polynomial rings over a regular semi-local ring, essentially of finite type over a characteristic zero field.
{"title":"Zero-cycles with modulus and relative\u0000K-theory","authors":"Rahul Gupta, A. Krishna","doi":"10.2140/akt.2020.5.757","DOIUrl":"https://doi.org/10.2140/akt.2020.5.757","url":null,"abstract":"We construct a cycle class map from the higher Chow groups of 0-cycles to the relative $K$-theory of a modulus pair. We show that this induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and relative $K$-theory of truncated polynomial rings over a regular semi-local ring, essentially of finite type over a characteristic zero field.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42302765","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}
For X a product of Severi-Brauer varieties, we conjecture: if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to: if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E/B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E/B is an isomorphism. In certain cases we verify this conjecture. Notation and Conventions. We fix a field k throughout. All of our objects are defined over k unless stated otherwise. Sometimes we use k as an index when no confusion will occur. For any field F , we fix an algebraic closure F . A variety X is a separated scheme of finite type over a field. Let X = X1 × · · · ×Xr be a product of varieties with projections πi : X → Xi. Let F1, ...,Fr be sheaves of modules on X1, ..., Xr. We use F1 · · · Fr for the external product π∗ 1F1⊗ · · ·⊗π∗ rFr. For a ring R with a Z-indexed descending filtration F • ν , (e.g. ν = γ or τ as in Section 2), we write grνR for the corresponding quotient F i ν/F i+1 ν . We write grνR = ⊕ i∈Z gr i νR for the associated graded ring. A semisimple algebraic group G is of type AC if its Dynkin diagram is a union of diagrams of type A and type C. Similarly a semisimple group G is of type AA if its Dynkin diagram is a union of diagrams of type A. For an index set I, two elements i, j ∈ I, we write δij for the function which is 0 when i 6= j and 1 if i = j. Given two r-tuples of integers, say I, J , we write I < J if the ith component of I is less than the ith component of J for any 1 ≤ i ≤ r.
{"title":"On the K-theory coniveau epimorphism for\u0000products of Severi–Brauer varieties","authors":"N. Karpenko, Eoin Mackall","doi":"10.2140/AKT.2019.4.317","DOIUrl":"https://doi.org/10.2140/AKT.2019.4.317","url":null,"abstract":"For X a product of Severi-Brauer varieties, we conjecture: if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to: if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E/B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E/B is an isomorphism. In certain cases we verify this conjecture. Notation and Conventions. We fix a field k throughout. All of our objects are defined over k unless stated otherwise. Sometimes we use k as an index when no confusion will occur. For any field F , we fix an algebraic closure F . A variety X is a separated scheme of finite type over a field. Let X = X1 × · · · ×Xr be a product of varieties with projections πi : X → Xi. Let F1, ...,Fr be sheaves of modules on X1, ..., Xr. We use F1 · · · Fr for the external product π∗ 1F1⊗ · · ·⊗π∗ rFr. For a ring R with a Z-indexed descending filtration F • ν , (e.g. ν = γ or τ as in Section 2), we write grνR for the corresponding quotient F i ν/F i+1 ν . We write grνR = ⊕ i∈Z gr i νR for the associated graded ring. A semisimple algebraic group G is of type AC if its Dynkin diagram is a union of diagrams of type A and type C. Similarly a semisimple group G is of type AA if its Dynkin diagram is a union of diagrams of type A. For an index set I, two elements i, j ∈ I, we write δij for the function which is 0 when i 6= j and 1 if i = j. Given two r-tuples of integers, say I, J , we write I < J if the ith component of I is less than the ith component of J for any 1 ≤ i ≤ r.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2019.4.317","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42888028","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 K(Fq) be the algebraic K-theory spectrum of the finite field with q elements and let p ≥ 5 be a prime number coprime to q. In this paper we study the mod p and v1 topological Hochschild homology of K(Fq), denoted V (1)∗ THH(K(Fq)), as an Fp-algebra. The computations are organized in four different cases, depending on the mod p behaviour of the function q−1. We use different spectral sequences, in particular the Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the Fp-algebras THH∗(K(Fq);HFp), and we compute V (1)∗ THH(K(Fq)) in the first two cases.
{"title":"The topological Hochschild homology of\u0000algebraic K-theory of finite fields","authors":"E. Honing","doi":"10.2140/akt.2021.6.29","DOIUrl":"https://doi.org/10.2140/akt.2021.6.29","url":null,"abstract":"Let K(Fq) be the algebraic K-theory spectrum of the finite field with q elements and let p ≥ 5 be a prime number coprime to q. In this paper we study the mod p and v1 topological Hochschild homology of K(Fq), denoted V (1)∗ THH(K(Fq)), as an Fp-algebra. The computations are organized in four different cases, depending on the mod p behaviour of the function q−1. We use different spectral sequences, in particular the Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the Fp-algebras THH∗(K(Fq);HFp), and we compute V (1)∗ THH(K(Fq)) in the first two cases.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42685978","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 $C^*$-algebra $A$ is said to have the ideal property if each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. $C^*$-algebras with the ideal property are generalization and unification of real rank zero $C^*$-algebras and unital simple $C^*$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see the introduction), consisting of scaled ordered total $K$-group $(underline{K}(A), underline{K}(A)^{+},Sigma A)_{Lambda}$ (used in the real rank zero case), the tracial state space $T(pAp)$ of cutting down algebra $pAp$ as part of Elliott invariant of $pAp$ (for each $[p]inSigma A$) with a certain compatibility, is the complete invariant for certain well behaved class of $C^*$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we will construct two non isomorphic $Amathbb{T}$ algebras $A$ and $B$ with the ideal property such that $Inv^0(A)cong Inv^0(B)$. The invariant to differentiate the two algebras is the Hausdorffifized algebraic $K_1$-groups $U(pAp)/overline{DU(pAp)}$ (for each $[p]inSigma A$) with a certain compatibility condition. It will be proved in [GJL] that, adding this new ingredients, the invariant will become the complete invariant for $AH$ algebras (of no dimension growth) with the ideal property.
{"title":"Hausdorffified algebraic K1-groups and\u0000invariants for C∗-algebras with the ideal property","authors":"G. Gong, Chunlan Jiang, Liangqing Li","doi":"10.2140/akt.2020.5.43","DOIUrl":"https://doi.org/10.2140/akt.2020.5.43","url":null,"abstract":"A $C^*$-algebra $A$ is said to have the ideal property if each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. $C^*$-algebras with the ideal property are generalization and unification of real rank zero $C^*$-algebras and unital simple $C^*$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see the introduction), consisting of scaled ordered total $K$-group $(underline{K}(A), underline{K}(A)^{+},Sigma A)_{Lambda}$ (used in the real rank zero case), the tracial state space $T(pAp)$ of cutting down algebra $pAp$ as part of Elliott invariant of $pAp$ (for each $[p]inSigma A$) with a certain compatibility, is the complete invariant for certain well behaved class of $C^*$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we will construct two non isomorphic $Amathbb{T}$ algebras $A$ and $B$ with the ideal property such that $Inv^0(A)cong Inv^0(B)$. The invariant to differentiate the two algebras is the Hausdorffifized algebraic $K_1$-groups $U(pAp)/overline{DU(pAp)}$ (for each $[p]inSigma A$) with a certain compatibility condition. It will be proved in [GJL] that, adding this new ingredients, the invariant will become the complete invariant for $AH$ algebras (of no dimension growth) with the ideal property.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47749785","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}
If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $nge 1$ is an integer coprime to $|G|$ and such that $ncdot |G|in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra [ K^G(R)/nstackrel{simeq}{longrightarrow} K^G(R/I)/n] is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.
{"title":"Rigidity in equivariant algebraic\u0000K-theory","authors":"N. Naumann, Charanya Ravi","doi":"10.2140/akt.2020.5.141","DOIUrl":"https://doi.org/10.2140/akt.2020.5.141","url":null,"abstract":"If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $nge 1$ is an integer coprime to $|G|$ and such that $ncdot |G|in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra [ K^G(R)/nstackrel{simeq}{longrightarrow} K^G(R/I)/n] is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2020.5.141","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47867577","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 compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.
{"title":"A Dolbeault–Hilbert complex for a variety with\u0000isolated singular points","authors":"J. Lott","doi":"10.2140/akt.2019.4.707","DOIUrl":"https://doi.org/10.2140/akt.2019.4.707","url":null,"abstract":"Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.707","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47700039","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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