Pub Date : 2011-06-01DOI: 10.1017/IS011003012JKT149
J. Jardine
This paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n -type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.
{"title":"Model structures for pro-simplicial presheaves","authors":"J. Jardine","doi":"10.1017/IS011003012JKT149","DOIUrl":"https://doi.org/10.1017/IS011003012JKT149","url":null,"abstract":"This paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n -type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"7 1","pages":"499-525"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011003012JKT149","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665665","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}
Pub Date : 2011-06-01DOI: 10.1017/IS011003012JKT150
P. Hu, I. Kríz, K. Ormsby
We prove convergence of the motivic Adams spectral sequence to completions at p and η under suitable conditions. We also discuss further conditions under which η can be removed from the statement.
{"title":"Convergence of the Motivic Adams Spectral Sequence","authors":"P. Hu, I. Kríz, K. Ormsby","doi":"10.1017/IS011003012JKT150","DOIUrl":"https://doi.org/10.1017/IS011003012JKT150","url":null,"abstract":"We prove convergence of the motivic Adams spectral sequence to completions at p and η under suitable conditions. We also discuss further conditions under which η can be removed from the statement.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"7 1","pages":"573-596"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011003012JKT150","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665699","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}
Pub Date : 2011-06-01DOI: 10.1017/IS011003002JKT147
B. Kahn
{"title":"Relatively unramified elements in cycle modules","authors":"B. Kahn","doi":"10.1017/IS011003002JKT147","DOIUrl":"https://doi.org/10.1017/IS011003002JKT147","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"22 1","pages":"409-427"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89432347","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}
Pub Date : 2011-06-01DOI: 10.1017/IS011004009JKT152
C. Weibel
We show that the groups Kn(RG; Z/m) are Bott-periodic for n≥1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∈ R. For any positive integer m there is a Bott element bK ∈ Kp(Z[1/m]; Z/m), where the period p = p(m) is: 2(l−1)l if m = l and l is odd; max{8, 2} if m = 2; and ∏ p(mi) if m = ∏ mi is the factorization of m into primary factors. In this appendix we consider a finite group G of order prime to m, and consider the Bott periodicity map x 7→ x · bK from Kn(R[G]; Z/m) to Kn+p(R[G]; Z/m) for rings of integers R in local and global fields. Theorem 0.1. Assume that m is relatively prime to |G|, and that R is a ring of S-integers in a number field with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 1. Theorem 0.2. Assume that m is relatively prime to |G|, and that R is the ring of integers in a local field F with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 0, Theorems 0.1 and 0.2 are used in [BKO] to show that KQn(R[G]; Z/m) also satisfies Bott periodicity. Since this is immediate if m is odd, when the non-Witt part of KQn(A; Z/m) is a summand of Kn(A; Z/m), this result is primarily interesting for m even and G a (solvable) group of odd order. Remark. The proofs show that we may replace R[G] by any order in F [G]. The oldest result of this kind is due to Browder, who proved in [1, 2.6] that the Bott periodicity map bK is an isomorphism for finite fields and n ≥ 0 (when m is prime, which implies periodicity for all m). Almost as old is the following folklore result, which includes finite group rings. Lemma 0.3. If B is a finite ring and 1/m ∈ B, the the Bott periodicity map Kn(B, Z/m) → Kn+p(B, Z/m) is an isomorphism for all n ≥ 0. Proof. If m is the nilradical of B, then Bred = B/m is semisimple. As such it is a product of matrix rings over finite fields. Now K∗(B; Z/m) ∼= K∗(Bred; Z/m) by [9, 1.4]. By Morita invariance, we are reduced to the Browder’s theorem that the Bott periodicity map is an isomorphism for finite fields. Remark 0.4. The finite groups Kn(F2[C2]; Z/8) were computed by Hesselholt and Madsen in [4]; they are not Bott periodic as their order goes to infinity with n. The next step is to consider the semisimple group ring F [G] when F is a number field. We will use the fact that multiplication by bK is an isomorphism on
{"title":"Bott Periodicity for group rings An Appendix to “Periodicity of Hermitian K-groups”","authors":"C. Weibel","doi":"10.1017/IS011004009JKT152","DOIUrl":"https://doi.org/10.1017/IS011004009JKT152","url":null,"abstract":"We show that the groups Kn(RG; Z/m) are Bott-periodic for n≥1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∈ R. For any positive integer m there is a Bott element bK ∈ Kp(Z[1/m]; Z/m), where the period p = p(m) is: 2(l−1)l if m = l and l is odd; max{8, 2} if m = 2; and ∏ p(mi) if m = ∏ mi is the factorization of m into primary factors. In this appendix we consider a finite group G of order prime to m, and consider the Bott periodicity map x 7→ x · bK from Kn(R[G]; Z/m) to Kn+p(R[G]; Z/m) for rings of integers R in local and global fields. Theorem 0.1. Assume that m is relatively prime to |G|, and that R is a ring of S-integers in a number field with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 1. Theorem 0.2. Assume that m is relatively prime to |G|, and that R is the ring of integers in a local field F with 1/m ∈ R. Then the Bott periodicity maps bK : Kn(R[G]; Z/m) → Kn+p(R[G]; Z/m) are isomorphisms for all n ≥ 0, Theorems 0.1 and 0.2 are used in [BKO] to show that KQn(R[G]; Z/m) also satisfies Bott periodicity. Since this is immediate if m is odd, when the non-Witt part of KQn(A; Z/m) is a summand of Kn(A; Z/m), this result is primarily interesting for m even and G a (solvable) group of odd order. Remark. The proofs show that we may replace R[G] by any order in F [G]. The oldest result of this kind is due to Browder, who proved in [1, 2.6] that the Bott periodicity map bK is an isomorphism for finite fields and n ≥ 0 (when m is prime, which implies periodicity for all m). Almost as old is the following folklore result, which includes finite group rings. Lemma 0.3. If B is a finite ring and 1/m ∈ B, the the Bott periodicity map Kn(B, Z/m) → Kn+p(B, Z/m) is an isomorphism for all n ≥ 0. Proof. If m is the nilradical of B, then Bred = B/m is semisimple. As such it is a product of matrix rings over finite fields. Now K∗(B; Z/m) ∼= K∗(Bred; Z/m) by [9, 1.4]. By Morita invariance, we are reduced to the Browder’s theorem that the Bott periodicity map is an isomorphism for finite fields. Remark 0.4. The finite groups Kn(F2[C2]; Z/8) were computed by Hesselholt and Madsen in [4]; they are not Bott periodic as their order goes to infinity with n. The next step is to consider the semisimple group ring F [G] when F is a number field. We will use the fact that multiplication by bK is an isomorphism on","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"185 1","pages":"495-498"},"PeriodicalIF":0.0,"publicationDate":"2011-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011004009JKT152","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665763","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}
Pub Date : 2011-04-21DOI: 10.1017/IS013001030JKT212
Rob de Jeu, James D. Lewis
Let U /ℂ be a smooth quasi-projective variety of dimension d , CH r ( U,m ) Bloch's higher Chow group, and cl r,m : CH r ( U,m ) ⊗ ℚ → hom MHS (ℚ(0), H 2 r−m ( U , ℚ( r ))) the cycle class map. Beilinson once conjectured cl r,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of cl r,m in more detail (as well as at the “generic point” of U ) in terms of kernels of Abel-Jacobi mappings. When r = m , we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of cl m,m at the generic point is the same for integral or rational coefficients.
{"title":"Beilinson's Hodge conjecture for smooth varieties","authors":"Rob de Jeu, James D. Lewis","doi":"10.1017/IS013001030JKT212","DOIUrl":"https://doi.org/10.1017/IS013001030JKT212","url":null,"abstract":"Let U /ℂ be a smooth quasi-projective variety of dimension d , CH r ( U,m ) Bloch's higher Chow group, and cl r,m : CH r ( U,m ) ⊗ ℚ → hom MHS (ℚ(0), H 2 r−m ( U , ℚ( r ))) the cycle class map. Beilinson once conjectured cl r,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of cl r,m in more detail (as well as at the “generic point” of U ) in terms of kernels of Abel-Jacobi mappings. When r = m , we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of cl m,m at the generic point is the same for integral or rational coefficients.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"243-282"},"PeriodicalIF":0.0,"publicationDate":"2011-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013001030JKT212","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667221","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}
Pub Date : 2011-04-08DOI: 10.1017/IS012003003JKT184
Aurélien Djament
On generalise les resultats de l'auteur et C. Vespa (Ann. Sci. ENS 2010) a l'homologie stable des groupes unitaires sur un anneau quelconque a coefficients tordus par un foncteur polynomial, dont on montre qu'elle peut s'exprimer a partir de l'homologie a coefficients constants et de groupes d'homologie des foncteurs qu'on peut calculer explicitement dans les cas favorables.
{"title":"Sur l'homologie des groupes unitaires à coefficients polynomiaux","authors":"Aurélien Djament","doi":"10.1017/IS012003003JKT184","DOIUrl":"https://doi.org/10.1017/IS012003003JKT184","url":null,"abstract":"On generalise les resultats de l'auteur et C. Vespa (Ann. Sci. ENS 2010) a l'homologie stable des groupes unitaires sur un anneau quelconque a coefficients tordus par un foncteur polynomial, dont on montre qu'elle peut s'exprimer a partir de l'homologie a coefficients constants et de groupes d'homologie des foncteurs qu'on peut calculer explicitement dans les cas favorables.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"10 1","pages":"87-139"},"PeriodicalIF":0.0,"publicationDate":"2011-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS012003003JKT184","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666783","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}
Pub Date : 2011-04-01DOI: 10.1017/IS010002021JKT109
P. Chattopadhyay, R. A. Rao
It is shown that the set of orbits of the action of the elementary symplectic group on all unimodular rows over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary general linear group. This result is used to improve injective stability for K1 of the symplectic group over non-singular affine algebras.
{"title":"Elementary symplectic orbits and improved K 1 -stability","authors":"P. Chattopadhyay, R. A. Rao","doi":"10.1017/IS010002021JKT109","DOIUrl":"https://doi.org/10.1017/IS010002021JKT109","url":null,"abstract":"It is shown that the set of orbits of the action of the elementary symplectic group on all unimodular rows over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary general linear group. This result is used to improve injective stability for K1 of the symplectic group over non-singular affine algebras.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"35 1","pages":"389-403"},"PeriodicalIF":0.0,"publicationDate":"2011-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74242247","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}
Pub Date : 2011-03-31DOI: 10.1017/IS012001012JKT180
Martin Grensing
Combining Kasparov's theorem of Voiculesu and Cuntz's description of $KK$-theory in terms of quasihomomorphisms, we give a simple construction of the Kasparov product. This will be used in a more general context of locally convex algebras in order to treat products of certain universal cycles.
{"title":"A note on Kasparov products","authors":"Martin Grensing","doi":"10.1017/IS012001012JKT180","DOIUrl":"https://doi.org/10.1017/IS012001012JKT180","url":null,"abstract":"Combining Kasparov's theorem of Voiculesu and Cuntz's description of $KK$-theory in terms of quasihomomorphisms, we give a simple construction of the Kasparov product. This will be used in a more general context of locally convex algebras in order to treat products of certain universal cycles.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"23 1","pages":"233-240"},"PeriodicalIF":0.0,"publicationDate":"2011-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90447694","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}
Pub Date : 2011-03-30DOI: 10.1017/IS011004009JKT155
Gonçalo Tabuada
We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.
{"title":"The fundamental theorem via derived Morita invariance, localization, and A(1)-homotopy invariance","authors":"Gonçalo Tabuada","doi":"10.1017/IS011004009JKT155","DOIUrl":"https://doi.org/10.1017/IS011004009JKT155","url":null,"abstract":"We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"407-420"},"PeriodicalIF":0.0,"publicationDate":"2011-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011004009JKT155","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665769","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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