We study the moduli space of stable sheaves of Euler characteristic one, supported on curves of degree (3,3) contained in a smooth quadric surface. We compute its Hodge numbers by investigating the variation of the moduli spaces of α-semi-stable coherent systems. We find the classification of the semi-stable sheaves by means of extensions or resolutions.

Let G be a group. Let X be a connected algebraic group over an algebraically closed field K. Denote by A=X(K) the set of K-points of X. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over (G,X,K). They are cellular automata τ:AG→AG whose local defining map is induced by a homomorphism of algebraic groups XM→X where M⊂G is a finite memory set of τ

Let k be an algebraically closed field. Let C be an irreducible smooth projective curve over k. Let E be a locally free sheaf on C of rank ≥2. Fix an integer d≥2. Let Q denote the Quot scheme parameterizing torsion quotients of E of degree d. In this article we compute the S-fundamental group scheme of Q.

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions

We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a proof for the known fact that any abelian category is extension full in its ind-completion.

For a positive integer N, let C(N) be the subgroup of J0(N) generated by the equivalence classes of cuspidal divisors of degree 0 and C(N)(Q):=C(N)∩J0(N)(Q) be its Q-rational subgroup. Let also CQ(N) be the subgroup of C(N)(Q) generated by Q-rational cuspidal divisors. We prove that when N=n2M for some integer n dividing 24 and some squarefree integer M, the two groups C(N)(Q) and CQ(N) are equal.

Over a commutative noetherian ring R of finite Krull dimension, we show that every complex of flat cotorsion R-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion

Let k be an algebraically closed field and A a Z-graded finitely generated simple k-algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z-graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove

For an n-dimensional Leibniz/Lie algebra h over a field k we introduce a new invariant A(h), called the universal algebra of h, as a quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by n3 polynomials. We prove that A(h) admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h. The new object A(h) is the key tool

We consider a block B of a finite group with defect group D≅(C2m)n and inertial quotient E containing a Singer cycle (an element of order 2n−1). This implies E=E⋊F, where E≅C2n−1, F≤Cn, and E acts transitively on the elements in D of order 2. We classify the basic Morita equivalence classes of B over a complete discrete valuation ring O: when m=1, B is basic Morita equivalent to the principal block

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products A↪A⋈H↩H, double cross coproducts , biproducts and bicrossproducts on the assumption that the factors have strongly bicovariant calculi Ω(A),Ω(H) (or a braided version Ω(B)). We use super versions of each of the constructions. Moreover,

In the present article, we investigate the following deformation problem. Let (R,m) be a local (graded local) Noetherian ring with a (homogeneous) regular element y∈m and assume that R/yR is quasi-Gorenstein. Then is R quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition

In this paper, we study derived versions of the fusion category associated to Lusztig's quantum group Uq. The categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to Grothendieck rings. On the derived level it turns out that it is possible to define fusion for Uq without using the notion of tilting modules. Hence, we arrive at a definition of the fusion

We show that a profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x∈G there is a prime p (depending on x) such that CG(x) is virtually pro-p. Further, if G is a profinite group in which each element has either finite or prime power (possibly infinite) order, then G is either torsion or virtually pro-p for some prime p. A detailed description of profinite groups

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only

In this paper, we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group GLn(C). We then use the properties of a generalized version of the Hodge dual operator to determine an explicit basis for each of these algebras, and by restriction, we obtain an explicit basis for each of the irreducible rational representations

An ideal I of a commutative ring R with identity is called an SFT (strong finite type) ideal if there exist a finitely generated ideal J of R with J⊆I and a positive integer k such that ak∈J for each a∈I. A ring R is called an SFT ring if every ideal of R is an SFT ideal. Arnold showed that if R is a non-SFT ring, then the Krull dimension of the power series ring R〚X〛 is infinite. Let C be the class

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a “large” family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s=1 and irreducible singular fibre with 4 nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe

We determine the structure of the fibered biset functor sending a finite group G to the complex vector space of complex valued class functions of G. Previously, it is studied as a biset functor by Bouc and as a C×-fibered biset functor by Boltje and the second author. In this paper, we complete the study by considering proper non-trivial subgroups of C× as the fiber group. As a corollary, we obtain

Let k be an infinite field and I⊂k[x1,…,xn] be a non-zero ideal such that dim V(I)=q≥0. Denote by (f1,…,fs) a set of generators of I. One can see that in the set I∩k[x1,...,xq+1] there exist non-zero polynomials, depending only on these q+1 variables. We aim to bound the minimal degree of the polynomials of this type, and of a Bézout (i.e. membership) relation expressing such a polynomial as a combination

We give sufficient conditions to ensure that the ideal Φ(E′) of E′-phantom maps in a locally λ-presentable exact category (A,E) is a (special) (pre)covering ideal, where E′ is an exact substructure of (A,E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular, we deal with a Zariski-local notion of

We show basic results on super-manifolds and super Lie groups over a complete field of characteristic ≠2, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category equivalence between super Lie groups and Harish-Chandra pairs, which is applied especially to construct the Hopf super-algebra of all analytic representative functions on a super

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs G whose toric edge ideal IG is generated by quadrics, showing that every cycle of G of length at least 6 must have a chord. This corresponds to the Green-Lazarsfeld condition N1. In this paper, we investigate the higher syzygies of IG and give combinatorial descriptions of the Green-Lazarsfeld conditions Np of toric edge

We prove a conjecture of Lecouvey, which proposes a closed, positive combinatorial formula for symplectic Kostka–Foulkes polynomials, in the case of rows of arbitrary weight. To show this, we construct a new algorithm for computing cocyclage in terms of which the conjecture is described. Our algorithm is free of local constraints, which were the main obstacle in Lecouvey's original construction. In

We show that pure monomorphisms are cofibrantly generated—generated from a set of morphisms by pushouts, transfinite composition, and retracts—in any locally finitely presentable additive category. In particular, this is true in any category of R-modules. On the other hand, the classes of all monomorphisms and regular monomorphisms in a locally finitely presentable additive category need not be well-behaved

For a monoid M, we denote by G(M) the group of units, E(M) the submonoid generated by the idempotents, and GL(M) and GR(M) the submonoids consisting of all left or right units. Writing M for the (monoidal) category of monoids, G, E, GL and GR are all (monoidal) functors M→M. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units

We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme X, the graded ring that our functor associates to X is related to the associated graded ring of the γ-filtration on the Grothendieck ring of finite rank locally free sheaves on X via a Grothendieck-Riemann-Roch

We give the algebraic classification of alternative, left alternative, Jordan, bicommutative, left commutative, assosymmetric, Novikov and left symmetric central extensions of null-filiform associative algebras.

Let X be a smooth algebraic variety over an arbitrary field. Let φ be the canonical surjective homomorphism of the Chow ring of X onto the ring associated with the Chow filtration on the Grothendieck ring K(X). We remark that φ is injective if and only if the connective K-theory CK(X) coincides with the terms of the Chow filtration on K(X). As a consequence, CK(X) turns out to be computed for numerous

We consider the deformed versions of the classical Howe dual pairs (O(r,C),sl(2,C)) and (O(r,C),spo(2|2,C)) in the context of a rational Cherednik algebra Hc=Hc(W,h) associated to a finite Coxeter group W at the parameters c and t=1. For the first pair, we compute the centraliser of the well-known copy of s≅sl(2,C) inside Hc. For the second pair, we show that the classical copy of g≅spo(2|2,C) inside

We classify the finitely presented restricted Lie algebras among the finitely generated restricted Lie algebras L that are split extension of abelian-by-abelian restricted Lie algebras.

We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra.

In this paper, an explicit construction of a countable parafree Lie algebra with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over Z is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial H2.

We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete intersection of pure codimension two in a smooth projective variety of dimension d over an infinite field is the degeneracy locus of d−1 sections of a rank d bundle.

For all sufficiently large odd integers n, the following version of Higman's embedding theorem is proved in the variety Bn of all groups satisfying the identity xn=1. A finitely generated group G from Bn has a presentation G=〈A|R〉 with a finite set of generators A and a recursively enumerable set R of defining relations if and only if it is a subgroup of a group H finitely presented in the variety

Let R(h) denote the polynomial ring in variables x1,…,xh over a specified field K. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with x1>⋯>xh. Given a fixed homogeneous ideal I in R(h), for each d there is unique lex ideal generated in degree at most d whose Hilbert function agrees with the Hilbert function of I up to degree d. When we consider IR(N)

Let F be a field containing a primitive m-th root of the unit. We characterize the actions of a Taft's algebra Hm of a certain order m on finite dimensional arbitrary algebras. We describe the action in terms of gradings and actions by skew-derivations. Moreover we prove the associative algebra UT2 of 2×2 upper triangular matrices with entries from F does not generate a variety of Hm-module algebras

We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that

Let A be a finite dimensional unital commutative associative algebra and let B be a finite dimensional vertex A-algebroid such that its Levi factor is isomorphic to sl2. Under suitable conditions, we construct an indecomposable non-simple N-graded vertex algebra VB‾ from the N-graded vertex algebra VB associated with the vertex A-algebroid B. We show that this indecomposable non-simple N-graded vertex

We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them.

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra grv(R) of a subring (R,m) of a valuation ring Ov, for which Kv:=Ov/mv=R/m, is isomorphic to Kv[tv(R)], where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases

We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar

Let p be a prime, K/Qp a finite extension, G a uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by CG(Z2(G)), the centraliser of the second term in the upper central series for G.

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice Xn which is transversal to the nilpotent orbit of Jordan type [2n−2,1,1] in the nilpotent cone of sp2n for some n≥2. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds.

Let K be an algebraically closed field and G=SL2(K). Let E be the natural module and SrE the rth symmetric power. We consider here, for r,s≥0, the tensor product of SrE and the dual of SsE. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when K is a field of positive characteristic. We show that each indecomposable component

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group G and the same for a subgroup H, often the normalizer of a p-subgroup of G. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence

In this paper we describe solvable Leibniz algebras whose quotient algebra by one-dimensional ideal is a Lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical. We prove that such Leibniz algebra is unique and centerless. Also it is proved that the first and the second cohomology groups of the algebra with coefficients in the adjoint representation is trivial

Let G and A be finite groups with A acting on G by automorphisms. In this paper we introduce the concept of “good action”; namely we say the action of A on G is good, if H=[H,B]CH(B) for every subgroup B of A and every B-invariant subgroup H of G. This definition allows us to prove a new noncoprime Hall-Higman type theorem. If A is a nilpotent group acting on the finite solvable group G with CG(A)=1

In this paper we present a matricial result that generalizes Hironaka's game and Perron transforms simultaneously. We also show how one can deduce the various forms in which the algorithm of Perron appears in proofs of local uniformization from our main result.

An even Artin group is a group which has a presentation with relations of the form (st)n=(ts)n with n≥1. With a group G we associate a Lie Z-algebra TGr(G). This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TGr(G). By means of this presentation we obtain information about the diagram of G. We

Given an integer n>0 we give a computable injective listing of the isomorphism types of all computable abelian p-groups of Ulm type ≤n. We prove a similar result for certain classes of profinite groups.

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, C a stably quasi-serial component (i.e. its stable part is a tube) of rank n of the Auslander-Reiten quiver of A, and S be a simple-minded system of the stable module category mod_A. We show that the intersection S∩C is of size strictly less than n, and consists only of modules with quasi-length strictly less than

Let Ω be a finite set and T(Ω) be the full transformation monoid on Ω. The rank of a transformation t∈T(Ω) is the natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 the semigroup generated by A. Let k be a fixed natural number such that 2≤k≤|Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformations t∈T(Ω), every element in St:=〈G,t〉

In this paper we develop a structure theory for a class of table algebras, i.e. the so-called quasi-thin table algebras. They are direct generalizations of quasi-thin association schemes. Although this class of table algebras is rather close to finite groups, its structural properties are quite different from those of finite groups. The thin radical and thin residue give rise to two group structures

The purpose of this paper is to show that for a complete intersection curve C in projective space (other than a few exceptions stated below), any morphism f:C→Pr satisfying deg⁡f⁎OPr(1)

We study the finite groups all the Schmidt subgroups (the minimal non-nilpotent subgroups) of which are σ-subnormal. Thus an affirmative answer is obtained to Problem 19.85 from the Kourovka Notebook.

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic p different from 2. Given an involution φ on E, denote by Id(E,φ) and C(E,φ) the set of all ⁎-polynomial identities and ⁎-central polynomials of (E,φ) respectively. In this paper we describe Id(E,φ) and C(E,φ). Moreover, we prove that C(E,φ) is not finitely generated as a T(⁎)-space if p>2.

We study Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of Sturmfels and Zelevinsky in [27]. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of