Dinatural transformations, which generalise the ubiquitous natural transformations to the case where the domain and codomain functors are of mixed variance, fail to compose in general; this has been known since they were discovered by Dubuc and Street in 1970. Many ad hoc solutions to this remarkable shortcoming have been found, but a general theory of compositionality was missing until Petric, in

We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual dual of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra

The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular

Let $\mathcal{F}$ be a vector bundle on a complex projective algebraic variety $X$. If $\mathcal{F}$ deforms along a first order deformation of $X$, its Mukai vector remains of Hodge type along this deformation. We prove an analogous statement for all polyvector fields, not only for those in $H^1(X,T_X)$ corresponding to deformations of the complex structure. This answers a question of Markman. We

Abstract We further the techniques developed by Etingof, Nikshych, and Ostrik in [7] to classify the C -based equivalences between two G-graded extensions of C , in terms of cohomological data. As a warm up for the proof of our classification result we reprove the classification of graded extensions of a fusion category, making extensive use of graphical calculus. We end by providing several practical

For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed by Greenberg, we deduce the existence of Galois extensions of $\mathbf{Q}$ with Galois group isomorphic to an open subgroup of $GL_n(\mathbf{Z_p})$, for $n =4$ and

For finite semidistributive lattices the map $\kappa$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $\kappa$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $\kappa$-map is well-defined

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \Bbbk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving

In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein

We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We establish an adjunction

Abstract For a finite group G, a subgroup P of G is 2-minimal if B P , where B = N G ( S ) for some Sylow 2-subgroup S of G, and B is contained in a unique maximal subgroup of P. Here we give a detailed and explicit description of all the 2-minimal subgroups for finite symplectic groups defined over a field of odd characteristic.

It is known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions

In this paper, we study the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ of a Gorenstein monomial algebra $A$. Firstly, for a general positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits a silting object. Secondly, for $A=kQ/I$ being a $1$-Gorenstein monomial algebra, which

Let $ G$ be a finite group and $p$ be a prime. Let $ \mathrm{Vo}(G) $ denote the set of the orders of vanishing elements, $\mathrm{Vo}_{p} (G)$ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements divisible by $p$ and $\mathrm{Vo}_{p'} (G) $ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements not divisible by $p$. Dolfi, Pacifi, Sanus

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. The

In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of $\mathcal A(\mathcal S)$; (ii) there exists a bijection from the set of cluster variables of a generalized

Abstract A new method is established to construct recollements of derived categories of rings from exact contexts and noncommutative tensor products. This method is applicable to a large variety of situations, including Milnor squares, localizations, ring extensions and ring epimorphisms.

Abstract One of the main goals of this paper is to present the relation between limit key polynomials and MacLane-Vaquie limit key polynomials. This is a continuation of the work started in [3] , where the equivalent result for key polynomials (which are not limit) is proved. Moreover, we present a result (Theorem 1.1) that generalizes various results in the literature.

We characterize the $k$-torsion freeness of the module of differentials of order $n$ of a point of a hypersurface in terms of the singular locus of the corresponding local ring.

We compute the $\mathbb{A}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of $\

Let $p$ be a prime number, and let $A$ be a ring in which $p$ is nilpotent. In this paper, we consider the maps $$K_{q+1}(A[x]/(x^m), (x))\to K_{q+1}(A[x]/(x^{mn}), (x)),$$induced by the ring homomorphism $A[x]/(x^{m})\to A[x]/(x^{mn})$, $x\mapsto x^n$. We evaluate these maps, up to extension, for general $A$ in terms of topological Hochschild homology, and for regular $\mathbb{F}_p$-algebras $A$,

We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra

Abstract Automorphisms of free bicommutative algebras over an arbitrary field are studied. Wild automorphisms are constructed in two-generated and three-generated free bicommutative algebras. Moreover, for any n ≥ 2 , a wild automorphism is constructed in the n-generated free associative bicommutative algebra which is not stably tame and can not be lifted to an automorphism of the n-generated free

In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this form is equal to the number of positive divisors of $q+1$ and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plucker algebras. We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Plucker ideal, hence a toric degeneration for the flag variety.

Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*:V\to V$ such that (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i+ V_{i+1}$ for $0 \leq i

Abstract The Cohn localization is a process of adjoining universal inverse morphisms to the category of R-modules over a noncommutative ring R. Finding specific models for universal localizations is, in general, difficult. For certain classes of rings, however, there exist constructions that are more accessible. We define such a class of rings, the generalized triangular matrix rings, and provide complete

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero two-sided ideal $I$ of $R$. In this paper, we show

We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.

Let $Z$ be the center of a nonnoetherian dimer algebra on a torus. Although $Z$ itself is also nonnoetherian, we show that it has Krull dimension $3$, and is locally noetherian on an open dense set of $\operatorname{Max}Z$. Furthermore, we show that the reduced center $Z/\operatorname{nil}Z$ is depicted by a Gorenstein singularity, and contains precisely one closed point of positive geometric dimension

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur

We examine the utility of a new family of basis functions for use with the Complex Variable Boundary Element Method (CVBEM) and other mesh-free numerical methods for solving partial differential equations. The family of polygamma functions have found use in mathematics since as early as 1730 when James Stirling related the digamma function to the factorial function [1]. Now, we propose using the digamma

In this paper, wave scattering by an H-type porous barrier having nonlinear pressure drop boundary condition is analysed within the framework of small amplitude water wave theory. The H-type barrier is constructed using multiple thin (near zero thickness) rigid porous plates which are termed degenerate boundaries. The boundary value problem is solved using an iterative dual boundary element method

We propose a counterexample to the Gröbner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V[X,Y] equipped with some irrational monomial order <, such that the ideal LT<(J) generated by the leading terms of the elements of J is not finitely generated.

By work of Belyĭ [2], the absolute Galois group GQ=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ2), the automorphism group of the free profinite group Fˆ2 on two generators. The image of GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge

In this work, an algebraic method to prove the existence of left eigenvalues for the quaternionic matrix is investigated. The left eigenvalues of a n×n quaternionic matrix can be derived by solving the zeros of a general quaternionic polynomial of degree 2n. Using the Study’s determinant, it can be found by solving the zeros of quaternionic polynomials of degree at most n or of rational functions.

ABSTRACT Identifiability holds for the k-secant variety σk(X) of an embedded variety X⊂Pr if a general q∈σk(X) is in the linear span of a unique subset of X with cardinality k. Identifiability is true if the general tangential k-contact locus Γk⊂X has dimension 0. Under certain assumptions on dim⁡σx(X) for some specific x>k, we get dim⁡Γk=0. Here we give some conditions which exclude the case Γx a

In this paper, we study 2-local superderivations on the Lie superalgebra of Block type S(q), which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local superderivations on S(q) are superderivations.

Let L be a finite-dimensional nilpotent Lie algebra of class two over a field F of characteristic different from two. The aim of the present paper is to give an upper bound for the dimension of the center factor L/Z(L) of a capable Lie algebra L. As a result, we also find a lower bound for the dimension of the derived subalgebra L2 for a capable Lie algebra L. Furthermore, we also give some criteria

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A∗NX∗MB=C with a tensor inequality constraint D∗NX∗ME≥F. The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm.

Abstract We introduce a concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via a cohomology theory.

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K[S]. We define the type of S, type(S), in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K[S] is Cohen-Macaulay

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [Lib20b]. We hint that 0-dimensional adjunction inhabits the categorified one.

Let p be a prime integer and F the function field in two algebraically independent variables over a smaller field F0. We prove that if char(F0)=p⩾3 then there exist p2−1 cyclic algebras of degree p over F that have no maximal subfield in common, and if char(F0)=0 then there exist p2 cyclic algebras of degree p over F that have no maximal subfield in common.

We prove that given C a presentably symmetric monoidal ∞-category, and any essentially small ∞-operad O, the ∞-category of O-algebras in C is enriched, tensored and cotensored over the presentably symmetric monoidal ∞-category of O-coalgebras in C. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, López

We give a complete description of the primary degenerations and non-degenerations between the 3-dimensional nilpotent algebras, the 4-dimensional nilpotent commutative algebras and the 5-dimensional nilpotent anticommutative algebras over C. It follows that all the varieties under consideration are irreducible and determined by the rigid algebra N2, the family of algebras C19(α) and the rigid algebra

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units R×, we study the algebra AR(G,σ) consisting of locally constant compactly supported R-valued

We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of Toën [32]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category SH(k) in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ΣP1∞X+, where X+=X⨿⁎, for a smooth scheme X∈Smk over an infinite perfect field k, is computed. The computation by Garkusha, Neshitov

In a triangulated category, cofibre fill-ins always exist. Neeman showed that there is always at least one “good” fill-in, i.e., one whose mapping cone is exact. Verdier constructed a fill-in of a particular form in his proof of the 4×4 lemma, which we call “Verdier good”. We show that for several classes of morphisms of exact triangles, the notions of good and Verdier good agree. We prove a lifting

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let q be a prime power, n be a positive integer and σ be a generator of Gal(Fqn:Fq). In this paper we provide closed formulas for the coefficients of a σ-trinomial f over Fqn which ensure that the dimension of the kernel of f equals its σ-degree, that is linearized polynomials

In this paper, a construction of \((n,k,\delta )\) LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an \((n,k,\delta )\) convolutional code, where \(n\in {\mathbb

Let R be a commutative ring with 1≠0. A proper ideal I of R is said to be a strongly quasi-primary ideal if, whenever a,b∈R with ab∈I, then either a2∈I or b∈I. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of R is prime. Many examples

The b-Exact Multicover problem takes a universe U of n elements, a family \(\mathcal {F}\) of m subsets of U, a function \({\textsf {dem}}: U \rightarrow \{1,\ldots ,b\}\) and a positive integer k, and decides whether there exists a subfamily(set cover) \(\mathcal {F}^{\prime }\) of size at most k such that each element u ∈ U is covered by exactly dem(u) sets of \(\mathcal {F}^{\prime }\). The b-Exact

Let V be a finite-dimensional vector space over a finite field, and suppose G≤ΓL(V) is a group with a unique subnormal quasisimple subgroup E(G) that is absolutely irreducible on V. A base for G is a set of vectors B⊆V with pointwise stabiliser GB=1. If G has a base of size 1, we say that it has a regular orbit on V. In this paper we investigate the minimal base size of groups G with E(G)/Z(E(G))≅PSLn(q)

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its

Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known—it is what is known as a Tararin group. By defining semiconjugacy of circular orderings in a general setting (that is, for arbitrary circular orderings of groups that may not act on S1), we can view the subspace of left-orderings

Let F be a field of characteristic different from two. Suppose that N2 is the category of finite-dimensional nilpotent Lie algebras of class two over the field F and that ALT is the category of alternating bilinear maps of F-vector spaces. We establish a relation between the category N2 and the category ALT. Then we show that the problem of determining the capability of these Lie algebras reduces to

For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover,

Let V be a valuation domain with quotient field K. We show how to describe all extensions of V to K(X) when the V-adic completion Kˆ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced