Strongly (respectively strictly) A1-invariant sheaves are foundational for motivic homotopy theory over fields. They are sheaves of (abelian) groups on the Nisnevich site of smooth varieties over a field k, with the property that their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich cohomology groups) are invariant under replacing a variety X by the affine line over X. A celebrated theorem of Fabien Morel states that if the base field k is perfect, then any strongly A1-invariant sheaf of abelian groups is automatically strictly A1-invariant. The aim of these lecture notes is twofold: (1) provide a complete proof if this result, and (2) outline some of its applications.
{"title":"Strongly A^1-invariant sheaves (after F. Morel)","authors":"Tom Bachmann","doi":"arxiv-2406.11526","DOIUrl":"https://doi.org/arxiv-2406.11526","url":null,"abstract":"Strongly (respectively strictly) A1-invariant sheaves are foundational for\u0000motivic homotopy theory over fields. They are sheaves of (abelian) groups on\u0000the Nisnevich site of smooth varieties over a field k, with the property that\u0000their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich\u0000cohomology groups) are invariant under replacing a variety X by the affine line\u0000over X. A celebrated theorem of Fabien Morel states that if the base field k is\u0000perfect, then any strongly A1-invariant sheaf of abelian groups is\u0000automatically strictly A1-invariant. The aim of these lecture notes is twofold: (1) provide a complete proof if\u0000this result, and (2) outline some of its applications.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511489","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 study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection $mathsf{Ham}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles $mathsf{CycHam}$ that forms a right module over the contractad $mathsf{Ham}$. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.
{"title":"Operadic structure on Hamiltonian paths and cycles","authors":"Denis Lyskov","doi":"arxiv-2406.06931","DOIUrl":"https://doi.org/arxiv-2406.06931","url":null,"abstract":"We study Hamiltonian paths and cycles in undirected graphs from an operadic\u0000viewpoint. We show that the graphical collection $mathsf{Ham}$ encoding\u0000directed Hamiltonian paths in connected graphs admits an operad-like structure,\u0000called a contractad. Similarly, we construct the graphical collection of\u0000Hamiltonian cycles $mathsf{CycHam}$ that forms a right module over the\u0000contractad $mathsf{Ham}$. We use the machinery of contractad generating series\u0000for counting Hamiltonian paths/cycles for particular types of graphs.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511490","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 $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let $F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of support. Then $G_0(X)congmathbb{Z}oplus F^1G_0(X)$ and $F^1G_0(X)$ is a finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$. We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions.
{"title":"$G_0$ of affine, simplicial toric varieties","authors":"Zeyu Shen","doi":"arxiv-2406.05562","DOIUrl":"https://doi.org/arxiv-2406.05562","url":null,"abstract":"Let $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote\u0000the Grothendieck group of coherent sheaves on a Noetherian scheme and let\u0000$F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of\u0000support. Then $G_0(X)congmathbb{Z}oplus F^1G_0(X)$ and $F^1G_0(X)$ is a\u0000finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite\u0000cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined\u0000up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$.\u0000We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on\u0000the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"180 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511492","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, if one allows for curved deformations, the canonical map introduced in [KL09] between Morita deformations and second Hochschild cohomology of a dg algebra becomes a bijection. We also show that a bimodule induces an equivalence of curved deformations precisely when it induces an equivalence between the respective 1-derived categories. These results, together with arXiv:2402.08660, solve the curvature problem for first order deformations.
{"title":"Hochschild cohomology parametrizes curved Morita deformations","authors":"Alessandro Lehmann","doi":"arxiv-2406.04945","DOIUrl":"https://doi.org/arxiv-2406.04945","url":null,"abstract":"We show that, if one allows for curved deformations, the canonical map\u0000introduced in [KL09] between Morita deformations and second Hochschild\u0000cohomology of a dg algebra becomes a bijection. We also show that a bimodule\u0000induces an equivalence of curved deformations precisely when it induces an\u0000equivalence between the respective 1-derived categories. These results,\u0000together with arXiv:2402.08660, solve the curvature problem for first order\u0000deformations.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511496","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 $k$ be a field of positive characteristic $p$, and $X$ be a separated of finite type $k$-scheme of dimension $d$. We construct a cycle map from the additive cycle complex to the residual complex of Serre-Grothendieck coherent duality theory. This map is compatible with a cubical version of the map constructed in [Ren23] arXiv:2104.09662 when $k$ is perfect. As a corollary, we get injectivity statements for (additive) higher Chow groups as well as for motivic cohomology (with modulus) with $mathbb{Z}/p$ coefficients when $k$ is algebraically closed.
{"title":"Additive cycle complex and coherent duality","authors":"Fei Ren","doi":"arxiv-2406.01212","DOIUrl":"https://doi.org/arxiv-2406.01212","url":null,"abstract":"Let $k$ be a field of positive characteristic $p$, and $X$ be a separated of\u0000finite type $k$-scheme of dimension $d$. We construct a cycle map from the\u0000additive cycle complex to the residual complex of Serre-Grothendieck coherent\u0000duality theory. This map is compatible with a cubical version of the map\u0000constructed in [Ren23] arXiv:2104.09662 when $k$ is perfect. As a corollary, we\u0000get injectivity statements for (additive) higher Chow groups as well as for\u0000motivic cohomology (with modulus) with $mathbb{Z}/p$ coefficients when $k$ is\u0000algebraically closed.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259545","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 homotopy theory of pre-Calabi-Yau morphisms, viewing them as Maurer-Cartan elements of an $L_{infty}$-algebra. We give two different notions of homotopy: a notion of weak homotopy for morphisms between $d$-pre-Calabi-Yau categories whose underlying graded quivers on the domain (resp. codomain) are the same, and a notion of homotopy for morphisms between fixed pre-Calabi-Yau categories $(mathcal{A},s_{d+1}M_{mathcal{A}})$ and $(mathcal{B},s_{d+1}M_{mathcal{B}})$. Then, we show that the notion of homotopy is stable under composition and that homotopy equivalences are quasi-isomorphisms. Finally, we prove that the functor constructed by the author in a previous article between the category of pre-Calabi-Yau categories and the partial category of $A_{infty}$-categories of the form $mathcal{A}oplusmathcal{A}^*[d-1]$, for $mathcal{A}$ a graded quiver, together with hat morphisms sends homotopic $d$-pre-Calabi-Yau morphisms to weak homotopic $A_{infty}$-morphisms.
{"title":"Homotopy theory of pre-Calabi-Yau morphisms","authors":"Marion Boucrot","doi":"arxiv-2405.20854","DOIUrl":"https://doi.org/arxiv-2405.20854","url":null,"abstract":"In this article we study the homotopy theory of pre-Calabi-Yau morphisms,\u0000viewing them as Maurer-Cartan elements of an $L_{infty}$-algebra. We give two\u0000different notions of homotopy: a notion of weak homotopy for morphisms between\u0000$d$-pre-Calabi-Yau categories whose underlying graded quivers on the domain\u0000(resp. codomain) are the same, and a notion of homotopy for morphisms between\u0000fixed pre-Calabi-Yau categories $(mathcal{A},s_{d+1}M_{mathcal{A}})$ and\u0000$(mathcal{B},s_{d+1}M_{mathcal{B}})$. Then, we show that the notion of\u0000homotopy is stable under composition and that homotopy equivalences are\u0000quasi-isomorphisms. Finally, we prove that the functor constructed by the\u0000author in a previous article between the category of pre-Calabi-Yau categories\u0000and the partial category of $A_{infty}$-categories of the form\u0000$mathcal{A}oplusmathcal{A}^*[d-1]$, for $mathcal{A}$ a graded quiver,\u0000together with hat morphisms sends homotopic $d$-pre-Calabi-Yau morphisms to\u0000weak homotopic $A_{infty}$-morphisms.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259324","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}
David Gepner, Mee Seong Im, Mikhail Khovanov, Nitu Kitchloo
This paper proposes a connection between algebraic K-theory and foam cobordisms, where foams are stratified manifolds with singularities of a prescribed form. We consider $n$-dimensional foams equipped with a flat bundle of finitely-generated projective $R$-modules over each facet of the foam, together with gluing conditions along the subfoam of singular points. In a suitable sense which will become clear, a vertex (or the smallest stratum) of an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of vertices. We show that the first K-theory group of a ring $R$ can be identified with the cobordism group of decorated 1-foams embedded in the plane. A similar relation between the $n$-th algebraic K-theory group of a ring $R$ and the cobordism group of decorated $n$-foams embedded in $mathbb{R}^{n+1}$ is expected for $n>1$. An analogous correspondence is proposed for arbitrary exact categories. Modifying the embedding and other conditions on the foams may lead to new flavors of K-theory groups.
本文提出了代数 K 理论与泡沫共线性之间的联系，其中泡沫是具有规定形式奇点的分层流形。我们考虑了 $n$ 维泡沫，泡沫的每个面上都有一个有限生成的投影 $R$ 模块的平束，以及奇点子泡沫的粘合条件。在合适的意义上，一个 $n$ 维泡沫的顶点（或最小层）取代了一个具有顶点总排序的 $(n+1)$ 复数。我们证明，环 $R$ 的第一 K 理论群可以与嵌入平面的装饰 1 泡沫的共线性群相提并论。当 $n>1$ 时，环 $R$ 的第 $n$ 个代数 K 理论群与嵌入 $mathbb{R}^{n+1}$ 的装饰 $n$ 泡沫的共线性群之间也有类似的关系。对于任意精确范畴，也提出了类似的对应关系。修改泡沫的嵌入和其他条件可能会导致新的K理论群。
{"title":"Foams with flat connections and algebraic K-theory","authors":"David Gepner, Mee Seong Im, Mikhail Khovanov, Nitu Kitchloo","doi":"arxiv-2405.14465","DOIUrl":"https://doi.org/arxiv-2405.14465","url":null,"abstract":"This paper proposes a connection between algebraic K-theory and foam\u0000cobordisms, where foams are stratified manifolds with singularities of a\u0000prescribed form. We consider $n$-dimensional foams equipped with a flat bundle\u0000of finitely-generated projective $R$-modules over each facet of the foam,\u0000together with gluing conditions along the subfoam of singular points. In a\u0000suitable sense which will become clear, a vertex (or the smallest stratum) of\u0000an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of\u0000vertices. We show that the first K-theory group of a ring $R$ can be identified\u0000with the cobordism group of decorated 1-foams embedded in the plane. A similar\u0000relation between the $n$-th algebraic K-theory group of a ring $R$ and the\u0000cobordism group of decorated $n$-foams embedded in $mathbb{R}^{n+1}$ is\u0000expected for $n>1$. An analogous correspondence is proposed for arbitrary exact\u0000categories. Modifying the embedding and other conditions on the foams may lead\u0000to new flavors of K-theory groups.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148696","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}
Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for $A_infty$-algebras, we give an explicit proof that the universal enveloping algebra $UL$ of a differential graded Lie algebra $L$ is Koszul, via an explicit contracting homotopy from the cobar construction $Omega CL$ of the Chevalley-Eilenberg chain coalgebra $CL$ of $L$ to $UL$.
{"title":"Koszul duality and the Poincaré-Birkhoff-Witt theorem","authors":"Ezra Getzler","doi":"arxiv-2405.14798","DOIUrl":"https://doi.org/arxiv-2405.14798","url":null,"abstract":"Using a homotopy introduced by de Wilde and Lecomte and homological\u0000perturbation theory for $A_infty$-algebras, we give an explicit proof that the\u0000universal enveloping algebra $UL$ of a differential graded Lie algebra $L$ is\u0000Koszul, via an explicit contracting homotopy from the cobar construction\u0000$Omega CL$ of the Chevalley-Eilenberg chain coalgebra $CL$ of $L$ to $UL$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148829","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 investigate the surjectivity of the real cycle class map from $I$-cohomology to classical intergral cohomology for some real smooth varieties, in particular surfaces. This might be considered as one of several possible incarnations of real integral Hodge theory.
{"title":"A few computations about the real cycle class map in low dimensions","authors":"Jens Hornbostel","doi":"arxiv-2405.14348","DOIUrl":"https://doi.org/arxiv-2405.14348","url":null,"abstract":"We investigate the surjectivity of the real cycle class map from\u0000$I$-cohomology to classical intergral cohomology for some real smooth\u0000varieties, in particular surfaces. This might be considered as one of several\u0000possible incarnations of real integral Hodge theory.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148650","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 give a generalization of Quillen's $S^{-1}S$ construction for arbitrary $E_n$-monoids as an $E_{n-1}$-monoidal $infty$-category and show that its realization models the group completion provided that $n geq 2$. We will also show how this construction is related to a variety of other constructions of the group completion.
{"title":"Group completion via the action $infty$-category","authors":"Georg Lehner","doi":"arxiv-2405.12118","DOIUrl":"https://doi.org/arxiv-2405.12118","url":null,"abstract":"We give a generalization of Quillen's $S^{-1}S$ construction for arbitrary\u0000$E_n$-monoids as an $E_{n-1}$-monoidal $infty$-category and show that its\u0000realization models the group completion provided that $n geq 2$. We will also\u0000show how this construction is related to a variety of other constructions of\u0000the group completion.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148855","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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