A novel, highly general construction of integration, function calculus, and exterior calculus was achieved in this paper, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann integral, geometric product integral, and Lebesgue integral were all shown as special cases. Notions similar to chain complexes were developed to allow this general form of integration to define notions of exterior derivative for differential forms, and of derivatives of functions too. Fundamental realizations, some quite surprising, were achieved on the deepest natures of key concepts of analysis including integration, orientation, differentiation, and more. It's clear that further applications such as calculus on fractals, stochastic calculus, discrete calculus, and many other novel forms of analysis can all be achieved as special cases of this theory.
{"title":"A Topological Reimagining of Integration and Exterior Calculus","authors":"Petal B. Mokryn","doi":"arxiv-2407.11689","DOIUrl":"https://doi.org/arxiv-2407.11689","url":null,"abstract":"A novel, highly general construction of integration, function calculus, and\u0000exterior calculus was achieved in this paper, allowing for integration of\u0000unital magma valued functions against (compactified) unital magma valued\u0000measures over arbitrary topological spaces. The Riemann integral, geometric\u0000product integral, and Lebesgue integral were all shown as special cases.\u0000Notions similar to chain complexes were developed to allow this general form of\u0000integration to define notions of exterior derivative for differential forms,\u0000and of derivatives of functions too. Fundamental realizations, some quite\u0000surprising, were achieved on the deepest natures of key concepts of analysis\u0000including integration, orientation, differentiation, and more. It's clear that\u0000further applications such as calculus on fractals, stochastic calculus,\u0000discrete calculus, and many other novel forms of analysis can all be achieved\u0000as special cases of this theory.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719986","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}
In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence, we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also obtain a comparison of the absolute cohomology with the equivariant higher Chow groups in certain special cases.
在本文中，我们考虑了光滑代数栈的 K 理论，建立了兰姆达运算和伽马运算，并证明了这类栈的高 K 理论总是前兰姆达环，而且如果每个相干剪子都是向量束的商，那么高 K 理论就是兰姆达环。因此，我们能够为满足这一假设的光滑代数堆栈定义亚当斯迭代和绝对同调。我们还得到了在某些特殊情况下绝对同调与后向高周群的比较。
{"title":"Lambda-ring structures on the K-theory of algebraic stacks","authors":"Roy Joshua, Pablo Pelaez","doi":"arxiv-2407.10394","DOIUrl":"https://doi.org/arxiv-2407.10394","url":null,"abstract":"In this paper we consider the K-theory of smooth algebraic stacks, establish\u0000lambda and gamma operations, and show that the higher K-theory of such stacks\u0000is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is\u0000the quotient of a vector bundle. As a consequence, we are able to define Adams\u0000operations and absolute cohomology for smooth algebraic stacks satisfying this\u0000hypothesis. We also obtain a comparison of the absolute cohomology with the\u0000equivariant higher Chow groups in certain special cases.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719988","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 certain Ext and Tor groups in the category of all functors from an Z/p-linear additive category A to vector spaces in terms of Ext and Tor computed in the full subcategory of additive functors from A to vector spaces. We thus obtain group homology computations for general lineargroups.
我们在从 Z/p 线性加法类别 A 到向量空间的所有函数类别中，以从 A 到向量空间的加法函数全子类中计算的 Ext 和 Tor 来计算某些 Ext 和 Tor 群。
{"title":"The homology of additive functors in prime characteristic","authors":"Aurélien DjamentLAGA, Antoine TouzéLPP","doi":"arxiv-2407.10522","DOIUrl":"https://doi.org/arxiv-2407.10522","url":null,"abstract":"We compute certain Ext and Tor groups in the category of all functors from an\u0000Z/p-linear additive category A to vector spaces in terms of Ext and Tor\u0000computed in the full subcategory of additive functors from A to vector spaces.\u0000We thus obtain group homology computations for general lineargroups.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719987","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 complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category.
{"title":"The Hodge structure on the singularity category of a complex hypersurface","authors":"Michael K. Brown, Mark E. Walker","doi":"arxiv-2407.09988","DOIUrl":"https://doi.org/arxiv-2407.09988","url":null,"abstract":"Given a complex affine hypersurface with isolated singularity determined by a\u0000homogeneous polynomial, we identify the noncommutative Hodge structure on the\u0000periodic cyclic homology of its singularity category with the classical Hodge\u0000structure on the primitive cohomology of the associated projective\u0000hypersurface. As a consequence, we show that the Hodge conjecture for the\u0000projective hypersurface is equivalent to a dg-categorical analogue of the Hodge\u0000conjecture for the singularity category.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719989","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}
To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we identify connective analytic K-theory in the unstable homotopy category with both $mathbb{Z}timesmathrm{BGL}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra.
{"title":"Towards $mathbb{A}^1$-homotopy theory of rigid analytic spaces","authors":"Christian Dahlhausen, Can Yaylali","doi":"arxiv-2407.09606","DOIUrl":"https://doi.org/arxiv-2407.09606","url":null,"abstract":"To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an\u0000$mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in\u0000any presentable category. We show some functorial properties of this assignment\u0000as a functor on the category of rigid analytic spaces. Moreover, we show that\u0000there exists a full six functor formalism for the precomposition with the\u0000analytification functor by evoking Ayoub's thesis. As an application, we\u0000identify connective analytic K-theory in the unstable homotopy category with\u0000both $mathbb{Z}timesmathrm{BGL}$ and the analytification of connective\u0000algebraic K-theory. As a consequence, we get a representability statement for\u0000coefficients in light condensed spectra.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719990","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}
Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens
In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $infty$-categories of connective genuine $G$-spectra and space-valued Mackey functors, first proven by Guillou-May, and use this to give a description of connective normed equivariant ring spectra as space-valued Tambara functors. In more detail, we first introduce and study a general notion of homotopy-coherent normed (semi)rings, and identify these with product-preserving functors out of a corresponding $infty$-category of bispans. In the equivariant setting, this identifies space-valued Tambara functors with normed algebras with respect to a certain normed monoidal structure on grouplike $G$-commutative monoids in spaces. We then show that the latter is canonically equivalent to the normed monoidal structure on connective $G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we produce normed ring structures on equivariant algebraic K-theory spectra.
在本文中，我们扩展了等变无限环空间理论，以考虑乘法规范：对于每一个有限群 $G$，我们都构建了连接真正 $G$ 谱的 $/infty$ 类别与空间值麦基函子之间比较的乘法细化（这是吉鲁-梅首次证明的），并以此给出了连接规范等价环谱作为空间值坦巴拉函子的描述。更详细地说，我们首先引入并研究了同位相干规范（半）环的一般概念，并将其与相应的$infty$-category of bispans中的保积函子相鉴别。在等价设定中，这将空间值的坦巴拉函数与关于空间中的类群$G$-交换单元上的某个规范单元结构的规范代数相识别。然后我们证明，这一结构与希尔-霍普金斯-拉文尔规范给出的连接$G$谱上的规范单元结构具有典型等价性。结合我们与埃尔曼托-豪森（Elmanto-Haugseng）和巴威克-格拉斯曼-马修-尼古拉斯（Barwick-Glasman-Mathew-Nikolaus）的比较结果，我们得出了等变代数 K 理论谱上的规范环结构。
{"title":"Normed equivariant ring spectra and higher Tambara functors","authors":"Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens","doi":"arxiv-2407.08399","DOIUrl":"https://doi.org/arxiv-2407.08399","url":null,"abstract":"In this paper we extend equivariant infinite loop space theory to take into\u0000account multiplicative norms: For every finite group $G$, we construct a\u0000multiplicative refinement of the comparison between the $infty$-categories of\u0000connective genuine $G$-spectra and space-valued Mackey functors, first proven\u0000by Guillou-May, and use this to give a description of connective normed\u0000equivariant ring spectra as space-valued Tambara functors. In more detail, we first introduce and study a general notion of\u0000homotopy-coherent normed (semi)rings, and identify these with\u0000product-preserving functors out of a corresponding $infty$-category of\u0000bispans. In the equivariant setting, this identifies space-valued Tambara\u0000functors with normed algebras with respect to a certain normed monoidal\u0000structure on grouplike $G$-commutative monoids in spaces. We then show that the\u0000latter is canonically equivalent to the normed monoidal structure on connective\u0000$G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison\u0000with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we\u0000produce normed ring structures on equivariant algebraic K-theory spectra.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141608714","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 labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically into a boundedly acyclic group of type $F_n$ that has no proper finite index subgroups. This improves a result of Bridson cite{Br98} and a theorem of Fournier-Facio--L"oh--Moraschini cite[Theorem 2]{FFCM21}. Second, every group of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of type $F_n$. Third, using Belk--Zaremsky's construction of twisted Brin--Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group.
{"title":"Embedding groups into boundedly acyclic groups","authors":"Fan Wu, Xiaolei Wu, Mengfei Zhao, Zixiang Zhou","doi":"arxiv-2407.07703","DOIUrl":"https://doi.org/arxiv-2407.07703","url":null,"abstract":"We show that the labeled Thompson groups and the twisted Brin--Thompson\u0000groups are boundedly acyclic. This allows us to prove several new embedding\u0000results for groups. First, every group of type $F_n$ embeds quasi-isometrically\u0000into a boundedly acyclic group of type $F_n$ that has no proper finite index\u0000subgroups. This improves a result of Bridson cite{Br98} and a theorem of\u0000Fournier-Facio--L\"oh--Moraschini cite[Theorem 2]{FFCM21}. Second, every group\u0000of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of\u0000type $F_n$. Third, using Belk--Zaremsky's construction of twisted\u0000Brin--Thompson groups, we show that every finitely generated group embeds\u0000quasi-isometrically into a finitely generated boundedly acyclic simple group.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588080","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 $C$ be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of $C$ for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.
{"title":"Bounding generators for the kernel and cokernel of the tame symbol for curves","authors":"Rob de Jeu","doi":"arxiv-2407.07974","DOIUrl":"https://doi.org/arxiv-2407.07974","url":null,"abstract":"Let $C$ be a regular, irreducible curve that is projective over a field. We\u0000obtain bounds in terms of the arithmetic genus of $C$ for the generators that\u0000are required for the cokernel of the tame symbol, as well as, under a\u0000simplifying assumption, its kernel. We briefly discuss a potential application\u0000to Chow groups.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141608711","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}
In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the Chow line of the motivic cohomology of $X$ the pushforward of a fundamental class along a projective derived-lci morphism? If $X$ is a smooth variety over a field of characteristic $p geq 0$, then a positive answer to this question follows up to $p$-torsion from resolution of singularities by alterations. However, if $X$ is singular, then this is no longer necessarily so: we give examples of motivic cohomology classes of a singular scheme $X$ that are not $p$-torsion and are not expressible as such pushforwards. A consequence of our result is that the Chow ring of a singular variety cannot be expressed as a quotient of its algebraic cobordism ring, as suggested by the first-named-author in his thesis.
{"title":"Motivic Steenrod problem away from the characteristic","authors":"Toni Annala, Tobias Shin","doi":"arxiv-2407.07194","DOIUrl":"https://doi.org/arxiv-2407.07194","url":null,"abstract":"In topology, the Steenrod problem asks whether every singular homology class\u0000is the pushforward of the fundamental class of a closed oriented manifold.\u0000Here, we introduce an analogous question in algebraic geometry: is every\u0000element on the Chow line of the motivic cohomology of $X$ the pushforward of a\u0000fundamental class along a projective derived-lci morphism? If $X$ is a smooth\u0000variety over a field of characteristic $p geq 0$, then a positive answer to\u0000this question follows up to $p$-torsion from resolution of singularities by\u0000alterations. However, if $X$ is singular, then this is no longer necessarily\u0000so: we give examples of motivic cohomology classes of a singular scheme $X$\u0000that are not $p$-torsion and are not expressible as such pushforwards. A\u0000consequence of our result is that the Chow ring of a singular variety cannot be\u0000expressed as a quotient of its algebraic cobordism ring, as suggested by the\u0000first-named-author in his thesis.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"231 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588081","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 `pro-cdh descent' result for suitably connective localizing invariants and the cotangent complex on arbitrary qcqs derived schemes. As an application, we deduce a generalised Weibel vanishing for negative $K$-groups of non-Noetherian schemes.
{"title":"On pro-cdh descent on derived schemes","authors":"Shane Kelly, Shuji Saito, Georg Tamme","doi":"arxiv-2407.04378","DOIUrl":"https://doi.org/arxiv-2407.04378","url":null,"abstract":"We prove a `pro-cdh descent' result for suitably connective localizing\u0000invariants and the cotangent complex on arbitrary qcqs derived schemes. As an\u0000application, we deduce a generalised Weibel vanishing for negative $K$-groups\u0000of non-Noetherian schemes.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"147 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573544","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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