We investigate stability properties of the reductive Borel-Serre categories; these were introduced as a model for unstable algebraic K-theory in previous work. We see that they exhibit better homological stability properties than the general linear groups. We also show that they provide an explicit model for Yuan's partial algebraic K-theory.
我们研究了还原伯勒-塞雷范畴的稳定性；这些范畴是在以前的工作中作为不稳定代数 K 理论的模型引入的。我们发现它们比一般线性群表现出更好的同调稳定性。我们还证明它们为袁氏部分代数 K 理论提供了一个明确的模型。
{"title":"Unstable algebraic K-theory: homological stability and other observations","authors":"Mikala Ørsnes Jansen","doi":"arxiv-2405.02065","DOIUrl":"https://doi.org/arxiv-2405.02065","url":null,"abstract":"We investigate stability properties of the reductive Borel-Serre categories;\u0000these were introduced as a model for unstable algebraic K-theory in previous\u0000work. We see that they exhibit better homological stability properties than the\u0000general linear groups. We also show that they provide an explicit model for\u0000Yuan's partial algebraic K-theory.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140882591","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 a commutative orthogonal $C_2$-ring spectrum, $mathrm{MSpin}^c_{mathbb{R}}$, along with a $C_2$-$E_{infty}$-orientation $mathrm{MSpin}^c_{mathbb{R}} to mathrm{KU}_{mathbb{R}}$ of Atiyah's Real K-theory. Further, we define $E_{infty}$-maps $mathrm{MSpin} to (mathrm{MSpin}^c_{mathbb{R}})^{C_2}$ and $mathrm{MU}_{mathbb{R}} to mathrm{MSpin}^c_{mathbb{R}}$, which are used to recover the three well-known orientations of topological $mathrm{K}$-theory, $mathrm{MSpin}^c to mathrm{KU}$, $mathrm{MSpin} to mathrm{KO}$, and $mathrm{MU}_{mathbb{R}} to mathrm{KU}_{mathbb{R}}$, from the map $mathrm{MSpin}^c_{mathbb{R}} to mathrm{KU}_{mathbb{R}}$. We also show that the integrality of the $hat{A}$-genus on spin manifolds provides an obstruction for the fixed points $(mathrm{MSpin}^c_{mathbb{R}})^{C_2}$ to be equivalent to $mathrm{MSpin}$, using the Mackey functor structure of $underline{pi}_*mathrm{MSpin}^c_{mathbb{R}}$. In particular, the usual map $mathrm{MSpin} to mathrm{MSpin}^c$ does not arise as the inclusion of fixed points for any $C_2$-$E_{infty}$-ring spectrum.
{"title":"Real spin bordism and orientations of topological $mathrm{K}$-theory","authors":"Zachary Halladay, Yigal Kamel","doi":"arxiv-2405.00963","DOIUrl":"https://doi.org/arxiv-2405.00963","url":null,"abstract":"We construct a commutative orthogonal $C_2$-ring spectrum,\u0000$mathrm{MSpin}^c_{mathbb{R}}$, along with a $C_2$-$E_{infty}$-orientation\u0000$mathrm{MSpin}^c_{mathbb{R}} to mathrm{KU}_{mathbb{R}}$ of Atiyah's Real\u0000K-theory. Further, we define $E_{infty}$-maps $mathrm{MSpin} to\u0000(mathrm{MSpin}^c_{mathbb{R}})^{C_2}$ and $mathrm{MU}_{mathbb{R}} to\u0000mathrm{MSpin}^c_{mathbb{R}}$, which are used to recover the three well-known\u0000orientations of topological $mathrm{K}$-theory, $mathrm{MSpin}^c to\u0000mathrm{KU}$, $mathrm{MSpin} to mathrm{KO}$, and $mathrm{MU}_{mathbb{R}}\u0000to mathrm{KU}_{mathbb{R}}$, from the map $mathrm{MSpin}^c_{mathbb{R}} to\u0000mathrm{KU}_{mathbb{R}}$. We also show that the integrality of the\u0000$hat{A}$-genus on spin manifolds provides an obstruction for the fixed points\u0000$(mathrm{MSpin}^c_{mathbb{R}})^{C_2}$ to be equivalent to $mathrm{MSpin}$,\u0000using the Mackey functor structure of\u0000$underline{pi}_*mathrm{MSpin}^c_{mathbb{R}}$. In particular, the usual map\u0000$mathrm{MSpin} to mathrm{MSpin}^c$ does not arise as the inclusion of fixed\u0000points for any $C_2$-$E_{infty}$-ring spectrum.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"125 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833945","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 is the first in a sequence of articles exploring the relationship between commutative algebras and $E_infty$-algebras in characteristic $p$ and mixed characteristic. In this paper we lay the groundwork by defining a new class of cohomology operations over $mathbb F_p$ called cotriple products, generalising Massey products. We compute the secondary cohomology operations for a strictly commutative dg-algebra and the obstruction theories these induce, constructing several counterexamples to characteristic 0 behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We construct some families of higher cotriple products and comment on their behaviour. Finally, we distingush a subclass of cotriple products that we call higher Steenrod operation and conclude with our main theorem, which says that $E_infty$-algebras can be rectified if and only if the higher Steenrod operations vanish coherently.
{"title":"An obstruction theory for strictly commutative algebras in positive characteristic","authors":"Oisín Flynn-Connolly","doi":"arxiv-2404.16681","DOIUrl":"https://doi.org/arxiv-2404.16681","url":null,"abstract":"This is the first in a sequence of articles exploring the relationship\u0000between commutative algebras and $E_infty$-algebras in characteristic $p$ and\u0000mixed characteristic. In this paper we lay the groundwork by defining a new\u0000class of cohomology operations over $mathbb F_p$ called cotriple products,\u0000generalising Massey products. We compute the secondary cohomology operations\u0000for a strictly commutative dg-algebra and the obstruction theories these\u0000induce, constructing several counterexamples to characteristic 0 behaviour, one\u0000of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We\u0000construct some families of higher cotriple products and comment on their\u0000behaviour. Finally, we distingush a subclass of cotriple products that we call\u0000higher Steenrod operation and conclude with our main theorem, which says that\u0000$E_infty$-algebras can be rectified if and only if the higher Steenrod\u0000operations vanish coherently.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"138 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798728","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}
The flip Stiefel manifolds (FV_{m,2s}) are defined as the quotient of the real Stiefel manifolds (V_{m,2s}) induced by the simultaneous pairwise flipping of the co-ordinates by the cyclic group of order 2. We calculate the complex (K)-ring of the flip Stiefel manifolds, $K^ast(FV_{m,2s})$, for $s$ even. Standard techniques involve the representation theory of $Spin(m),$ and the Hodgkin spectral sequence. However, the non-trivial element inducing the action doesn't readily yield the desired homomorphisms. Hence, by performing additional analysis, we settle the question for the case of (s equiv 0 pmod 2.)
{"title":"The complex K ring of the flip Stiefel manifolds","authors":"Samik Basu, Shilpa Gondhali, Fathima Safikaa","doi":"arxiv-2404.15803","DOIUrl":"https://doi.org/arxiv-2404.15803","url":null,"abstract":"The flip Stiefel manifolds (FV_{m,2s}) are defined as the quotient of the\u0000real Stiefel manifolds (V_{m,2s}) induced by the simultaneous pairwise flipping\u0000of the co-ordinates by the cyclic group of order 2. We calculate the complex\u0000(K)-ring of the flip Stiefel manifolds, $K^ast(FV_{m,2s})$, for $s$ even.\u0000Standard techniques involve the representation theory of $Spin(m),$ and the\u0000Hodgkin spectral sequence. However, the non-trivial element inducing the action\u0000doesn't readily yield the desired homomorphisms. Hence, by performing\u0000additional analysis, we settle the question for the case of (s equiv 0 pmod\u00002.)","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798707","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 the partial group (co)homology of partial group actions using simplicial methods. We introduce the concept of universal globalization of a partial group action on a $K$-module and prove that, given a partial representation of $G$ on $M$, the partial group homology $H^{par}_{bullet}(G, M)$ is naturally isomorphic to the usual group homology $H_{bullet}(G, KG otimes_{G_{par}} M)$, where $KG otimes_{G_{par}} M$ is the universal globalization of the partial group action associated to $M$. We dualize this result into a cohomological spectral sequence converging to $H^{bullet}_{par}(G,M)$.
{"title":"On the homology of partial group actions","authors":"Emmanuel Jerez","doi":"arxiv-2404.14650","DOIUrl":"https://doi.org/arxiv-2404.14650","url":null,"abstract":"We study the partial group (co)homology of partial group actions using\u0000simplicial methods. We introduce the concept of universal globalization of a\u0000partial group action on a $K$-module and prove that, given a partial\u0000representation of $G$ on $M$, the partial group homology $H^{par}_{bullet}(G,\u0000M)$ is naturally isomorphic to the usual group homology $H_{bullet}(G, KG\u0000otimes_{G_{par}} M)$, where $KG otimes_{G_{par}} M$ is the universal\u0000globalization of the partial group action associated to $M$. We dualize this\u0000result into a cohomological spectral sequence converging to\u0000$H^{bullet}_{par}(G,M)$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798773","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 describe the $T_{comp}$-equivariant topological $K$-ring of a $T$-{it cellular} simplicial toric variety. We further show that $K_{T_{comp}}^0(X)$ is isomorphic as an $R(T_{comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted $PLP(Delta)$. Furthermore, we compute a basis for $K_{T_{comp}}^0(X)$ as a $R(T_{comp})$-module and multiplicative structure constants with respect to this basis.
{"title":"Equivariant $K$-theory of cellular toric varieties","authors":"V. Uma","doi":"arxiv-2404.14201","DOIUrl":"https://doi.org/arxiv-2404.14201","url":null,"abstract":"In this article we describe the $T_{comp}$-equivariant topological $K$-ring of a $T$-{it cellular} simplicial toric variety. We further show that $K_{T_{comp}}^0(X)$ is isomorphic as an $R(T_{comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted $PLP(Delta)$. Furthermore, we compute a basis for $K_{T_{comp}}^0(X)$ as a $R(T_{comp})$-module and multiplicative structure constants with respect to this basis.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798724","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 the present paper, we discuss applications of the derived completion theorems proven in our previous two papers. One of the main applications is to Riemann-Roch problems for forms of higher equivariant K-theory, which we are able to establish in great generality both for equivariant G-theory and equivariant homotopy K-theory with respect to actions of linear algebraic groups on normal quasi-projective schemes over a given field. We show such Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point schemes with respect to actions of diagonalizable group schemes. We also show the existence of certain spectral sequences that compute the homotopy groups of the derived completions of equivariant G-theory starting with equivariant Borel-Moore motivic cohomology.
在本文中，我们讨论了前两篇论文中证明的派生完备定理的应用。其中一个主要应用是高等等式 K 理论形式的黎曼-罗赫（Riemann-Roch）问题，我们可以就给定域上正态准投影方案上的线性代数群的作用，在等式 G 理论和等式同调 K 理论中普遍建立黎曼-罗赫定理。我们证明这样的黎曼-罗赫定理适用于所有环状和球状变体。我们还得到了涉及可对角化群方案作用的定点化学的莱夫谢茨-黎曼-罗赫定理。我们还证明了某些谱序列的存在，这些谱序列从等变伯尔莫尔动机同调开始计算等变 G 理论的派生完备的同调群。
{"title":"Equivariant Algebraic K-Theory and Derived completions III: Applications","authors":"Gunnar Carlsson, Roy Joshua, Pablo Pelaez","doi":"arxiv-2404.13199","DOIUrl":"https://doi.org/arxiv-2404.13199","url":null,"abstract":"In the present paper, we discuss applications of the derived completion\u0000theorems proven in our previous two papers. One of the main applications is to\u0000Riemann-Roch problems for forms of higher equivariant K-theory, which we are\u0000able to establish in great generality both for equivariant G-theory and\u0000equivariant homotopy K-theory with respect to actions of linear algebraic\u0000groups on normal quasi-projective schemes over a given field. We show such\u0000Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point\u0000schemes with respect to actions of diagonalizable group schemes. We also show\u0000the existence of certain spectral sequences that compute the homotopy groups of\u0000the derived completions of equivariant G-theory starting with equivariant\u0000Borel-Moore motivic cohomology.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806556","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 the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. In an earlier work by the first two authors, we solved this conjecture by providing a derived completion theorem for equivariant G-theory. In the present paper, we provide a similar derived completion theorem for the homotopy Algebraic K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any normal quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). With finite coefficients invertible in the base fields, we are also able to obtain such derived completion theorems for equivariant algebraic K-theory but with respect to actions of diagonalizable group schemes. These enable us to obtain a wide range of applications, several of which are also explored.
20 世纪 80 年代中期，已故的罗伯特-托马森（RobertThomason）在研究建立类似于著名的等变拓扑 K 理论的阿蒂亚-西格尔完备定理（Atiyah-Segalcompletion theorem）的等变代数 K 理论完备定理时，发现这类定理所要求的强有限性条件限制性太强。然后，他提出了一个猜想，即存在阿蒂亚和西格尔意义上的等变代数G理论的补全定理，适用于线性代数群在方案上的作用。在前两位作者的早期研究中，我们为等变 G 理论提供了衍生完备性定理，从而解决了这一猜想。在本文中，我们为等变完备复数的同调代数 K 理论提供了一个类似的推导完备定理，而且是在不需要规则的方案上。我们的解决方案足够宽泛，允许所有线性代数群的作用，无论它们是否连通，并且作用于有限类型的域上的任何正则准投影方案，无论它们是正则的还是投影的。这样，我们就可以考虑大类变项的等变同调代数 K 理论，如全多角变项（对于环的作用）和全球面变项（对于还原群的作用）。由于有限系数在基域中是可逆的，我们还能得到这种衍生完备定理，即关于可对角化群方案作用的前变代数 K 理论。这些定理使我们能够获得广泛的应用，其中一些应用也得到了探讨。
{"title":"Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory","authors":"Gunnar Carlsson, Roy Joshua, Pablo Pelaez","doi":"arxiv-2404.13196","DOIUrl":"https://doi.org/arxiv-2404.13196","url":null,"abstract":"In the mid 1980s, while working on establishing completion theorems for\u0000equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal\u0000completion theorem for equivariant topological K-theory, the late Robert\u0000Thomason found the strong finiteness conditions that are required in such\u0000theorems to be too restrictive. Then he made a conjecture on the existence of a\u0000completion theorem in the sense of Atiyah and Segal for equivariant Algebraic\u0000G-theory, for actions of linear algebraic groups on schemes that holds without\u0000any of the strong finiteness conditions that are required in such theorems\u0000proven by him, and also appearing in the original Atiyah-Segal theorem. In an\u0000earlier work by the first two authors, we solved this conjecture by providing a\u0000derived completion theorem for equivariant G-theory. In the present paper, we\u0000provide a similar derived completion theorem for the homotopy Algebraic\u0000K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups,\u0000irrespective of whether they are connected or not, and acting on any normal\u0000quasi-projective scheme of finite type over a field, irrespective of whether\u0000they are regular or projective. This allows us therefore to consider the\u0000Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all\u0000toric varieties (for the action of a torus) and all spherical varieties (for\u0000the action of a reductive group). With finite coefficients invertible in the\u0000base fields, we are also able to obtain such derived completion theorems for\u0000equivariant algebraic K-theory but with respect to actions of diagonalizable\u0000group schemes. These enable us to obtain a wide range of applications, several\u0000of which are also explored.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806655","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}
Kostiantyn Iusenko, Eduardo do Nascimento Marcos, Victor do Valle Pretti
We investigate the relationship between smoothness and the relative global dimension. We prove that a smooth ring map $Bto A$ between commutative rings implies the finiteness of the relative global dimension $operatorname{gldim}(A,B)$. Conversely, we identify a sufficient condition on $B$ such that the finiteness of $operatorname{gldim}(A,B)$ implies the smoothness of the map $Bto A$.
{"title":"A relative homology criteria of smoothness","authors":"Kostiantyn Iusenko, Eduardo do Nascimento Marcos, Victor do Valle Pretti","doi":"arxiv-2404.08534","DOIUrl":"https://doi.org/arxiv-2404.08534","url":null,"abstract":"We investigate the relationship between smoothness and the relative global\u0000dimension. We prove that a smooth ring map $Bto A$ between commutative rings\u0000implies the finiteness of the relative global dimension\u0000$operatorname{gldim}(A,B)$. Conversely, we identify a sufficient condition on\u0000$B$ such that the finiteness of $operatorname{gldim}(A,B)$ implies the\u0000smoothness of the map $Bto A$.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"299 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140596759","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}
The compactly supported $mathbb{A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an anologue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $mathrm{K}_0(mathrm{Var}_k)$ taking values in the Grothendieck-Witt ring $mathrm{GW}(k)$ of the base field $k$. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, Pajwani and P'al construct a power structure on $mathrm{GW}(k)$ and show that the compactly supported $mathbb{A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently 'etale $k$-algebras. In this paper, we define the class $mathrm{Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $mathbb{A}^1$-Euler characteristic respects the power structures and study the algebraic properties of $mathrm{K}_0(mathrm{Sym}_k)$. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $mathbb{A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.
{"title":"Compactly supported $mathbb{A}^1$-Euler characteristics of symmetric powers of cellular varieties","authors":"Jesse Pajwani, Herman Rohrbach, Anna M. Viergever","doi":"arxiv-2404.08486","DOIUrl":"https://doi.org/arxiv-2404.08486","url":null,"abstract":"The compactly supported $mathbb{A}^1$-Euler characteristic, introduced by\u0000Hoyois and later refined by Levine and others, is an anologue in motivic\u0000homotopy theory of the classical Euler characteristic of complex topological\u0000manifolds. It is an invariant on the Grothendieck ring of varieties\u0000$mathrm{K}_0(mathrm{Var}_k)$ taking values in the Grothendieck-Witt ring\u0000$mathrm{GW}(k)$ of the base field $k$. The former ring has a natural power\u0000structure induced by symmetric powers of varieties. In a recent preprint,\u0000Pajwani and P'al construct a power structure on $mathrm{GW}(k)$ and show that\u0000the compactly supported $mathbb{A}^1$-Euler characteristic respects these two\u0000power structures for $0$-dimensional varieties, or equivalently 'etale\u0000$k$-algebras. In this paper, we define the class $mathrm{Sym}_k$ of\u0000symmetrisable varieties to be those varieties for which the compactly supported\u0000$mathbb{A}^1$-Euler characteristic respects the power structures and study the\u0000algebraic properties of $mathrm{K}_0(mathrm{Sym}_k)$. We show that it\u0000includes all cellular varieties, and even linear varieties as introduced by\u0000Totaro. Moreover, we show that it includes non-linear varieties such as\u0000elliptic curves. As an application of our main result, we compute the compactly\u0000supported $mathbb{A}^1$-Euler characteristics of symmetric powers of\u0000Grassmannians and certain del Pezzo surfaces.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"239 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564927","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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