Recent zbMATH articles in MSC 18Ghttps://zbmath.org/atom/cc/18G2021-06-15T18:09:00+00:00WerkzeugTruncated derived functors and spectral sequences.https://zbmath.org/1460.550222021-06-15T18:09:00+00:00"Baues, Hans-Joachim"https://zbmath.org/authors/?q=ai:baues.hans-joachim"Blanc, David"https://zbmath.org/authors/?q=ai:blanc.david-a"Chorny, Boris"https://zbmath.org/authors/?q=ai:chorny.borisThe \(E_2\)-term of the Adams spectral sequence [\textit{J. F. Adams}, Comment. Math. Helv. 32, 180--214 (1958; Zbl 0083.17802)] may be identified with certain derived functors, and this also holds for a number of other spectral sequences.
The authors show how the higher \(E_{n+2}\)-terms for \(n\ge 0\) of such spectral sequences are determined by truncations of relative derived functors, defined in terms of certain simplicial functors called mapping algebras.
The first author carried out this program for the \(E_3\)-term of the stable Adams spectral sequence in [\textit{H.-J. Baues}, The algebra of secondary cohomology operations. Basel: Birkhäuser (2006; Zbl 1091.55001)] and [\textit{H.-J. Baues} and \textit{M. Jibladze}, J. \(K\)-Theory 7, No. 2, 203--347 (2011; Zbl 1231.18009)], showing that extended calculations may be made using such a construction.
Reviewer: Marek Golasiński (Olsztyn)Non-abelian Galois cohomology via descent cohomology.https://zbmath.org/1460.180022021-06-15T18:09:00+00:00"Mesablishvili, Bachuki"https://zbmath.org/authors/?q=ai:mesablishvili.bachukiFrom Ohkawa to strong generation via approximable triangulated categories -- a variation on the theme of Amnon Neeman's Nagoya lecture series.https://zbmath.org/1460.140462021-06-15T18:09:00+00:00"Minami, Norihiko"https://zbmath.org/authors/?q=ai:minami.norihikoSummary: This survey stems from Amnon Neeman's lecture series at Ohakawa's memorial workshop. Starting with Ohakawa's theorem, this survey intends to supply enough motivation, background and technical details to read Neeman's recent papers on his ``approximable triangulated categories'' and his \(\mathbf{D}^b_{\mathrm{coh}}(X)\) strong generation sufficient criterion via de Jong's regular alteration, even for non-experts.
For the entire collection see [Zbl 1460.55001].Punctual gluing of \(t\)-structures and weight structures.https://zbmath.org/1460.140472021-06-15T18:09:00+00:00"Vaish, Vaibhav"https://zbmath.org/authors/?q=ai:vaish.vaibhavSummary: We formulate a notion of ``punctual gluing'' of \(t\)-structures and weight structures. As our main application we show that the relative version of Ayoub's 1-motivic \(t\)-structure restricts to compact motives. We also demonstrate the utility of punctual gluing by recovering several constructions in literature. In particular we construct the weight structure on the category of motivic sheaves over any base \(X\) and we also construct the relative Artin motive and the relative Picard motive of any variety \(Y/X\).Stringy Kähler moduli for the Pfaffian-Grassmannian correspondence.https://zbmath.org/1460.140432021-06-15T18:09:00+00:00"Donovan, Will"https://zbmath.org/authors/?q=ai:donovan.willSummary: The Pfaffian-Grassmannian correspondence relates certain pairs of derived equivalent non-birational Calabi-Yau 3-folds. Given such a pair, I construct a set of derived equivalences corresponding to mutations of an exceptional collection on the relevant Grassmannian, and give a mirror symmetry interpretation, following a physical analysis of \textit{R. Eager} et al. [Chin. Ann. Math., Ser. B 38, No. 4, 901--912 (2017; Zbl 1373.32019)].Flat model structures under excellent extensions.https://zbmath.org/1460.180142021-06-15T18:09:00+00:00"Ren, Wei"https://zbmath.org/authors/?q=ai:ren.wei.3|ren.wei.1|ren.wei.4|ren.wei.2|ren.wei.5The aim of the paper is to study Gillespie's flat model structure under excellent extensions of rings. The author shows that the weak equivalences, cofibrations and fibrations in this structure are invariant under excellent extensions of rings. Several applications related to the flat and cotorsion dimensions of unbounded complexes are given.
Reviewer: Constantin Năstăsescu (Bucureşti)Hochschild cohomology of \(m\)-cluster tilted algebras of type \(\widetilde{\mathbb{A}} \).https://zbmath.org/1460.160112021-06-15T18:09:00+00:00"Gubitosi, Viviana"https://zbmath.org/authors/?q=ai:gubitosi.vivianaRepresentation theory of quivers and finite dimensional algebras. Abstracts from the workshop held January 19--25, 2020.https://zbmath.org/1460.000312021-06-15T18:09:00+00:00"Amiot, Claire (ed.)"https://zbmath.org/authors/?q=ai:amiot.claire"Crawley-Boevey, William (ed.)"https://zbmath.org/authors/?q=ai:crawley-boevey.william"Iyama, Osamu (ed.)"https://zbmath.org/authors/?q=ai:iyama.osamu"Krause, Henning (ed.)"https://zbmath.org/authors/?q=ai:krause.henningSummary: Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold.https://zbmath.org/1460.140312021-06-15T18:09:00+00:00"Lo, Jason"https://zbmath.org/authors/?q=ai:lo.jason.1|lo.jasonGiven a Fourier-Mukai transform \(\Phi: D^b(X) \xrightarrow{\sim} D^b(Y)\) between the bounded derived categories of coherent sheaves on two smooth projective varieties \(X\) and \(Y\), the author asks a natural question:
\textit{What is a notion of stability on \(Y\) for the Mumford slope stability on \(X\) under the Fourier-Mukai transform \(\Phi\), without fixing Chern classes of sheaves on \(X\)? }
In this article, the author answers above question in the set-up that \(X\) is a product elliptic threefold \(\pi: X=C\times S \to S\), where \(C\) is an elliptic curve, \(S\) is a \(K3\) surface of Picard rank one with ample generator \(H_S\), and \(\pi\) is the natural projection map, see also [\textit{J. Lo} and \textit{Z. Zhang}, Geom. Dedicata 193, 89--119 (2018: Zbl 1454.14029)]. In this case, the Picard rank of \(X\) is two, with generators \(H\) (which is the zero section of \(\pi\)) and \(D\) (which is the vertical divisor \(\pi^*H_S\)). The fiberwise Fourier-Mukai transform \(D^b(C) \xrightarrow{\sim} D^b(C)\) induces a Fourier-Mukai transform \(\Phi\) above with \(Y=X\). Moreover, \(\Phi\circ\Phi=\mathrm{id}_X[-1]\).
There is a notion of tilt stability \((\mathcal{B}_\omega, \nu_\omega)\) [\textit{A. Bayer} et al., J. Algebr. Geom. 23, No. 1, 117--163 (2014: Zbl 1306.14005)], where \(\omega\) is a fixed ample divisor, \(\mathcal{B}_\omega\) is the tilting of coherent heart \(\mathrm{Coh}(X)\) according to Mumford slope \(\mu_\omega(\cdot):=\omega^2\mathrm{ch}_1(\cdot)/\mathrm{ch}_0(\cdot)\) with respect to the value \(0\), and \[\nu_\omega(\cdot):=\frac{\omega\mathrm{ch}_2(\cdot)-\omega^3\mathrm{ch}_0(\cdot)/6}{\omega^2\mathrm{ch}_1(\cdot)}\] is the slope function on \(\mathcal{B}_\omega\). For any fixed positive number \(\alpha\), the author defines a limit \((\mathcal{B}^l,\nu^l)\) of tilt stability \((\mathcal{B}_\omega, \nu_\omega)\) by taking \[\omega=tH+sD,\] with the constraint curve \[ ts=\alpha, \] and \(s\to +\infty\). More precisely, it is a polynomial stability condition in the sense of \textit{A. Bayer} [Geom. Topol. 13, No. 4, 2389--2425 (2009: Zbl 1171.14011)].
The main result Theorem 5.1 is a comparison of the notion of Mumford slope stability \(\mu_{\bar{\omega}}\) and the limit tilt stability \(\nu^l\) before and after the Fourier-Mukai transform \(\Phi[1]\). Here \[\bar{\omega}=\frac{\lambda}{\alpha}H+\lambda D,\] with fixed positive numbers \(\alpha\) and \(\lambda\). In particular, the value of the number \(\lambda\) is not essential for the notion of \(\mu_{\bar{\omega}}\). Actually, the constraint curve \(ts=\alpha\) is obtained by comparing \(\mu_{\bar{\omega}}(E)\) and \(\nu_{\omega}(\Phi(E)[1])\) via cohomological Fourier-Mukai transform.
Fix any positive number \(\alpha\). The first part of Theorem 5.1 states that if \(E\) is a \(\mu_{\bar{\omega}}\)-stable sheaf with a codimension 2 vanishing condition, then \(\Phi(E)[1]\) is a \(\nu^l\)-stable object in \(\mathcal{B}^l\). The second part of the theorem says that if \(F\in \mathcal{B}^l\) is a \(\nu^l\)-semistable object with \(\mathrm{ch}_{10}(F)\neq 0\), then there is a codimension 2 modification \(F'\) of \(F\) in \(\mathcal{B}^l\) such that \(\Phi(F')\) is \(\mu_{\bar{\omega}}\)-semistable sheaf.
In section 6, the author shows that the Harder-Narasimhan property holds for the limit tilt stability condition. The proof is a modification of an approach of \textit{Y. Toda} [Adv. Math. 217, No. 6, 2736--2781 (2008: Zbl 1136.14007)], that there is a torsion quadruple \[(\mathcal{A}_\bullet, \mathcal{A}_{1/2}, \mathcal{A}_{0}, \mathcal{A}_{-1/2}) \] in the heart \(\mathcal{B}^l\), and in each \( \mathcal{A}_{i}\), \(i=1/2, 0, -1/2\), there are finiteness properties.
In the last section of the article, the author compares tilt stability and limit tilt stability and gives an example that the structure sheaf \(\mathcal{O}_X\) is limit tilt stable.
In a subsequent paper [\textit{J. Lo}, ``Fourier-Mukai transforms of slope stable torsion-free sheaves and stable 1-dimensional sheaves on Weierstrass elliptic threefolds'', Preprint, \url{arXiv:1710.03771}], the author generalizes the main result of this paper to the case that \(X\) is an Weierstrass elliptic threefold. A similar result for elliptic surface case also holds [\textit{W. Liu} et al., ``Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces'', Preprint, \url{arXiv:1910.02477}], where the notion of limit tilt stability is replaced by the notion of limit Bridgeland stability.
Reviewer: Wanmin Liu (Uppsala)Topological Hochschild homology and higher characteristics.https://zbmath.org/1460.160102021-06-15T18:09:00+00:00"Campbell, Jonathan A."https://zbmath.org/authors/?q=ai:campbell.jonathan-a"Ponto, Kate"https://zbmath.org/authors/?q=ai:ponto.kateSummary: We show that an important classical fixed-point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows (see [the second author, Fixed point theory and trace for bicategories. Paris: Société Mathématique de France (SMF) (2010; Zbl 1207.18001)]), topological Hochschild homology and Morita-invariance in bicategorical generality.Tilting objects on tubular weighted projective lines: a cluster tilting approach.https://zbmath.org/1460.140042021-06-15T18:09:00+00:00"Chen, Jianmin"https://zbmath.org/authors/?q=ai:chen.jianmin"Lin, Yanan"https://zbmath.org/authors/?q=ai:lin.yanan"Liu, Pin"https://zbmath.org/authors/?q=ai:liu.pin"Ruan, Shiquan"https://zbmath.org/authors/?q=ai:ruan.shiquanSummary: Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type \((2, 2, 2, 2)\). More precisely, a tilting object consisting of rank-two bundles is constructed via the cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces.https://zbmath.org/1460.141202021-06-15T18:09:00+00:00"Allegretti, Dylan G. L."https://zbmath.org/authors/?q=ai:allegretti.dylan-g-lThe author considers two spaces associated to a quiver with potential: a complex manifold parametrizing Bridgeland stability conditions on a certain \(3\)-Calabi-Yau triangulated category, and a cluster variety. They can be interpreted as moduli spaces of geometric structures on surfaces. \textit{T. Bridgeland} and \textit{I. Smith} [Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)] have shown that the space of stability conditions is isomorphic to a moduli space of meromorphic quadratic differentials, while \textit{V. V. Fock} and \textit{A. B. Goncharov} [Invent. Math. 175, No. 2, 223--286 (2009; Zbl 1183.14037)] have proved that the cluster variety is birational to a moduli space of local systems equipped with additional framing data. The author shows that if the quiver with potential arises from an ideal triangulation of a marked bordered surface, then one can construct a natural map from a dense subset of the space of stability conditions to the cluster variety. Using this construction, he gives solutions to a family of Riemann-Hilbert problems arising in Donaldson-Thomas theory.
Reviewer: Vladimir P. Kostov (Nice)Remarks on the homological mirror symmetry for tori.https://zbmath.org/1460.140882021-06-15T18:09:00+00:00"Kobayashi, Kazushi"https://zbmath.org/authors/?q=ai:kobayashi.kazushiSummary: Let us consider an \(n\)-dimensional complex torus \(T_{J=T}^{2 n}:=\mathbb{C}^n \slash 2\pi(\mathbb{Z}^n\oplus T\mathbb{Z}^n)\). Here, \(T\) is a complex matrix of order \(n\) whose imaginary part is positive definite. In particular, when we consider the case \(n=1\), the complexified symplectic form of a mirror partner of \(T_{J=T}^2\) is defined by using \(-\frac{1}{T}\) or \(T\). However, if we assume \(n\geq 2\) and that \(T\) is a singular matrix, we cannot define a mirror partner of \(T_{J=T}^{2n}\) as a natural generalization of the case \(n=1\) to the higher dimensional case. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.Matrix factorizations for self-orthogonal categories of modules.https://zbmath.org/1460.130212021-06-15T18:09:00+00:00"Bergh, Petter Andreas"https://zbmath.org/authors/?q=ai:bergh.petter-andreas"Thompson, Peder"https://zbmath.org/authors/?q=ai:thompson.pederA functorial approach to categorical resolutions.https://zbmath.org/1460.180122021-06-15T18:09:00+00:00"Hafezi, Rasool"https://zbmath.org/authors/?q=ai:hafezi.rasool"Keshavarz, Mohammad Hossein"https://zbmath.org/authors/?q=ai:keshavarz.mohammad-hSummary: Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by \textit{M. Auslander} [Representation dimension of Artin algebras. With the assistance of Bernice Auslander. London: Queen Mary College (1971; Zbl 0331.16026)], to categorical resolutions.Curved Rickard complexes and link homologies.https://zbmath.org/1460.570152021-06-15T18:09:00+00:00"Cautis, Sabin"https://zbmath.org/authors/?q=ai:cautis.sabin"Lauda, Aaron D."https://zbmath.org/authors/?q=ai:lauda.aaron-d"Sussan, Joshua"https://zbmath.org/authors/?q=ai:sussan.joshuaA Rickard complex is a certain complex of bimodules which, in the context of BGG category \(\mathcal{O}\) and, more generally, categorified quantum groups, generates a categorical braid group action. The paper under review studies certain deformations of Rickard complexes, which are called curved Rickard complexes in the paper. This gives rise to deformations of various types of link homologies and allows one to conceptually connect such deformations to the classical (undeformed) versions of these homology theories.
Reviewer: Volodymyr Mazorchuk (Uppsala)On the extension dimension of module categories.https://zbmath.org/1460.180132021-06-15T18:09:00+00:00"Peng, Yeyang"https://zbmath.org/authors/?q=ai:peng.yeyang"Zhao, Tiwei"https://zbmath.org/authors/?q=ai:zhao.tiweiLet \(\mathcal{A}\) be an abelian \textit{length} category -- that is, which is essentially small and such that each object has finite length. There are several notions of \textit{dimension} for such a category. This article deals with the \textit{extension dimension}, defined to be \(\le n\) if there is an object \(T\) in \(\mathcal{A}\) such that each object of this category has a filtration of length \(\le n+1\) whose subquotients are direct summands of finite direct sums of copies of \(T\).
The main result of the article (\textit{Theorem 1.1}) gives an upper bound for the extension dimension of the category of finite modules over an Artin algebra which involves a class of simple modules with finite projective and injective dimensions.
Reviewer: Aurelien Djament (Villeneuve d'Ascq)