We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

Let $\mathscr{C}$ be a $(d+2)$ -angulated category with d-suspension functor $\Sigma^d$ . Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$ -angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander–Reiten $(d+2)$ -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is d-Auslander–Reiten translation. These results generalize work by Bondal–Kapranov and Reiten–Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$ , the quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$ -angulated category if and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$ -mutation pair, and if and only if $\tau_d\mathscr{X} =\mathscr{X}$ .

We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

We say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups.

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, Kn, $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each Ki is scattered.

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

Let G be a group and A a set equipped with a collection of finitary operations. We study cellular automata $$\tau :{A^G} \to {A^G}$$ that preserve the operations AG of induced componentwise from the operations of A. We show τ that is an endomorphism of AG if and only if its local function is a homomorphism. When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting of all such endomorphic cellular automata, is isomorphic to the direct limit of Hom(AS, A), where S runs among all finite subsets of G. In particular, when A is an R-module, we show that EndCA(G;A) is isomorphic to the group algebra $${\rm{End}}(A)[G]$$ . Moreover, when A is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over AG admitting a memory set S is precisely $${(k|S|)^k}$$ , where k is the number of atoms of A.

It is conjectured that the central quotient of any irreducible Artin–Tits group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin–Tits groups that are known to be CAT(0) groups by a result of Brady and McCammond, that is, Artin–Tits groups associated with graphs having no 3-cycles and Artin–Tits groups of almost large type associated with graphs admitting appropriate directions. In particular, the latter family contains Artin–Tits groups of large type associated with cones over square-free bipartite graphs.

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimal d-Auslander–Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.

The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).

We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two.

Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.