Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a1
Sergey Arkhipov, Daria Poliakova
For a Hopf DG‑algebra corresponding to a derived algebraic group, we compute the homotopy limit of the associated cosimplicial system of DG‑algebras given by the classifying space construction. The homotopy limit is taken in the model category of DG‑categories. The objects of the resulting DG‑category are Maurer–Cartan elements of $operatorname{Cobar}(A)$, or 1‑dimensional $A_infty$-comodules over $A$. These can be viewed as characters up to homotopy of the corresponding derived group. Their tensor product is interpreted in terms of Kadeishvili’s multibraces. We also study the coderived category of DG‑modules over this DG‑category.
{"title":"Homotopy characters as a homotopy limit","authors":"Sergey Arkhipov, Daria Poliakova","doi":"10.4310/hha.2024.v26.n2.a1","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a1","url":null,"abstract":"For a Hopf DG‑algebra corresponding to a derived algebraic group, we compute the homotopy limit of the associated cosimplicial system of DG‑algebras given by the classifying space construction. The homotopy limit is taken in the model category of DG‑categories. The objects of the resulting DG‑category are Maurer–Cartan elements of $operatorname{Cobar}(A)$, or 1‑dimensional $A_infty$-comodules over $A$. These can be viewed as characters up to homotopy of the corresponding derived group. Their tensor product is interpreted in terms of Kadeishvili’s multibraces. We also study the coderived category of DG‑modules over this DG‑category.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a2
Makoto Sakagaito
Let $R$ be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic $(0, p)$ and $k$ the residue field of $R$. In this paper, we prove an isomorphism of étale hypercohomology groups $H^{n+1}_{textrm{ét}} (R, mathfrak{T}_r (n)) simeq H^{1}_{textrm{ét}} (k, W_r Omega^n_{log})$ for any integers $n geqslant 0$ and $r gt 0$ where $mathfrak{T}_r (n)$ is the p-adic Tate twist and $W_r Omega^n_{log}$ is the logarithmic Hodge–Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic.
{"title":"On étale hypercohomology of henselian regular local rings with values in $p$-adic étale Tate twists","authors":"Makoto Sakagaito","doi":"10.4310/hha.2024.v26.n2.a2","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a2","url":null,"abstract":"Let $R$ be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic $(0, p)$ and $k$ the residue field of $R$. In this paper, we prove an isomorphism of étale hypercohomology groups $H^{n+1}_{textrm{ét}} (R, mathfrak{T}_r (n)) simeq H^{1}_{textrm{ét}} (k, W_r Omega^n_{log})$ for any integers $n geqslant 0$ and $r gt 0$ where $mathfrak{T}_r (n)$ is the p-adic Tate twist and $W_r Omega^n_{log}$ is the logarithmic Hodge–Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a4
Y. Volkov, S. Witherspoon
For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and to Hermann, involves loops in extension categories. The algebraic definition, due to the first author, involves homotopy liftings of maps. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in this monoidal category setting. This answers a question of Hermann for those exact monoidal categories in which the unit object has a particular type of resolution that is called power flat. For use in proofs, we generalize $A_infty$-coderivation and homotopy lifting techniques from bimodule categories to these exact monoidal categories.
{"title":"Graded Lie structure on cohomology of some exact monoidal categories","authors":"Y. Volkov, S. Witherspoon","doi":"10.4310/hha.2024.v26.n2.a4","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a4","url":null,"abstract":"For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and to Hermann, involves loops in extension categories. The algebraic definition, due to the first author, involves homotopy liftings of maps. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in this monoidal category setting. This answers a question of Hermann for those exact monoidal categories in which the unit object has a particular type of resolution that is called power flat. For use in proofs, we generalize $A_infty$-coderivation and homotopy lifting techniques from bimodule categories to these exact monoidal categories.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a5
An-Khuong Doan
We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semiprorepresentable is produced. This can be seen as an analogue of Schlessinger’s conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a $G$ equivariant structure in a sense specified below, where $G$ is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of ($G$-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds.
{"title":"Semi-prorepresentability of formal moduli problems and equivariant structures","authors":"An-Khuong Doan","doi":"10.4310/hha.2024.v26.n2.a5","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a5","url":null,"abstract":"We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semiprorepresentable is produced. This can be seen as an analogue of Schlessinger’s conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a $G$ equivariant structure in a sense specified below, where $G$ is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of ($G$-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a6
Yuki Minowa
Let $mathcal{G}_alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $alpha : X to Bmathcal{G}_1$. We compute the integral cohomology of $Bmathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $Bmathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n leqslant 4$.
{"title":"On the cohomology of the classifying spaces of $SO(n)$-gauge groups over $S^2$","authors":"Yuki Minowa","doi":"10.4310/hha.2024.v26.n2.a6","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a6","url":null,"abstract":"Let $mathcal{G}_alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $alpha : X to Bmathcal{G}_1$. We compute the integral cohomology of $Bmathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $Bmathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n leqslant 4$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-18DOI: 10.4310/hha.2024.v26.n2.a3
Matteo Barucco
$defT{mathbb{T}}defCPV{mathbb{C}P(V)}$ We prove a splitting result between the algebraic models for rational $T^2$- and $T$-equivariant elliptic cohomology, where $T$ is the circle group and $T^2$ is the $2$-torus. As an application we compute rational $T$-equivariant elliptic cohomology of $CPV$: the $T$-space of complex lines for a finite dimensional complex $T$-representation $V$. This is achieved by reducing the computation of $T$-elliptic cohomology of $CPV$ to the computation of $T^2$-elliptic cohomology of certain spheres of complex representations.
{"title":"Rational circle-equivariant elliptic cohomology of CP(V)","authors":"Matteo Barucco","doi":"10.4310/hha.2024.v26.n2.a3","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n2.a3","url":null,"abstract":"$defT{mathbb{T}}defCPV{mathbb{C}P(V)}$ We prove a splitting result between the algebraic models for rational $T^2$- and $T$-equivariant elliptic cohomology, where $T$ is the circle group and $T^2$ is the $2$-torus. As an application we compute rational $T$-equivariant elliptic cohomology of $CPV$: the $T$-space of complex lines for a finite dimensional complex $T$-representation $V$. This is achieved by reducing the computation of $T$-elliptic cohomology of $CPV$ to the computation of $T^2$-elliptic cohomology of certain spheres of complex representations.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-29DOI: 10.4310/hha.2024.v26.n1.a19
Florian Kranhold
It is a classical result that configuration spaces of labelled particles in $mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses an additional verticality constraint. In the last section, we apply these results in order to calculate the stable homology of certain vertical configuration spaces.
{"title":"Configuration spaces of clusters as $E_d$-algebras","authors":"Florian Kranhold","doi":"10.4310/hha.2024.v26.n1.a19","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a19","url":null,"abstract":"It is a classical result that configuration spaces of labelled particles in $mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled <i>clusters</i> of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as <i>cellular</i> $E_1$-algebras, and the other one uses an additional <i>verticality</i> constraint. In the last section, we apply these results in order to calculate the stable homology of certain <i>vertical</i> configuration spaces.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-29DOI: 10.4310/hha.2024.v26.n1.a20
Mihail Hurmuzov
We develop a bundle theory of presheaves on small categories, based on similar work by Brent Everitt and Paul Turner. For a certain set of presheaves on posets, we produce a Leray–Serre type spectral sequence that gives a reduction property for the cohomology of the presheaf. This extends the usual cohomological reduction of posets with a unique maximum.
{"title":"A cohomological bundle theory for sheaf cohomology","authors":"Mihail Hurmuzov","doi":"10.4310/hha.2024.v26.n1.a20","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a20","url":null,"abstract":"We develop a bundle theory of presheaves on small categories, based on similar work by Brent Everitt and Paul Turner. For a certain set of presheaves on posets, we produce a Leray–Serre type spectral sequence that gives a reduction property for the cohomology of the presheaf. This extends the usual cohomological reduction of posets with a unique maximum.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-29DOI: 10.4310/hha.2024.v26.n1.a21
Daniel G. Davis, Vojislav Petrović
For a profinite group $G$, we define an $S[[G]]$-module to be a certain type of $G$-spectrum $X$ built from an inverse system ${lbrace X_i rbrace}_i$ of $G$-spectra, with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal subgroup and $G cong lim_i G/N_i$. We define the homotopy orbit spectrum $X_{hG}$ and its homotopy orbit spectral sequence. We give results about when its $E_2$-term satisfies $E^{p,q}_2 cong lim_i H_p (G / N_i , pi_q (X_i))$. Our main result is that this occurs if ${lbrace pi_ast (X_i) rbrace}_i$ degreewise consists of compact Hausdorff abelian groups and continuous homomorphisms, with each $G/N_i$ acting continuously on $pi_q (X_i)$ for all $q$. If $pi_q (X_i)$ is additionally always profinite, then the $E_2$-term is the continuous homology of $G$ with coefficients in the graded profinite $widehat{mathbb{Z}} [[G]]$ module $pi_ast (X)$. Other results include theorems about Eilenberg–Mac Lane spectra and about when homotopy orbits preserve weak equivalences.
{"title":"A homotopy orbit spectrum for profinite groups","authors":"Daniel G. Davis, Vojislav Petrović","doi":"10.4310/hha.2024.v26.n1.a21","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a21","url":null,"abstract":"For a profinite group $G$, we define an $S[[G]]$-module to be a certain type of $G$-spectrum $X$ built from an inverse system ${lbrace X_i rbrace}_i$ of $G$-spectra, with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal subgroup and $G cong lim_i G/N_i$. We define the homotopy orbit spectrum $X_{hG}$ and its homotopy orbit spectral sequence. We give results about when its $E_2$-term satisfies $E^{p,q}_2 cong lim_i H_p (G / N_i , pi_q (X_i))$. Our main result is that this occurs if ${lbrace pi_ast (X_i) rbrace}_i$ degreewise consists of compact Hausdorff abelian groups and continuous homomorphisms, with each $G/N_i$ acting continuously on $pi_q (X_i)$ for all $q$. If $pi_q (X_i)$ is additionally always profinite, then the $E_2$-term is the continuous homology of $G$ with coefficients in the graded profinite $widehat{mathbb{Z}} [[G]]$ module $pi_ast (X)$. Other results include theorems about Eilenberg–Mac Lane spectra and about when homotopy orbits preserve weak equivalences.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a13
Juliet Aygun, Jeremy Miller
The degree of a based graph is the number of essential non-basepoint vertices after generic perturbation. Hatcher–Vogtmann’s degree theorem states that the subcomplex of Auter Space of graphs of degree at most $d$ is $(d-1)$-connected. We extend the definition of degree to the simplicial closure of Auter Space and prove a version of Hatcher–Vogtmann’s result in this context.
{"title":"A degree theorem for the simplicial closure of Auter Space","authors":"Juliet Aygun, Jeremy Miller","doi":"10.4310/hha.2024.v26.n1.a13","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a13","url":null,"abstract":"The degree of a based graph is the number of essential non-basepoint vertices after generic perturbation. Hatcher–Vogtmann’s degree theorem states that the subcomplex of Auter Space of graphs of degree at most $d$ is $(d-1)$-connected. We extend the definition of degree to the simplicial closure of Auter Space and prove a version of Hatcher–Vogtmann’s result in this context.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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