We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_infty$-algebra of maps from $S^1$ to the disjoint union of classifying spaces $mathrm{BGL}_d(mathbb F)$, in terms of the $K$-theory of finite field extensions of $mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field $mathbb F$. We also discuss the topological cases of $mathbb F =mathbb C,mathbb R$.
{"title":"Segal K-theory of vector spaces with an automorphism","authors":"Andrea Bianchi, Florian Kranhold","doi":"arxiv-2407.01482","DOIUrl":"https://doi.org/arxiv-2407.01482","url":null,"abstract":"We describe the Segal $K$-theory of the symmetric monoidal category of\u0000finite-dimensional vector spaces over a perfect field $mathbb{F}$ together\u0000with an automorphism, or, equivalently, the group-completion of the\u0000$E_infty$-algebra of maps from $S^1$ to the disjoint union of classifying\u0000spaces $mathrm{BGL}_d(mathbb F)$, in terms of the $K$-theory of finite field\u0000extensions of $mathbb{F}$. A key ingredient for this is a computation of the\u0000Segal $K$-theory of the category of finite-dimensional vector spaces with a\u0000nilpotent endomorphism, which we do over any field $mathbb F$. We also discuss\u0000the topological cases of $mathbb F =mathbb C,mathbb R$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511482","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}
Notes on Commutative Alegbra and Algebraic Geometry covering rings, ideals, modules, presheaves, sheaves, schemes, homological algebra, 'etale cohomology and further topics that are more advanced.
{"title":"Notes on Siegfried Bosch's Algebraic Geometry and Commutative Algebra (dedicated to Grothendieck)","authors":"Eric Schmid","doi":"arxiv-2407.01829","DOIUrl":"https://doi.org/arxiv-2407.01829","url":null,"abstract":"Notes on Commutative Alegbra and Algebraic Geometry covering rings, ideals,\u0000modules, presheaves, sheaves, schemes, homological algebra, 'etale cohomology\u0000and further topics that are more advanced.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511491","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 for nice enough $mathbb{N}$-graded $mathbb{E}_2$-algebras, a diagonal vanishing line in $mathbb{E}_1$-homology of gives rise to slope $1$ homological stability. This is an integral version of a result by Kupers-Miller-Patzt.
{"title":"A criterion for slope 1 homological stability","authors":"Mikala Ørsnes Jansen, Jeremy Miller","doi":"arxiv-2407.01124","DOIUrl":"https://doi.org/arxiv-2407.01124","url":null,"abstract":"We show that for nice enough $mathbb{N}$-graded $mathbb{E}_2$-algebras, a\u0000diagonal vanishing line in $mathbb{E}_1$-homology of gives rise to slope $1$\u0000homological stability. This is an integral version of a result by\u0000Kupers-Miller-Patzt.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511481","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 compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied $RO(G)$-graded coefficient groups of the equivariant Eilenberg--MacLane spectrum $Hunderline{mathbb Z}$. Our comparison between the equivariant $K$-theory spectrum and $Hunderline{mathbb Z}$ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where $G$ has prime order, we provide an explicit presentation of the equivariant $K$-groups.
{"title":"The Galois-equivariant $K$-theory of finite fields","authors":"David Chan, Chase Vogeli","doi":"arxiv-2406.19481","DOIUrl":"https://doi.org/arxiv-2406.19481","url":null,"abstract":"We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite\u0000field with an action by its Galois group $G$. Specifically, we show these\u0000$K$-groups split as the sum of an explicitly computable term and the\u0000well-studied $RO(G)$-graded coefficient groups of the equivariant\u0000Eilenberg--MacLane spectrum $Hunderline{mathbb Z}$. Our comparison between\u0000the equivariant $K$-theory spectrum and $Hunderline{mathbb Z}$ further shows\u0000they share the same Tate spectra and geometric fixed point spectra. In the case\u0000where $G$ has prime order, we provide an explicit presentation of the\u0000equivariant $K$-groups.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511483","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 a transformation formula for the Goresky-Hingston loop coproduct in string topology under homotopy equivalences of manifolds. The formula involves the trace of the Whitehead torsion of the homotopy equivalence. In particular, it implies that the loop coproduct is invariant under simple homotopy equivalences. In a sense, our results determine the Dennis trace of the simple homotopy type of a closed manifold from its framed configuration spaces of $leq 2$ points. We also explain how the loop coproduct arises as a secondary operation in a 2-dimensional TQFT which elucidates a topological origin of the transformation formula.
{"title":"Simple homotopy invariance of the loop coproduct","authors":"Florian Naef, Pavel Safronov","doi":"arxiv-2406.19326","DOIUrl":"https://doi.org/arxiv-2406.19326","url":null,"abstract":"We prove a transformation formula for the Goresky-Hingston loop coproduct in\u0000string topology under homotopy equivalences of manifolds. The formula involves\u0000the trace of the Whitehead torsion of the homotopy equivalence. In particular,\u0000it implies that the loop coproduct is invariant under simple homotopy\u0000equivalences. In a sense, our results determine the Dennis trace of the simple\u0000homotopy type of a closed manifold from its framed configuration spaces of\u0000$leq 2$ points. We also explain how the loop coproduct arises as a secondary\u0000operation in a 2-dimensional TQFT which elucidates a topological origin of the\u0000transformation formula.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511484","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 H be a coFrobenius Hopf algebra over a field k. Let A be a right H-comodule algebra over k. We recall that the category of right H-comodules admits a certain model structure whose homotopy category is equivalent to the stable category of right H-comodules given in Farina's paper. In the first part of this paper, we show that the category of left A-module objects in the category of right H-comodules admits a model structure, which becomes a model subcategory of the category of H*-equivariant A-modules endowed with a model structure given in the author's previous paper if H is finite dimensional with a certain assumption. Note that this category is not a Frobenius category in general. We also construct a functorial cofibrant replacement by proceeding the similar argument as in Qi's paper. In the latter half of this paper, we see that cyclic H-comodules which give Hopf-cyclic (co)homology with coefficients in Hopf H-modules are contructible in the homotopy category of right H-comodules, and we investigate a Hopf-cyclic (co)homology in slightly modified setting by assuming A a right H-comodule k-Hopf algebra with H-colinear bijective antipode in stable category of right H-comodules and give an analogue of the characteristic map. We remark that, as an expansion of an idea of taking trivial comodule k as the coefficients, if we take an A-coinvariant part of M assuming M a Hopf A-module in the category of right H-comodules, we have the degree shift of cyclic modules.
让H是k域上的共弗罗贝纽斯-霍普夫代数，让A是k域上的右H-模代数。我们回顾一下，右H-模范畴包含某种模型结构，它的同调范畴等价于法利纳论文中给出的右H-模的稳定范畴。在本文的第一部分，我们证明了右H-模子范畴中的左A-模子对象范畴包含一个模型结构，如果H是有限维的，并有一定的假设，这个模型结构就会成为作者上一篇论文中给出的禀赋了模型结构的H*-后变A-模子范畴的一个模型子范畴。请注意，这个范畴一般不是弗罗贝尼斯范畴。我们还通过与齐氏论文类似的论证，构造了一个矢量共纤替换。在本文的后半部分，我们发现在右 H-模子的同调范畴中，给出霍普夫 H-模子中系数的霍普夫循环（同）同调的循环 H-模子是可构造的，并且我们通过假设 A 是右 H-模子的 k-Hopf 代数，在右 H-模子的稳定范畴中具有 H-线性双射反节点，在稍作修改的情况下研究了霍普夫循环（同）同调，并给出了特征映射的类似物。我们注意到，作为以琐碎组合数 k 为系数的思想的扩展，如果我们假定 M 是右 H-组合数范畴中的霍普夫 A-组合数，取 M 的 A-币变部分，就会得到循环组合数的度移。
{"title":"A model structure and Hopf-cyclic theory on the category of coequivariant modules over a comodule algebra","authors":"Mariko Ohara","doi":"arxiv-2406.16329","DOIUrl":"https://doi.org/arxiv-2406.16329","url":null,"abstract":"Let H be a coFrobenius Hopf algebra over a field k. Let A be a right\u0000H-comodule algebra over k. We recall that the category of right H-comodules admits a certain model\u0000structure whose homotopy category is equivalent to the stable category of right\u0000H-comodules given in Farina's paper. In the first part of this paper, we show\u0000that the category of left A-module objects in the category of right H-comodules\u0000admits a model structure, which becomes a model subcategory of the category of\u0000H*-equivariant A-modules endowed with a model structure given in the author's\u0000previous paper if H is finite dimensional with a certain assumption. Note that\u0000this category is not a Frobenius category in general. We also construct a\u0000functorial cofibrant replacement by proceeding the similar argument as in Qi's\u0000paper. In the latter half of this paper, we see that cyclic H-comodules which\u0000give Hopf-cyclic (co)homology with coefficients in Hopf H-modules are\u0000contructible in the homotopy category of right H-comodules, and we investigate\u0000a Hopf-cyclic (co)homology in slightly modified setting by assuming A a right\u0000H-comodule k-Hopf algebra with H-colinear bijective antipode in stable category\u0000of right H-comodules and give an analogue of the characteristic map. We remark\u0000that, as an expansion of an idea of taking trivial comodule k as the\u0000coefficients, if we take an A-coinvariant part of M assuming M a Hopf A-module\u0000in the category of right H-comodules, we have the degree shift of cyclic\u0000modules.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511485","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}
Dung Phuong PhanGAATI, UPF, Tuan Anh BuiHCMUS, Alexander D. RahmGAATI, UPF
This article investigates the torsion homology behaviour in towers of Oeljeklaus-Toma (OT) manifolds. This adapts an idea of Silver and Williams from knot theory to OT-manifolds and extends it to higher degree homology groups.In the case of surfaces, i.e. Inoue surfaces of type $S^{0}$, the torsion grows exponentially in both $H_1$ and $H_2$ according to a parameters which already plays a role in Inoue's classical paper. This motivates running example calculations in all homological degrees.
{"title":"Computations regarding the torsion homology of Oeljeklaus-Toma manifolds","authors":"Dung Phuong PhanGAATI, UPF, Tuan Anh BuiHCMUS, Alexander D. RahmGAATI, UPF","doi":"arxiv-2406.14942","DOIUrl":"https://doi.org/arxiv-2406.14942","url":null,"abstract":"This article investigates the torsion homology behaviour in towers of\u0000Oeljeklaus-Toma (OT) manifolds. This adapts an idea of Silver and Williams from\u0000knot theory to OT-manifolds and extends it to higher degree homology groups.In\u0000the case of surfaces, i.e. Inoue surfaces of type $S^{0}$, the torsion grows\u0000exponentially in both $H_1$ and $H_2$ according to a parameters which already\u0000plays a role in Inoue's classical paper. This motivates running example\u0000calculations in all homological degrees.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511486","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 paper by Haynes Miller shows that there is a filtration on the unitary groups that splits in the stable homotopy category, where the stable summands are certain Thom spaces over Grassmannians. We give an algebraic version of this result in the context of Voevodsky's tensor triangulated category of stable motivic complexes $textbf{DM}(k,R)$, where $k$ is a field. Specifically, we show that there are algebraic analogs of the Thom spaces appearing in Miller's splitting that give rise to an analogous splitting of the motive $M(textrm{GL}_n)$ in $textbf{DM}(k,R)$, where $textrm{GL}_n$ is the general linear group scheme over $k$.
{"title":"A Geometric Splitting of the Motive of $textrm{GL}_n$","authors":"W. Sebastian Gant","doi":"arxiv-2406.14687","DOIUrl":"https://doi.org/arxiv-2406.14687","url":null,"abstract":"A paper by Haynes Miller shows that there is a filtration on the unitary\u0000groups that splits in the stable homotopy category, where the stable summands\u0000are certain Thom spaces over Grassmannians. We give an algebraic version of\u0000this result in the context of Voevodsky's tensor triangulated category of\u0000stable motivic complexes $textbf{DM}(k,R)$, where $k$ is a field.\u0000Specifically, we show that there are algebraic analogs of the Thom spaces\u0000appearing in Miller's splitting that give rise to an analogous splitting of the\u0000motive $M(textrm{GL}_n)$ in $textbf{DM}(k,R)$, where $textrm{GL}_n$ is the\u0000general linear group scheme over $k$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511487","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 a decomposition of the Hochschild homology groups of the equivariant dg category $mathscr{C}^G$ associated to a small dg category $mathscr{C}$ with direct sums on which a finite group $G$ acts. When the ground field is $mathbb{C}$ this decomposition is related to a categorical action of $text{Rep}(G)$ on $mathscr{C}^G$ and the resulting action of the representation ring $R_mathbb{C}(G)$ on $HH_bullet(mathscr{C}^G)$.
{"title":"Finite group actions on dg categories and Hochschild homology","authors":"Ville Nordstrom","doi":"arxiv-2406.13866","DOIUrl":"https://doi.org/arxiv-2406.13866","url":null,"abstract":"We prove a decomposition of the Hochschild homology groups of the equivariant\u0000dg category $mathscr{C}^G$ associated to a small dg category $mathscr{C}$\u0000with direct sums on which a finite group $G$ acts. When the ground field is\u0000$mathbb{C}$ this decomposition is related to a categorical action of\u0000$text{Rep}(G)$ on $mathscr{C}^G$ and the resulting action of the\u0000representation ring $R_mathbb{C}(G)$ on $HH_bullet(mathscr{C}^G)$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511488","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 classify the localizing tensor ideals of the derived categories of mixed Tate motives over certain algebraically closed fields. More precisely, we prove that these categories are stratified in the sense of Barthel, Heard and Sanders. A key ingredient in the proof is the development of a new technique for transporting stratification between categories by means of Brown--Adams representability, which may be of independent interest.
{"title":"Stratification of Derived Categories of Tate Motives","authors":"David Rubinstein","doi":"arxiv-2406.13088","DOIUrl":"https://doi.org/arxiv-2406.13088","url":null,"abstract":"We classify the localizing tensor ideals of the derived categories of mixed\u0000Tate motives over certain algebraically closed fields. More precisely, we prove\u0000that these categories are stratified in the sense of Barthel, Heard and\u0000Sanders. A key ingredient in the proof is the development of a new technique\u0000for transporting stratification between categories by means of Brown--Adams\u0000representability, which may be of independent interest.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511498","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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