Recent zbMATH articles in MSC 16Ehttps://zbmath.org/atom/cc/16E2024-09-27T17:47:02.548271ZWerkzeugProjective covers over local ringshttps://zbmath.org/1541.160042024-09-27T17:47:02.548271Z"Ercolanoni, Sofia"https://zbmath.org/authors/?q=ai:ercolanoni.sofia"Facchini, Alberto"https://zbmath.org/authors/?q=ai:facchini.albertoSummary: We describe the structure of the projective cover of a module \(M_R\) over a local ring \(R\) and its relation with minimal sets of generators of \(M_R\). The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect.Weighted homological regularitieshttps://zbmath.org/1541.160072024-09-27T17:47:02.548271Z"Kirkman, E."https://zbmath.org/authors/?q=ai:kirkman.ellen-e"Won, R."https://zbmath.org/authors/?q=ai:won.robert"Zhang, J. J."https://zbmath.org/authors/?q=ai:zhang.james-jLet \(A\) be a noetherian connected graded \(k\)-algebra with a balanced dualizing complex. For a weight \(\xi = (\xi_0,\xi_1) \in \mathbb{R}^2\) and a nonzero object \(X \in \mathsf{D}^\mathrm{b}_\mathrm{fg}(A\text{-Gr})\) the authors define the following invariants :
\begin{itemize}
\item The \(\xi\)-Castelnuovo-Mumford regularity, \(\mathrm{CMreg}_\xi(X)\)
\item The \(\xi\)-Ext-regularity, \(\mathrm{Extreg}_\xi(X)\)
\item The \(\xi\)-Tor-regularity, \(\mathrm{Torreg}_\xi(X)\)
\end{itemize}
They also define the \(\xi\)-Artin-Shelter-regularity of \(A\) as
\[
\mathrm{ASreg}_\xi(A) := \mathrm{Torreg}_\xi(k) + \mathrm{CMreg}_\xi(A).
\]
With weight \(\xi = (1,1)\) one obtains the (ordinary) Castelnuovo-Mumford, Tor, Ext, and Artin-Shelter regularities that have already been studied in the existing literature (the relevant references can be found in the paper). Main results in the paper give various (in)equalities between the weighted regularities introduced above. For example, if \(\xi_0>0\), then one has:
\[
\mathrm{Torreg}_\xi(X) \,=\, \mathrm{Extreg}_\xi(X) \,\leq\, \mathrm{CMreg}_\xi(X) + \mathrm{Torreg}_\xi(k)
\]
and
\[
\mathrm{CMreg}_\xi(X) \,\leq\, \mathrm{Extreg}_\xi(X) + \mathrm{CMreg}_\xi(A).
\]
With \(X=A\) it follows that \(\mathrm{ASreg}_\xi(A) \geq 0\) holds. Further, if \(X\) has finite projective dimension (and still \(\xi_0>0\)), then there is an equality,
\[
\mathrm{CMreg}_\xi(X) \,=\, \mathrm{Torreg}_\xi(X) + \mathrm{CMreg}_\xi(A),
\]
provided that \(0 \leq \xi_1 \leq \xi_0\) or that \(\xi_1 \ll 0\).
The authors also show that the \(\xi\)-Castelnuovo-Mumford regularity can be used to detect Artin-Shelter regularity of the algebra in various ways. For example, \(A\) is Artin-Shelter regular if and only if \(A\) is Cohen-Macaulay and there exists a \(\xi = (\xi_0,\xi_1)\) with \(\xi_0>0\) such that \(\mathrm{ASreg}_\xi(A) = 0\).
The results mentioned above generalize various known results from the literature (the relevant references can be found in the paper).
Reviewer: Henrik Holm (København)The wall-chamber structures of the real Grothendieck groupshttps://zbmath.org/1541.160082024-09-27T17:47:02.548271Z"Asai, Sota"https://zbmath.org/authors/?q=ai:asai.sotaSummary: For a finite-dimensional algebra \(A\) over a field \(K\) with \(n\) simple modules, the real Grothendieck group \(K_0 (\mathsf{proj}\ A)_{\mathbb{R}} := K_0(\mathsf{proj}\ A) \otimes_{\mathbb{Z}} \mathbb{R} \cong \mathbb{R}^n\) gives stability conditions of King. We study the associated wall-chamber structure of \(K_0 (\mathsf{proj}\ A)_{\mathbb{R}}\) by using the Koenig-Yang correspondences in silting theory. First, we introduce an equivalence relation on \(K_0 (\mathsf{proj}\ A)_{\mathbb{R}}\) called TF equivalence by using numerical torsion pairs of Baumann-Kamnitzer-Tingley. Second, we show that the open cone in \(K_0 (\mathsf{proj}\ A)_{\mathbb{R}}\) spanned by the g-vectors of each 2-term silting object gives a TF equivalence class, and this gives a one-to-one correspondence between the basic 2-term silting objects and the TF equivalence classes of full dimension. Finally, we determine the wall-chamber structure of \(K_0 (\mathsf{proj}\ A)_{\mathbb{R}}\) in the case that \(A\) is a path algebra of an acyclic quiver.The wall-chamber structures of the real Grothendieck groupshttps://zbmath.org/1541.160092024-09-27T17:47:02.548271Z"Asai, Sota"https://zbmath.org/authors/?q=ai:asai.sotaSummary: Let \(A\) be a finite-dimensional algebra over a field \(K\). By using stability conditions for modules introduced by King, we can define the wall-chamber structure on the real Grothendieck group \(K_0 (\mathrm{proj}A)_{\mathbb{R}} := K_0 (\mathrm{proj}A) \otimes_\mathbb{Z} \mathbb{R}\), as in the works of Brüstle, Smith and Treffinger, and of Bridgeland. In this article, we explain our result that the chambers in this wall-chamber structure are precisely the open cones associated to the basic 2-term silting objects in the perfect derived category \(\mathrm{K}^{\mathrm{b}}(\mathrm{proj}A)\). As one of the key steps, we introduce an equivalence relation, called TF equivalence, by using the numerical torsion pairs of Baumann-Kamnitzer-Tingley.
For the entire collection see [Zbl 1530.16002].Measurings of Hopf algebroids and morphisms in cyclic (co)homology theorieshttps://zbmath.org/1541.160102024-09-27T17:47:02.548271Z"Banerjee, Abhishek"https://zbmath.org/authors/?q=ai:banerjee.abhishek"Kour, Surjeet"https://zbmath.org/authors/?q=ai:kour.surjeetIn this work, the authors study coalgebra measurings in three contexts. First, they begin with Hopf algebroids, which are generalization of Hopf algebras to noncommutative base rings. They also study coalgebra measurings which induce maps on the homology of Lie-Rinehart algebras. Finally, they consider cyclic comp modules over non-symmetric operads with multiplication. In all these cases, the authors show how morphisms are induced on cyclic homology and cyclic cohomology.
Reviewer: Angela Gammella-Mathieu (Metz)Deformation cohomology for cyclic groups acting on polynomial ringshttps://zbmath.org/1541.160112024-09-27T17:47:02.548271Z"Lawson, Colin M."https://zbmath.org/authors/?q=ai:lawson.colin-m"Shepler, Anne V."https://zbmath.org/authors/?q=ai:shepler.anne-vHochschild cohomology governs deformations of algebras: Every deformation arises from a Hochschild \(2\)-cocycle, which is called an infinitesimal deformation. When the algebra is graded, the Hochschild cohomology inherits the grading, and graded deformations all arise from infinitesimal deformations of degree \(-1\). The main result of the present paper is the classification of infinitesimal deformations of degree \(-1\) of the skew group algebra \(S(V)\rtimes G\) when the characteristic of the underlying field is not \(2\). Here \(G\) is a finite cyclic group acting on a finite dimensional vector space \(V\), and \(S(V)\) is the symmetric algebra of \(V\).
In order to achieve the goal, the authors construct a twisted product resolution for \(S(V)\rtimes G\) using a periodic resolution for the cyclic group \(G\) and the Koszul resolution for \(S(V)\). With the help of this resolution, the authors decompose the space of infinitesimal deformations of degree \(-1\) into contributions from each group element. Then they discuss the cocycle conditions in terms of the codimension of fixed point spaces. It turns out that only group elements with fixed point spaces of codimension at most \(2\) (identity, reflections, and bireflections) contribute to this space of infinitesimal deformations.
The present paper recovers the known description of the space of infinitesimal deformations of degree \(-1\) in the nonmodular setting when the underlying field is algebraically closed. It also demonstrates that the main result can be used to lift an infinitesimal deformation to a deformation.
Reviewer: Yining Zhang (Singapura)The first Hochschild cohomology as a Lie algebrahttps://zbmath.org/1541.160122024-09-27T17:47:02.548271Z"Rubio y Degrassi, Lleonard"https://zbmath.org/authors/?q=ai:rubio-y-degrassi.lleonard"Schroll, Sibylle"https://zbmath.org/authors/?q=ai:schroll.sibylle"Solotar, Andrea"https://zbmath.org/authors/?q=ai:solotar.andrea-lThe question to describe the Hochschild cohomology of a finite dimensional algebra \(A\) as a Gerstenhaber algebra and in particular the first Hochschild cohomology as a Lie algebra has been studied in several recent papers. In this work, the authors contribute to this question. They start recalling some concepts about Lie algebras and Hochschild cohomology. Then, the authors obtain sufficient conditions for the solvability of the first Hochschild cohomology. They develop conditions of the Ext-quiver. The first condition is based on the Ext-quiver having no parallel arrows and no loops. They then apply the results to several examples of finite algebras. An example where the first Hochschild cohomology is semi-simple is also given.
Reviewer: Angela Gammella-Mathieu (Metz)On dimension of the space of derivations on commutative regular algebrashttps://zbmath.org/1541.160132024-09-27T17:47:02.548271Z"Ayupov, Shavkat"https://zbmath.org/authors/?q=ai:ayupov.sh-a"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Karimov, Khakimbek"https://zbmath.org/authors/?q=ai:karimov.khakimbekThe dimension of the space of derivations on a commutative von Neumann regular subalgebra of a measure space is determined. Specifically, \((\Omega,\Sigma,\mu)\) is a measure space with a finite countably additive measure, \(S(\Omega)\) is the algebra of \(\mathbb{F}\)-valued measurable functions on \((\Omega,\Sigma,\mu)\) (for \(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\)), and \(\mathcal{A}\) is a von Neumann regular subalgebra of \(S(\Omega)\). \textit{A. F. Ber} et al. [Extr. Math. 21, No. 2, 107--147 (2006; Zbl 1129.46056)] and \textit{A. G. Kusraev} [Sib. Mat. Zh. 47, No. 1, 97--107 (2006; Zbl 1113.46043); translation in Sib. Math. J. 47, No. 1, 77--85 (2006)] proved that \(S(\Omega)\) admits nonzero derivations if and only if \(\nabla(S(\Omega))\), the Boolean algebra of idempotents in \(S(\Omega)\), is not atomic. Also, \textit{A. F. Ber} [Mat. Tr. 13, No. 1, 3--14 (2010; Zbl 1249.13020); translation in Sib. Adv. Math. 21, No. 3, 161--169 (2011)] showed that \(\dim_{\mathbb{F}} \operatorname{Der} (S([0,1]))\) is uncountable, being at least \(\dim_{\mathbb{F}} S([0,1])^I\) for an uncountable set \(I\).
In the present paper, both \(S(\Omega)\) and \(\mathcal{A}\) are assumed to be homogeneous in the sense that the Boolean algebras \(\nabla(S(\Omega))\) and \(\nabla(\mathcal{A})\) are homogeneous. Moreover, the transcendence degree of \(\mathcal{A}\) over \(\mathbb{F}\), as introduced by the authors [Positivity 26, No. 1, Paper No. 11, 23 p. (2022; Zbl 1494.46055)], is assumed to be infinite, as is the weight \(\tau(\nabla(\mathcal{A}))\), that is, the least cardinality of a set generating \(\nabla(\mathcal{A})\). The authors prove that \(\dim_{\mathbb{F}} \operatorname{Der}(\mathcal{A}) = \tau(\nabla(\mathcal{A}))^{\operatorname{trdeg}(\mathcal{A})}\).
Reviewer: Kenneth R. Goodearl (Santa Barbara)Reflexive hull discriminants and applicationshttps://zbmath.org/1541.160142024-09-27T17:47:02.548271Z"Chan, Kenneth"https://zbmath.org/authors/?q=ai:chan.kenneth-s"Gaddis, Jason"https://zbmath.org/authors/?q=ai:gaddis.jason"Won, Robert"https://zbmath.org/authors/?q=ai:won.robert"Zhang, James J."https://zbmath.org/authors/?q=ai:zhang.james-jFor a noncommutative \(k\)-algebra (\(k\) is a field) that is a finite rank free module over its center, one can define the discriminant, which is powerful tool. If the algebra is not free over its center, then one can use the notion of modified discriminant ideals and define the discriminant to be the gcd of the elements in this ideal, provided that it exists. In this paper, the authors introduce the (extended) reflexive hull discriminant for certain types of \(k\)-algebras that are module-finite, but not necessarily free, over their centers. It has the advantage of not requiring the existence of gcd of the elements in the modified discriminant ideal.
In general, it is not clear for which algebras the extended reflexive hull discriminant exists; in fact, the authors raise this as Question 4.9 in the paper. However, one of their main results shows that the extended reflexive hull discriminant does exist for certain types of skew polynomial rings.
Another main result is the interaction between the reflexive hull discriminant of an algebra and its automorphisms as well as its locally nilpotent derivations. For certain types of algebras \(A\) with reflexive hull discriminant \(d \in A\) the following properties are proved: (1)\, Every automorphism \(g\) of \(A\) satisfies \(g(d) =_{A^\times} d\); this means that one has \(g(d) = cd\) for some unit \(c \in A\). (2)\, If char\,\(k = 0\) and \(A^\times = k^\times\), then every locally nilpotent derivation \(\delta\) of \(A\) satisfies \(\delta(d) = 0\).
The authors also give an in-depth study of the extended reflexive hull discriminant for a certain family of generalized Weyl algebras (GWAs) and, in some cases, the discriminant is explicitly computed.
Reviewer: Henrik Holm (København)Auslander-Reiten theory in extriangulated categorieshttps://zbmath.org/1541.160172024-09-27T17:47:02.548271Z"Iyama, Osamu"https://zbmath.org/authors/?q=ai:iyama.osamu"Nakaoka, Hiroyuki"https://zbmath.org/authors/?q=ai:nakaoka.hiroyuki"Palu, Yann"https://zbmath.org/authors/?q=ai:palu.yannAuslander-Reiten theory is a key tool to study the local structure of additive categories. The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. The authors develop Auslander-Reiten theory for extriangulated categories. This unifies Auslander-Reiten theories developed in exact categories and triangulated categories independently. The authors give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander-Reiten-Serre duality. They also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, Iyama, Nakaoka, and Palu prove that the stable category of an extriangulated category is a \(\tau\)-category if it has enough projectives, almost split extensions and source morphisms. Finally the authors prove that any locally finite symmetrizable \(\tau\)-quiver is an Auslander-Reiten quiver of some extriangulated category with sink morphisms and source morphisms.
Reviewer: Mee Seong Im (Annapolis)Corrigendum to: ``\(m\)-periodic Gorenstein objects''https://zbmath.org/1541.180152024-09-27T17:47:02.548271Z"Huerta, Mindy Y."https://zbmath.org/authors/?q=ai:huerta.mindy-y"Mendoza, Octavio"https://zbmath.org/authors/?q=ai:mendoza.octavio"Pérez, Marco A."https://zbmath.org/authors/?q=ai:perez.marco-aSummary: Let \((\mathcal{A}, \mathcal{B})\) be a GP-admissible pair and \((\mathcal{Z}, \mathcal{W})\) be a GI-admissible pair of classes of objects in an abelian category \(\mathcal{C}\), and consider the class \(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}\) of 1-periodic \((\omega,\mathcal{B})\)-Gorenstein projective objects, where \(\omega := \mathcal{A} \cap \mathcal{B}\) and \(\nu := \mathcal{Z} \cap \mathcal{W}\). We claimed in our paper [ibid. 621, 1--40 (2023; Zbl 1520.18011), Lem. 8.1] that the \((\mathcal{Z},\mathcal{W})\)-Gorenstein injective dimension of \(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}\) is bounded by the \((\mathcal{Z}, \mathcal{W})\)-Gorenstein injective dimension of \(\omega\), provided that: (1) \(\omega\) is closed under direct summands, (2) \(\operatorname{Ext}^1 (\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}, \nu) = 0\), and (3) every object in \(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}\) admits a \(\operatorname{Hom}(-, \nu)\)-acyclic \(\nu\)-coresolution. These conditions and their duals are part of what we called ``Setup 1''. Moreover, if we replace \(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}\) by the class \(\mathcal{GP}_{(\mathcal{A}, \mathcal{B})}\) of \((\mathcal{A}, \mathcal{B})\)-Gorenstein projective objects, the resulting inequality is claimed to be true under a set of conditions named ``Setup 2''.
The proof we gave for the claims \(\operatorname{Gid}_{(\mathcal{Z}, \mathcal{W})}(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}) \leq \operatorname{Gid}_{(\mathcal{Z},\mathcal{W})}(\omega)\) and \(\operatorname{Gid}_{(\mathcal{Z}, \mathcal{W})} (\mathcal{GP}_{(\mathcal{A}, \mathcal{B})}) \leq \operatorname{Gid}_{(\mathcal{Z}, \mathcal{W})}(\omega)\) is incorrect, and the purpose of this note is to exhibit a corrected proof of the first inequality, under the additional assumption that every object in \(\pi \mathcal{GP}_{(\omega, \mathcal{B}, 1)}\) has finite injective dimension relative to \(\mathcal{Z}\). Setup 2 is no longer required, and as a result the second inequality was removed. We also fix those results in {\S}8 of [loc. cit.] affected by Lemma 8.1, and comment some applications and examples.A compact non-formal closed \(\mathrm{G_2}\) manifold with \(b_1 =1\)https://zbmath.org/1541.530342024-09-27T17:47:02.548271Z"Martín-Merchán, Lucía"https://zbmath.org/authors/?q=ai:martin-merchan.luciaThe author develops a method of resolution for orbifolds to construct an example of \(G_2\)-orbifold which is non-formal and has first Betti number \(b_1 = 1\).
An \(n\)-dimensional orbifold is a Hausdorff and second countable space \(X\) endowed with an atlas \(\{(U_\alpha, V_\alpha, \psi_\alpha, \Gamma_\alpha)\}\) of local charts covering \(X=\cup_\alpha V_\alpha\) by quotients of Euclidean open neigbourhoods \(\psi_\alpha: U_\alpha \subseteq \mathbb R^n \to V_\alpha/\Gamma_\alpha\) via diffeomorphism \(\psi_\alpha \in \mathrm{Diff}(U_\alpha)\) with natural conditions like for a manifold structure. A \(G_2\)-structure is defined by a \(G_2\)-form \(\varphi\) on a 7-dimensional manifold \(M\) such that for every point \(p\) the isomorphism \(u : U \to \mathbb R^7\) defines a \(G_2\)-form \(\varphi_0(u,v,w) = (u \times v, w)\) on \(\mathbb R^7\) such that \(u^*\varphi_0 = \varphi\). A minimal commutative differential graded CDGA \((\wedge V, d)\) is formal if there exists a quasi-isomorphism of minimal commutative differential graded algebras CDGA \((\wedge V, d) \to (H^*(\wedge V, d), 0)\) (Definition 2.14).
The author proves in Theorem 1.2 and Proposition 5.6 that there exists a compact non-formal closed \(G_2\) manifold with \(b_1 = 1\) that cannot be endowed with a torsion-free \(G_2\)-structure. The essential method used in the paper is the resolution method (Theorem 1.1) which is stating that for a closed \(G_2\)-structure on compact manifold \((M,\varphi,g)\) and an involution \(j: M \to M\) with \(j^*(\varphi) = \varphi\) and singular locus \(L = \mathrm{Fix}(j)\) with a nowhere-vanishing closed 1-form \(\theta\) on \(L\), then the singular locus in the quotient \(X= M/j\) can be resolved in the category of \(G_2\)-manifolds.
Reviewer: Do Ngoc Diep (Hà Nội)A multiplicative Tate spectral sequence for compact Lie group actionshttps://zbmath.org/1541.550022024-09-27T17:47:02.548271Z"Hedenlund, Alice"https://zbmath.org/authors/?q=ai:hedenlund.alice"Rognes, John"https://zbmath.org/authors/?q=ai:rognes.johnLet \(G\) be a compact Lie group. The Tate construction \(X^{tG}\) of a \(G\)-spectrum \(X\) is a homotopical generalization of the classical Tate cohomology of a finite group. The topological version was first considered by \textit{J. P. C. Greenlees} [Proc. Edinb. Math. Soc., II. Ser. 30, 435--443 (1987; Zbl 0608.57029)] and \textit{J. P. C. Greenlees} and \textit{J. P. May} [Generalized Tate cohomology. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0876.55003)]. One important example comes from the action of the circle group \(\mathbb T\) on the topological Hochschild homology \(\mathrm{THH}(A)\) of an \(E_1\)-ring spectrum \(A\). In this case, the Tate construction \(\mathrm{THH}(A)^{t\mathbb T}\) is also known as the \textit{periodic topological cyclic homology} of \(A\) and is connected to crystalline cohomology.
In the memoir under review, the authors carefully construct and analyze a \(G\)-Tate spectral sequence for an (orthogonal) \(G\)-spectrum \(X\). It has an algebraically defined \(E_2\)-term and suitably converges to the the homotopy groups of the Tate construction \(X^{tG}\). The main point of their construction is that this spectral sequence is multiplicative: For a pairing \(X \wedge Y \to Z\) of orthogonal \(G\)-spectra, there is an induced pairing of \(E_2\)-terms whose induced pairing on \(E_{\infty}\)-terms is compatible with the induced pairing \(\pi_*(X^{tG}) \otimes \pi_*(Y^{tG}) \to \pi_*(Z^{tG})\) on the homotopy groups of the Tate constructions. For finite groups, such multiplicative Tate spectral sequences appear in work of Greenlees-May [loc. cit.] and \textit{L. Hesselholt} and \textit{I. Madsen} [Ann. Math. (2) 158, No. 1, 1--113 (2003; Zbl 1033.19002)]. However, as the authors point out, while being used in computations, there appears to be a lack of references where a detailed construction of a multiplicative Tate spectral sequence for actions of compact Lie groups is carried out.
The following is a more detailed description of the main results and the methods used: To cover examples beyond the circle group, the authors work with a compact Lie group \(G\) and a commutative orthogonal ring spectrum \(R\) such that \(R[G]_* = \pi_*(R[G])\) is finitely generated and projective over \(R_* = \pi_*(R)\), which includes the cases where \(G\) is topologically a product of spheres. The \(E_2\)-term of the spectral sequence to be constructed is then given by completed Ext-terms \(\widehat{\mathrm{Ext}}^{*}_{R[G]_*}(R_*,\pi_*(X))\) where \(X\) is an \(R\)-module in orthogonal \(G\)-spectra. To implement multiplicative structures on these Ext-terms, the authors develop a theory of Tate cohomology of finitely generated and projective Hopf algebras over a commutative ring.
For the construction of multiplicative spectral sequences, they work with Cartan-Eilenberg systems and set up a general result about multiplicative spectral sequences associated with sequences of orthogonal \(G\)-spectra. To apply this to the construction of Tate spectral sequences, they totalize filtrations on \(F(EG_+,X)\) and \(\widetilde{EG}\) to get a `Hesselholt-Madsen filtration' on the Tate construction \(X^{tG} = ( \widetilde{EG} \wedge F(EG_+,X))^G\). As the authors point out, this approach is related to the construction of a Tate spectral sequence for the sphere spectrum and the circle group that appears in unpublished work of \textit{A. J. Blumberg} and \textit{M. A. Mandell} [``The strong Künneth theorem for topological periodic cyclic homology'', Preprint, \url{arXiv:1706.06846}].
Reviewer: Steffen Sagave (Nijmegen)Formality of cochains on \(BG\)https://zbmath.org/1541.550132024-09-27T17:47:02.548271Z"Benson, David J."https://zbmath.org/authors/?q=ai:benson.david-john"Greenlees, John"https://zbmath.org/authors/?q=ai:greenlees.john-p-cIt is well known (see e.g., [\textit{T. V. Kadeishvili}, Soobshch. Akad. Nauk Gruz. SSR 108, 249--252 (1982; Zbl 0535.55005)]) that over a field \(K\) of characteristic zero, the algebra \(C^\ast(BG,K)\) of cochains on the classifying space of a connected compact Lie group is formal as an \(A_\infty\) algebra, or equivalently as a differential graded \((DG)\) algebra.
The authors prove that this is the case for a compact Lie group \(G\) that is not necessarily connected, over any field \(K\) in which the order of the Weyl group \(N_G(T)/T\) of \(G\) is invertible, where \(T\) is a maximal torus in \(G\). The well-known proof for the connected case in characteristic zero is included for the sake of convenience of comparison. At the end, some remarks are made that put the main result in context.
Reviewer: Marek Golasiński (Olsztyn)The Hochschild homology and cohomology of \(A(1)\)https://zbmath.org/1541.550182024-09-27T17:47:02.548271Z"Salch, A."https://zbmath.org/authors/?q=ai:salch.andrew|salch.alexandreThe algebra \(\mathscr A\) of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces, known as the 2-primary Steenrod algebra, is generated by \(Sq^{1}, Sq^{2}, \ldots,\) the Steenrod squares. Within \(\mathscr A\), there exists a subalgebra denoted \(\mathscr A(1)\), which is spanned by the first two Steenrod squares, \(Sq^{1}\) and \(Sq^{2}\). The objective of this paper is to determine the Hochschild homology and cohomology of \(\mathscr A(1)\), utilizing two separate filtrations on \(\mathscr A(1)\), and investigating the spectral sequences in Hochschild homology that arise from these filtrations. These spectral sequences parallel May's spectral sequence for computing Ext over the Steenrod algebra, thus one denotes them as ``May-type'' spectral sequences.
Reviewer: Đặng Võ Phúc (Nha Trang)