. For an A 1 -connected pointed simplicial sheaf X over a perfect field k , we prove that the Hurewicz map π A 1 1 ( X ) → H A 1 1 ( X ) is surjective. We also observe that the Hurewicz map for P 1 k is the abelianisation map. In the course of proving this result, we also show that for any morphism φ of strongly A 1 -invariant sheaves of groups, the image and kernel of φ are also strongly A 1 -invariant.
{"title":"The Hurewicz map in motivic homotopy theory","authors":"Utsav Choudhury, A. Hogadi","doi":"10.2140/akt.2022.7.179","DOIUrl":"https://doi.org/10.2140/akt.2022.7.179","url":null,"abstract":". For an A 1 -connected pointed simplicial sheaf X over a perfect field k , we prove that the Hurewicz map π A 1 1 ( X ) → H A 1 1 ( X ) is surjective. We also observe that the Hurewicz map for P 1 k is the abelianisation map. In the course of proving this result, we also show that for any morphism φ of strongly A 1 -invariant sheaves of groups, the image and kernel of φ are also strongly A 1 -invariant.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47398237","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 hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type $A$, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme $G$, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components $pi_0(G)$ in $G$.
{"title":"Hypersurface support and prime ideal spectra for stable categories","authors":"C. Negron, J. Pevtsova","doi":"10.2140/akt.2023.8.25","DOIUrl":"https://doi.org/10.2140/akt.2023.8.25","url":null,"abstract":"We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type $A$, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme $G$, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components $pi_0(G)$ in $G$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47822728","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}
{"title":"On the norm and multiplication principles for norm varieties","authors":"Shira Gilat, Eliyahu Matzri","doi":"10.2140/akt.2020.5.709","DOIUrl":"https://doi.org/10.2140/akt.2020.5.709","url":null,"abstract":"","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49482522","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 motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the motivic cohomology agrees with the Edidin-Graham-Totaro Chow groups with integer coefficients.
{"title":"The Nisnevich motive of an algebraic stack","authors":"Utsav Choudhury, Neeraj Deshmukh, A. Hogadi","doi":"10.2140/akt.2023.8.245","DOIUrl":"https://doi.org/10.2140/akt.2023.8.245","url":null,"abstract":"We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the motivic cohomology agrees with the Edidin-Graham-Totaro Chow groups with integer coefficients.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42540010","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 $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ etale motivic cohomology of the classifying stack $mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.
{"title":"Motivic cohomology and infinitesimal group schemes","authors":"Eric Primozic","doi":"10.2140/akt.2022.7.441","DOIUrl":"https://doi.org/10.2140/akt.2022.7.441","url":null,"abstract":"For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. \u0000For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ etale motivic cohomology of the classifying stack $mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42611506","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 a $C^{*}$-category with a strict $G$-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe categories of objects in $C^{*}$-categories which are controlled over bornological coarse spaces, and then apply a homological functor. These equivariant coarse homology theories are then employed to verify that certain functors on the orbit category are CP-functors. This fact has consequences for the injectivity of assembly maps.
{"title":"Topological equivariant coarse\u0000K-homology","authors":"U. Bunke, A. Engel","doi":"10.2140/akt.2023.8.141","DOIUrl":"https://doi.org/10.2140/akt.2023.8.141","url":null,"abstract":"For a $C^{*}$-category with a strict $G$-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe categories of objects in $C^{*}$-categories which are controlled over bornological coarse spaces, and then apply a homological functor. These equivariant coarse homology theories are then employed to verify that certain functors on the orbit category are CP-functors. This fact has consequences for the injectivity of assembly maps.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41775852","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}
B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki
We construct and study a triangulated category of motives with modulus $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $mathbf{DM}_{mathrm{gm}}^{mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $mathbf{DM}_{mathrm{gm}}^{mathrm{eff}}$ is constructed out of smooth $k$-varieties, $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$. In some cases the $mathrm{Hom}$ group in $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.
{"title":"Motives with modulus, III: The categories of motives","authors":"B. Kahn, Hiroyasu Miyazaki, S. Saito, Takao Yamazaki","doi":"10.2140/akt.2022.7.119","DOIUrl":"https://doi.org/10.2140/akt.2022.7.119","url":null,"abstract":"We construct and study a triangulated category of motives with modulus $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $mathbf{DM}_{mathrm{gm}}^{mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $mathbf{DM}_{mathrm{gm}}^{mathrm{eff}}$ is constructed out of smooth $k$-varieties, $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$. In some cases the $mathrm{Hom}$ group in $mathbf{MDM}_{mathrm{gm}}^{mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44921210","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 $Q$ denote MacLane's $Q$-construction, and $otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)simeq mathbb Zotimes R$ in the category of $A_infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schw"anzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)simeq mathbb Zotimes -$ as symmetric monoidal functors, from which the conjecture follows immediately. From this result, we obtain a new proof of the equivalence $mathrm{HML}(R,M)simeq mathrm{THH}(R,M)$ originally proved by Pirashvili and Waldaushen in "MacLane homology and topological Hochschild homology" (a different paper from the one cited above). This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $mathrm{HML}(R)simeq mathrm{THH}(R)$ as $E_infty$ ring spectra, when $R$ is a commutative ring.
{"title":"A multiplicative comparison of Mac Lane\u0000homology and topological Hochschild homology","authors":"Geoffroy Horel, Maxime Ramzi","doi":"10.2140/akt.2021.6.571","DOIUrl":"https://doi.org/10.2140/akt.2021.6.571","url":null,"abstract":"Let $Q$ denote MacLane's $Q$-construction, and $otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)simeq mathbb Zotimes R$ in the category of $A_infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schw\"anzl, Vogt and Waldhausen in \"MacLane homology and topological Hochschild homology\". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)simeq mathbb Zotimes -$ as symmetric monoidal functors, from which the conjecture follows immediately. From this result, we obtain a new proof of the equivalence $mathrm{HML}(R,M)simeq mathrm{THH}(R,M)$ originally proved by Pirashvili and Waldaushen in \"MacLane homology and topological Hochschild homology\" (a different paper from the one cited above). This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $mathrm{HML}(R)simeq mathrm{THH}(R)$ as $E_infty$ ring spectra, when $R$ is a commutative ring.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48675275","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 G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.
{"title":"Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary","authors":"Paolo Piazza, H. Posthuma","doi":"10.2140/akt.2021.6.713","DOIUrl":"https://doi.org/10.2140/akt.2021.6.713","url":null,"abstract":"Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42900969","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 the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first filter is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 filter, and when k is even, whose next graded piece is controlled by the Griffiths group of codimension k/2+1 cycles. The first filter is a generalization of the Artin-Mumford invariant (k=2) and the Colliot-Thelene-Voisin invariant (k=3). We also give a homological analogue.
{"title":"Unramified cohomology, integral coniveau filtration and Griffiths groups","authors":"Shouhei Ma","doi":"10.2140/akt.2022.7.223","DOIUrl":"https://doi.org/10.2140/akt.2022.7.223","url":null,"abstract":"We prove that the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first filter is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 filter, and when k is even, whose next graded piece is controlled by the Griffiths group of codimension k/2+1 cycles. The first filter is a generalization of the Artin-Mumford invariant (k=2) and the Colliot-Thelene-Voisin invariant (k=3). We also give a homological analogue.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48323622","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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