Artin's Theorem Research Articles

Graded group actions and generalized H-actions compatible with gradings

We introduce the notion of a graded group action on a graded algebra or, which is the same, a group action by graded pseudoautomorphisms. An algebra with such an action is a natural generalization of an algebra with a super- or a pseudoinvolution. We study groups of graded pseudoautomorphisms, show that the Jacobson radical of a group graded finite dimensional associative algebra A over a field of characteristic 0 is stable under graded pseudoautomorphisms, prove the invariant version of the Wedderburn–Artin Theorem and the analog of Amitsur's conjecture for the codimension growth of graded polynomial G-identities in such algebras A with a graded action of a group G.

Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures

Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures

Theory and applications of linearized multivariate skew polynomials

In this work, linearized multivariate skew polynomials over division rings are introduced. Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. Their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independent sets are those given by right linearly independent sets when partitioned into conjugacy classes. Hence finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces, extending Lam and Leroy's results on univariate skew polynomials. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions of linearized multivariate skew polynomials, which in turn translate into matrix products over the corresponding centralizers. Later, linearized multivariate Vandermonde matrices are introduced, which generalize multivariate Vandermonde, Moore and Wronskian matrices. The previous results explicitly give their ranks in general. P-Galois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. Three Galois-theoretic results are generalized to such extensions: Artin's theorem on extension degrees, the Galois correspondence and Hilbert's Theorem 90.

Numerical contraction for orbifold surfaces

We study singularities and Artin's contraction theorem for orbifold surfaces. Our main result has a consequence which is in the direction of the birational Minimal Model Program for b-terminal orbifold surfaces. For example, we ascertain the nature of extremal contractions for such $b$-terminal pairs.

A Generalization of the Artin Theorem

A Generalization of the Artin Theorem

Ring structure theorems and arithmetic comprehension

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their $$\Pi ^{0}_{1}$$ subsets and show that Schur’s Lemma is provable in $$\mathrm RCA_{0}$$ . A ring R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module $$_{R}R$$ is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics.

A Luna étale slice theorem for algebraic stacks

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is \'etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

TATE’S CONJECTURE AND THE TATE–SHAFAREVICH GROUP OVER GLOBAL FUNCTION FIELDS

AbstractLet ${\mathcal{X}}$ be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of ${\mathcal{X}}$ is finite if and only Tate’s conjecture for divisors on $X$ holds and the Tate–Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.

Artin theorem for semigroups

In this paper, we prove an Artin type theorem for semigroups. Namely, we consider the concepts of hyperalternative and hyperassociative semigroups and prove that every two elements in any hyperalternative semigroup generate a hyperassociative subsemigroup. As a consequence, we characterize all hyperalternative semigroups, and prove that these semigroups form a variety of semigroups with four identities.

The Wedderburn–Artin Theorem for Paragraded Rings

In this paper, we prove the paragraded version of the Wedderburn–Artin theorem. Following the methods known from the abstract case, we first prove the density theorem and observe the matrix rings whose entries are from a paragraded ring. However, in order to arrive at the desired structure theorem, we introduce the notion of a Jacobson radical of a paragraded ring and prove some properties which are analogous to the abstract case. In the process, we study the faithful and irreducible paragraded modules over noncommutative paragraded rings and prove the paragraded version of the well-known Schur lemma.

A short proof of the Wedderburn-Artin theorem

ABSTRACTWe give a short proof of the Wedderburn–Artin theorem from the viewpoint of completely reducible modules.

Weak associativity and deformation quantization

Non-commutativity and non-associativity are quite natural in string theory. For open strings it appears due to the presence of non-vanishing background two-form in the world volume of Dirichlet brane, while in closed string theory the flux compactifications with non-vanishing three-form also lead to non-geometric backgrounds. In this paper, working in the framework of deformation quantization, we study the violation of associativity imposing the condition that the associator of three elements should vanish whenever each two of them are equal. The corresponding star products are called alternative and satisfy important for physical applications properties like the Moufang identities, alternative identities, Artin's theorem, etc. The condition of alternativity is invariant under the gauge transformations, just like it happens in the associative case. The price to pay is the restriction on the non-associative algebra which can be represented by the alternative star product, it should satisfy the Malcev identity. The example of nontrivial Malcev algebra is the algebra of imaginary octonions. For this case we construct an explicit expression of the non-associative and alternative star product. We also discuss the quantization of Malcev–Poisson algebras of general form, study its properties and provide the lower order expression for the alternative star product. To conclude we define the integration on the algebra of the alternative star products and show that the integrated associator vanishes.

Some results on deformations of sections of vector bundles

Let E be a vector bundle on a smooth complex projective variety X. We study the family of sections $$s_t\in H^0(E\otimes L_t)$$ where $$L_t\in Pic^0(X)$$ is a family of topologically trivial line bundle and $$L_0={\mathcal {O}}_X,$$ that is, we study deformations of $$s=s_0$$ . By applying the approximation theorem of Artin (Invent Math 5:277–291, 1968) we give a transversality condition that generalizes the semi-regularity of an effective Cartier divisor. Moreover, we obtain another proof of the Severi–Kodaira–Spencer theorem (Bloch In Invent Math 17:51–66, 1972). We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja (Duke Math J 164(3):541–568, 2015) and a proof of a piece of the generic vanishing theorems (Green and Lazarsfeld, Invent Math 90:389–407, 1987) and (Green and Lazarsfeld, J Am Math Soc 4:87–103, 1991) for the canonical bundle. We extend also to higher dimension a result given in (Mendes-Lopes et al. In Geo Topol 17:1205:1223, 2013) on the base locus of the paracanonical base locus for surfaces.

Commutators, commutativity and dimension in the socle of a Banach algebra: A generalized Wedderburn–Artin and Shoda's theorem

As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra are obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn–Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. {xy−yx:x,y∈SocA}) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity.

Artin's theorem for f-rings

The main result states that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Induction formulae for Mackey functors with applications to representations of the twisted quantum double of a finite group

In the theory of canonical induction formulae for Mackey functors, Boltje [4] demonstrated that the plus constructions, together with the mark morphism, are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups. In this paper, we present a short exact sequence for the plus constructions derived from Cauchy–Frobenius lemma, and apply it to the proof of Boltje's integrality result for canonical induction formulae. The methods appearing in Boltje's theory, combined with the Dress construction for Mackey functors, are applicable to induction theorems on representations of the twisted quantum double of a finite group. As a sequel to such a research, we describe canonical versions of two induction theorems whose origins are Artin's induction theorem and Brauer's induction theorem on C-characters of a finite group.

Supertropical linear algebra

The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of “ghost surpasses”. Special attention is paid to the various notions of “base”, which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in various d-bases may differ, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of “critical” elements. Then we turn to orthogonality of vectors, which leads to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence. We also obtain a supertropical version of a theorem of Artin, which says that if g-orthogonality is a symmetric relation, then the underlying bilinear form is (supertropically) symmetric.

Wedderburn quasialgebras

The Wedderburn–Artin Theorem for G-graded quasialgebras is proved. This provides new examples of quasialgebras.

Crossed Burnside rings and Bouc's construction of Green functors

We use a formula for primitive idempotents of the crossed Burnside ring given by F. Oda and T. Yoshida, and a result from the theory of Green functors obtained by S. Bouc to prove an induction theorem for the Drinfel'd double for a finite group over the complex field. We obtain Artin's induction theorem for group algebras as a corollary of the theorem.

Identities of Conformal Algebras and Pseudoalgebras

For a given conformal algebra C, we write the correspondence between identities of the coefficient algebra Coeff C and identities of C itself as a pseudoalgebra. In particular, we write the defining relations of Jordan, alternative, and Mal'cev conformal algebras, and show that the analogue of the Artin's Theorem does not hold for alternative conformal algebras.