Let $Q$ be a quiver and $R$ an associative ring. A representation by $R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic injective resolution in the category of representations. It is shown that such representations possess many nice properties. We characterize strongly fp-injective representations under some mild assumptions, which is closely related to strongly fp-injective $R$-modules. Subsequently, we use such representations to define relative Gorenstein injective representations, called Gorenstein strongly fp-injective representations, and give an explicit characterization of the Gorenstein strongly fp-injective representations of right rooted quivers. As an application, a model structure in the category of representations is given.
{"title":"Homological theory of representations having pure acyclic injective resolutions","authors":"Gang Yang, Qihui Li, Junpeng Wang","doi":"arxiv-2407.21660","DOIUrl":"https://doi.org/arxiv-2407.21660","url":null,"abstract":"Let $Q$ be a quiver and $R$ an associative ring. A representation by\u0000$R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic\u0000injective resolution in the category of representations. It is shown that such\u0000representations possess many nice properties. We characterize strongly\u0000fp-injective representations under some mild assumptions, which is closely\u0000related to strongly fp-injective $R$-modules. Subsequently, we use such\u0000representations to define relative Gorenstein injective representations, called\u0000Gorenstein strongly fp-injective representations, and give an explicit\u0000characterization of the Gorenstein strongly fp-injective representations of\u0000right rooted quivers. As an application, a model structure in the category of\u0000representations is given.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868478","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 introduce the notion of Grothendieck heaps for unpointed Waldhausen categories and unpointed stable $infty$-categories. This allows an extension of the studies of $mathrm{K}_0$ to the homotopy category of unpointed topological spaces.
{"title":"Algebraic $K_0$ for unpointed homotopy Categories","authors":"Felix Küng","doi":"arxiv-2407.20911","DOIUrl":"https://doi.org/arxiv-2407.20911","url":null,"abstract":"We introduce the notion of Grothendieck heaps for unpointed Waldhausen\u0000categories and unpointed stable $infty$-categories. This allows an extension\u0000of the studies of $mathrm{K}_0$ to the homotopy category of unpointed\u0000topological spaces.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868479","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 mathematically show an equality between the index of a Dirac operator on a flat continuum torus and the $eta$ invariant of the Wilson Dirac operator with a negative mass when the lattice spacing is sufficiently small. Unlike the standard approach, our formulation using $K$-theory does not require the Ginsparg-Wilson relation or the modified chiral symmetry on the lattice. We prove that a one-parameter family of continuum massive Dirac operators and the corresponding Wilson Dirac operators belong to the same equivalence class of the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated by the spectral flow or equivalently by the $eta$ invariant at finite masses, are proved to be equal.
{"title":"The index of lattice Dirac operators and $K$-theory","authors":"Shoto Aoki, Hidenori Fukaya, Mikio Furuta, Shinichiroh Matsuo, Tetsuya Onogi, Satoshi Yamaguchi","doi":"arxiv-2407.17708","DOIUrl":"https://doi.org/arxiv-2407.17708","url":null,"abstract":"We mathematically show an equality between the index of a Dirac operator on a\u0000flat continuum torus and the $eta$ invariant of the Wilson Dirac operator with\u0000a negative mass when the lattice spacing is sufficiently small. Unlike the\u0000standard approach, our formulation using $K$-theory does not require the\u0000Ginsparg-Wilson relation or the modified chiral symmetry on the lattice. We\u0000prove that a one-parameter family of continuum massive Dirac operators and the\u0000corresponding Wilson Dirac operators belong to the same equivalence class of\u0000the $K^1$ group at a finite lattice spacing. Their indices, which are evaluated\u0000by the spectral flow or equivalently by the $eta$ invariant at finite masses,\u0000are proved to be equal.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775489","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}
This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results regarding the homotopy theory of operadic algebras.
{"title":"Enveloping operads and applications","authors":"Victor Carmona","doi":"arxiv-2407.18190","DOIUrl":"https://doi.org/arxiv-2407.18190","url":null,"abstract":"This work addresses the homotopical analysis of enveloping operads in a\u0000general cofibrantly generated symmetric monoidal model category. We show the\u0000potential of this analysis by obtaining, in a uniform way, several central\u0000results regarding the homotopy theory of operadic algebras.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775493","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}
Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.
{"title":"On split Steinberg modules and Steinberg modules","authors":"Daniel Armeanu, Jeremy Miller","doi":"arxiv-2407.18208","DOIUrl":"https://doi.org/arxiv-2407.18208","url":null,"abstract":"Answering a question of Randal-Williams, we show the natural maps from split\u0000Steinberg modules of a Dedekind domain to the associated Steinberg modules are\u0000surjective.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775492","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 article we study the first, the second and the third homology groups of the elementary group $textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1in (A^{times})^2$ or $|A^{times}/(A^{times})^2|leq 4$.
{"title":"The low dimensional homology groups of the elementary group of rank two","authors":"Behrooz Mirzaii, Elvis Torres Pérez","doi":"arxiv-2407.17632","DOIUrl":"https://doi.org/arxiv-2407.17632","url":null,"abstract":"In this article we study the first, the second and the third homology groups\u0000of the elementary group $textrm{E}_2(A)$, where $A$ is a commutative ring. In\u0000particular, we prove a refined Bloch-Wigner type exact sequence over a\u0000semilocal ring (with some mild restriction on its residue fields) such that\u0000$-1in (A^{times})^2$ or $|A^{times}/(A^{times})^2|leq 4$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775490","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 digraph $G$ without multisquares and a field $mathbb{F}$, we construct a basis of the vector space of path $n$-chains $Omega_n(G;mathbb{F})$ for $ngeq 0$, generalising the basis of $Omega_3(G;mathbb{F})$ constructed by Grigory'an. For a field $mathbb{F},$ we consider the $mathbb{F}$-path Euler characteristic $chi^mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $mathbb{F}.$ If $Omega_bullet(G;mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $chi^mathbb{F}(G)$. We provide an explicit example of a digraph $mathcal{G}$ whose $mathbb{F}$-path Euler characteristic depends on whether the characteristic of $mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;mathbb{K})cong {rm PH}_*(mathcal{G};mathbb{K})$ simultaneously for $mathbb{K}=mathbb{Z}$ and $mathbb{K}=mathbb{Z}/2mathbb{Z}.$
{"title":"Path homology of digraphs without multisquares and its comparison with homology of spaces","authors":"Xin Fu, Sergei O. Ivanov","doi":"arxiv-2407.17001","DOIUrl":"https://doi.org/arxiv-2407.17001","url":null,"abstract":"For a digraph $G$ without multisquares and a field $mathbb{F}$, we construct\u0000a basis of the vector space of path $n$-chains $Omega_n(G;mathbb{F})$ for\u0000$ngeq 0$, generalising the basis of $Omega_3(G;mathbb{F})$ constructed by\u0000Grigory'an. For a field $mathbb{F},$ we consider the $mathbb{F}$-path Euler\u0000characteristic $chi^mathbb{F}(G)$ of a digraph $G$ defined as the alternating\u0000sum of dimensions of path homology groups with coefficients in $mathbb{F}.$ If\u0000$Omega_bullet(G;mathbb{F})$ is a bounded chain complex, the constructed\u0000bases can be applied to compute $chi^mathbb{F}(G)$. We provide an explicit\u0000example of a digraph $mathcal{G}$ whose $mathbb{F}$-path Euler characteristic\u0000depends on whether the characteristic of $mathbb{F}$ is two, revealing the\u0000differences between GLMY theory and the homology theory of spaces. This allows\u0000us to prove that there is no topological space $X$ whose homology is isomorphic\u0000to path homology of the digraph $H_*(X;mathbb{K})cong {rm\u0000PH}_*(mathcal{G};mathbb{K})$ simultaneously for $mathbb{K}=mathbb{Z}$ and\u0000$mathbb{K}=mathbb{Z}/2mathbb{Z}.$","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775494","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 colimit formula for the K-theory spectra of reductive p-adic groups of rank one with regular coefficients in terms of the K-theory of certain compact open subgroups. Furthermore, in the complex case, we show, using the construction of types provided by Roche, that this result can be improved to obtain a formula for the K-theory spectrum of every principal series Bernstein block if the group is split.
我们根据某些紧凑开子群的 K 理论，证明了具有正则系数的一阶还原 p-adic 群的 K 理论谱的临界公式。此外，在复数情况下，我们利用罗氏提供的类型构造证明，如果群是分裂的，这个结果可以改进为得到每个主系伯恩斯坦块的 K 理论谱公式。
{"title":"K-theory of rank one reductive p-adic groups and Bernstein blocks","authors":"Maximilian Tönies","doi":"arxiv-2407.14929","DOIUrl":"https://doi.org/arxiv-2407.14929","url":null,"abstract":"We prove a colimit formula for the K-theory spectra of reductive p-adic\u0000groups of rank one with regular coefficients in terms of the K-theory of\u0000certain compact open subgroups. Furthermore, in the complex case, we show,\u0000using the construction of types provided by Roche, that this result can be\u0000improved to obtain a formula for the K-theory spectrum of every principal\u0000series Bernstein block if the group is split.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775495","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}
Algebra objects in $infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $infty$-categories. Additionally, for every $infty$-category with finite limits $mathcal{C}$, we introduce a notion of a birelative $2$-Segal object in $mathcal{C}$ and establish a similar equivalence with the $infty$-category of bimodule objects in spans. Examples of these concepts arise from algebraic and hermitian K-theory through the corresponding Waldhausen $S_{bullet}$-construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.
{"title":"An $infty$-Category of 2-Segal Spaces","authors":"Jonte Gödicke","doi":"arxiv-2407.13357","DOIUrl":"https://doi.org/arxiv-2407.13357","url":null,"abstract":"Algebra objects in $infty$-categories of spans admit a description in terms\u0000of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects\u0000and extend this correspondence to an equivalence of $infty$-categories.\u0000Additionally, for every $infty$-category with finite limits $mathcal{C}$, we\u0000introduce a notion of a birelative $2$-Segal object in $mathcal{C}$ and\u0000establish a similar equivalence with the $infty$-category of bimodule objects\u0000in spans. Examples of these concepts arise from algebraic and hermitian\u0000K-theory through the corresponding Waldhausen $S_{bullet}$-construction. Apart\u0000from their categorical relevance, these concepts can be used to construct\u0000homotopy coherent representations of Hall algebras.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743967","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 power operations for twisted KR-theory of topological stacks. Standard algebraic properties of Clifford algebras imply that these power operations preserve universal Thom classes. As a consequence, we show that the twisted Atiyah-Bott-Shapiro orientation commutes with power operations.
{"title":"Power operations preserve Thom classes in twisted equivariant Real K-theory","authors":"Daniel Berwick-Evans, Meng Guo","doi":"arxiv-2407.13031","DOIUrl":"https://doi.org/arxiv-2407.13031","url":null,"abstract":"We construct power operations for twisted KR-theory of topological stacks.\u0000Standard algebraic properties of Clifford algebras imply that these power\u0000operations preserve universal Thom classes. As a consequence, we show that the\u0000twisted Atiyah-Bott-Shapiro orientation commutes with power operations.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743968","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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