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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. [ABSTRACT FROM AUTHOR]

We explain how any Artin stack X over Q extends to a functor on non-negatively graded commutative cochain algebras, which we think of as functions on Lie algebroids or stacky affine schemes. There is a notion of étale morphisms for these CDGAs, and Artin stacks admit étale atlases by stacky affines, giving rise to a small étale site of stacky affines over X. This site has the same quasi-coherent sheaves as X and leads to efficient formulations of shifted Poisson structures, differential operators and deformation quantisations for Artin stacks. There are generalisations to higher and derived stacks. We also describe analogues for differentiable and analytic stacks; in particular, a Lie groupoid naturally gives a functor on NQ-manifolds which we can use to transfer structures. In those settings, local diffeomorphisms and biholomorphisms are the analogues of étale morphisms. This note mostly elaborates constructions scattered across several of the author's papers, but with an emphasis on the functor of points perspective. New results include consistency checks showing that the induced notions of structures such as vector bundles or torsors on a stacky affine scheme coincide with familiar definitions in terms of flat connections. [ABSTRACT FROM AUTHOR]

Let Λ be an Artin algebra and mod - ( Gprj _ - Λ) the category of finitely presented functors over the stable category Gprj _ - Λ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which mod - ( Gprj _ - Λ) is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations Gprj _ (Q , Λ) of a finite acyclic quiver Q to the category of representations rep (Q , Gprj _ - Λ) over Gprj _ - Λ , provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra Λ Q of the G-semisimple algebra Λ is CM -finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj (A n , Λ) of the linear quiver A n over a G-semisimple algebra Λ. We also determine almost split sequences in Gprj (A n , Λ) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj (A n , Λ). [ABSTRACT FROM AUTHOR]

Let Γ be a finite graph and let A (Γ) be the corresponding right-angled Artin group. We characterize the Hamiltonicity of Γ via the structure of the cohomology algebra of A (Γ). In doing so, we define and develop a new canonical graph associated to a matrix, which as a consequence provides a novel perspective on the matrix determinant. [ABSTRACT FROM AUTHOR]

Let T , U be two Artin algebras and M T U be a U - T -bimodule. In this paper, we study the extension dimension of the formal triangular matrix algebra Λ = ( T 0 M U ). It is proved that if M U , M T are projective and max { gl. dim T , dim mod U } ⩾ 1 , then max { dim mod T , dim mod U } ⩽ dim mod Λ ⩽ max { gl. dim T , dim mod U }. [ABSTRACT FROM AUTHOR]

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles. Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules. The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules. The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields. [ABSTRACT FROM AUTHOR]

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by the mapping cone of a chain map, which is homogeneous of degree 1 with respect to a certain grading. We also study a higher dimensional analogue of representation finite hereditary algebras called d -representation finite algebras. One source of d -representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily. [ABSTRACT FROM AUTHOR]

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension — rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types A and E are given. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, where q is a power of an odd prime p. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve y q − y = x ⋅ F (x) − λ , where F (x) is a F q -linearized polynomial and λ ∈ F q. Our results provide a characterization of the number of affine rational points of this curve in the extension F q r of F q , for gcd ⁡ (q , r) = 1. In the case F (x) = x q i − x we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the curves C k (p , a) : y p − y = x p k + 1 + a x defined over F p and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves C k (p , a). Our proof involves finding an exact formula for the number of F p n -rational points on C k (p , a) for all n , and uses a result we proved elsewhere about the number of rational points on supersingular curves. [ABSTRACT FROM AUTHOR]

For an ideal I m , n generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of R / I m , n to R / I m , n itself. The construction of this explicit embedding depends on a minimal free R -resolution of an ideal generated by I m , n . Using this embedding, we give a resolution of connected sums of several copies of certain Artin k -algebras where k is a field. [ABSTRACT FROM AUTHOR]

Let A be an Artin group and denote by SR(A) the subgroup of A which is generated by the finite type (spherical) standard parabolic subgroups on at least three standard generators. In this work we show that if the K (π , 1) Conjecture holds for SR(A) then it holds for A. The method of proof is by combinatorial curvature calculations in van Kampen diagrams. [ABSTRACT FROM AUTHOR]

We study the Artin Approximation property with constraints in a different frame. As a consequence we give a nested Artin Strong Approximation property for algebraic power series rings over a field. [ABSTRACT FROM AUTHOR]

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ϕ -dimension and ψ -dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ϕ -dimension and finite ψ -dimension. [ABSTRACT FROM AUTHOR]

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field has finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a nonzero one-sided ideal of bigger algebras. Our result can be formulated for an arbitrary ideal as follows: Let B ⊆ A be an extension of Artin algebras and I an ideal of B such that the full subcategory of B / I -modules is B -syzygy-finite. (1) If the extension is right-bounded (for example, Gpd ( A B ) < ∞ ), I A rad ( B ) ⊆ B and findim ( A ) < ∞ , then findim ( B ) < ∞ . (2) If I rad ( B ) is a left ideal of A and A is torsionless-finite, then findim ( B ) < ∞ . Particularly, if I is specified to a power of the radical of B , then our result not only generalizes some of results in the literature (see Corollary 1.2 ), but also provides new ways to detect algebras of finite finitistic dimensions. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of Q A are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type A n , D n and E 6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras. [ABSTRACT FROM AUTHOR]

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed. [ABSTRACT FROM AUTHOR]

We generalize the results on finite Frobenius rings of T. Honold (2001) [16] and some classical results of Nakayama (1939, 1941) [21,22] on Frobenius algebras over fields, and the results of J.A. Wood (2008, 1999) [31,32] on linear codes and finite Frobenius rings, to the setting of Artin algebras, and provide a unifying context for these results. We show that an Artin algebra is Frobenius if and only if its socle and top are isomorphic only as left modules (equivalently, as right modules). We show that an Artin algebra A satisfies the MacWilliams code equivalence property if and only if A is a product of a finite Frobenius ring and a quasi-Frobenius ring with no nontrivial finite representations. We use a blend of ring theoretic, combinatorial and compact group methods; in particular, inspired by the work of J.A. Wood, we show how the theory of compact groups can be used to yield ring theoretical results. [ABSTRACT FROM AUTHOR]

In this article we describe the structure of artin algebras A for which any cycle of indecomposable finitely generated (right) A -modules is finite and almost all indecomposable (finitely generated) A -modules do not lie on short paths with injective source and projective target. [ABSTRACT FROM AUTHOR]

A notion of graph-wreath product of groups is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type F n . Under various additional assumptions we show that these conditions are necessary. Our results generalise results of Bartholdi, Cornulier and Kochloukova about wreath products. Graph-wreath products of groups include classical permutational wreath products and semidirect products of right-angled Artin groups by certain groups of automorphisms amongst others. [ABSTRACT FROM AUTHOR]

In this paper we study Sigma-invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong n -link condition for a graph Γ and prove that it gives a sufficient condition for a character χ : A Γ → Z to satisfy [ χ ] ∈ Σ n (A Γ , Z). This implies that the kernel A Γ χ = ker ⁡ χ is of type FP n. We prove the homotopical version of this result as well and discuss partial results on the converse. We also provide a general formula for the free part of H n (A Γ χ ; F) as an F [ t ± 1 ] -module with the natural action induced by χ. This gives a characterization of when H n (A Γ χ ; F) is a finite dimensional vector space over F. [ABSTRACT FROM AUTHOR]

Abstract: We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type . Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture. [Copyright &y& Elsevier]

Abstract: We study and classify Artin–Schelter regular algebras of dimension five with two generators under an additional -grading by Hilbert driven Gröbner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen–Macaulay. One of the results provides an answer to Fløystad–Vatneʼs question in the context of -grading. Our results also achieve a connection between Lyndon words and Artin–Schelter regular algebras. [Copyright &y& Elsevier]

Abstract: The main objective of this paper is to study the relative derived categories from various points of view. Let be an abelian category and be a contravariantly finite subcategory of . One can define -relative derived category of , denoted by . The interesting case for us is when has enough projective objects and is the class of Gorenstein projective objects, where is known as the Gorenstein derived category of . We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickardʼs theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories. [Copyright &y& Elsevier]

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce "big objects" and prove an Auslander-type "splitting-big-objects" theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type. [ABSTRACT FROM AUTHOR]

Let Λ be an Artin algebra and T a τ -tilting Λ-module. We prove that T is a tilting module if and only if Ext Λ i (T , Fac T) = 0 for all i ≥ 1 , where Fac T is the full subcategory consisting of modules generated by T. Consequently, a τ -tilting module T of finite projective dimension is a tilting module if and only if Ext Λ i (T , T) = 0 for all i ≥ 1. Moreover, we also give an example to show that a support τ -tilting but not τ -tilting module M of finite projective dimension satisfying Ext Λ i (M , M) = 0 for all i ≥ 1 need not be a partial tilting module. [ABSTRACT FROM AUTHOR]

Abstract: We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group. [Copyright &y& Elsevier]

Abstract: Let Λ be an artin algebra. We are going to consider full subcategories of mod Λ closed under finite direct sums and under submodules with infinitely many isomorphism classes of indecomposable modules. The main result asserts that such a subcategory contains a minimal one and we exhibit some striking properties of these minimal subcategories. These results have to be considered as essential finiteness conditions for such module categories. [Copyright &y& Elsevier]

Abstract: We introduce Artin–Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin–Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases. [Copyright &y& Elsevier]

Abstract: We prove that the supersingular K3 surface of Artin invariant 1 in characteristic (where denotes an arbitrary prime) admits a model over with Picard number 21. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander–Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes. [Copyright &y& Elsevier]

Abstract: We prove that the set of n-Artin presentations has a group structure. It is a known result but it seems that it does not appear explicitly in the literature. As an application we consider a special class of integral framed links in such that and we calculate the fundamental group of the 3-manifolds obtained by integral Dehn surgery on these links. In some cases we say when the groups obtained cannot be trivial. [Copyright &y& Elsevier]

Abstract: Let and be Gorenstein Artin algebras of finite Cohen–Macaulay type. We prove that, if and are derived equivalent, then their Cohen–Macaulay Auslander algebras are also derived equivalent. [Copyright &y& Elsevier]

Abstract: We study necessary and sufficient conditions for the existence of irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the -power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary -category with tilting object, such irreducible morphisms exist if and only if is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms. [Copyright &y& Elsevier]

Abstract: We give a complete description of the Morita equivalence classes of all connected selfinjective artin algebras for which the Auslander–Reiten quiver admits a family of quasitubes having common composition factors, closed under composition factors, and consisting of modules not lying on infinite short cycles. [Copyright &y& Elsevier]

Abstract: A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel–Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary Artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel–Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of Artin algebras. As the first step, we are going to study the wild quiver with three vertices, labeled by 1, 2 and 3, and one arrow from 1 to 2 and two arrows from 2 to 3. The Gabriel–Roiter submodules of the indecomposable preprojective modules and quasi-simple modules , are described, where is a Kronecker module and is the Auslander–Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel–Roiter measures admitting no direct predecessors. [Copyright &y& Elsevier]

Abstract: We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings. [Copyright &y& Elsevier]

Abstract: We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra , denoted , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from to if is an indecomposable complex. In case is a selfinjective Artin algebra we show that for every irreducible morphism in either is split monomorphism for all or split epimorphism, for all . Moreover, we prove that all the non-trivial components of the Auslander–Reiten quiver of are of the form . [ABSTRACT FROM AUTHOR]

Abstract: Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin–Schelter regular algebras of dimension to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin–Schelter regular algebra or a category of dimension is shown to have common properties with the classical Artin–Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension is a finite length Frobenius category. [ABSTRACT FROM AUTHOR]

Abstract: Let be an artin algebra and a finitely generated -module. Iyama has shown that there exists a module such that the endomorphism ring of is quasi-hereditary, with a heredity chain of length , and that the global dimension of is bounded by this . In general, one only knows that a quasi-hereditary algebra with a heredity chain of length must have global dimension at most . We want to show that Iyama’s better bound is related to the fact that the ring he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1. [Copyright &y& Elsevier]

Abstract: We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds étale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space, generalizing the results of Pinkham and Rim. We provide a generalization and stack-theoretic proof of Luna’s étale slice theorem which shows that GIT quotient stacks are étale locally quotients stacks by the stabilizer. [Copyright &y& Elsevier]

Abstract: According to the Tits conjecture proved by Crisp and Paris (2001) , the subgroups of the braid group generated by proper powers of the Artin elements are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroups generated by powers of the band generators is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups. [Copyright &y& Elsevier]

Abstract: Bazzoni had given a simple characterization of infinitely generated -tilting modules. Though her method is even inapplicable to classical -tilting modules over Artin algebras, we show in this note that a similar characterization does hold for (finitely generated) relative -tilting modules introduced by Auslander and Solberg for Artin algebras, by using a different method. We also present some applications. [Copyright &y& Elsevier]

Abstract: For any prime and group , denote the pro- completion of by . Let be the class of all groups such that, for each natural number and prime number , , where is viewed as a discrete, trivial -module. In this article we identify certain kinds of groups that lie in . In particular, we show that right-angled Artin groups are in and that this class also contains some special types of free products with amalgamation. [Copyright &y& Elsevier]

Abstract: Let be an Artinian standard graded -algebra defined by quadrics. Assume that and that is algebraically closed, of characteristic . We show that is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by where is a complete intersection of three quadrics not containing a square of a linear form. [Copyright &y& Elsevier]

Abstract: Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations. [Copyright &y& Elsevier]

Each pointed topological space has an associated π-module , obtained from action of its first homotopy group on its second homotopy group. For the 3-ball with a trivial link with n -components removed from its interior, its π -module M n is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of M n. We give a topological interpretation of this injection, showing that it is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups. [ABSTRACT FROM AUTHOR]

Abstract: Let be a prime ring and let be a noncentral Lie ideal of . An additive map is called strong commutativity preserving (SCP) on if for all . In this paper we show that if f is SCP on , then there exist , and an additive map such that for all where is the extended centroid of , unless char and satisfies the standard identity of degree 4. [Copyright &y& Elsevier]

Abstract: Let be a ring and be an (infinite dimensional) partial tilting module. We show that the perpendicular category of is equivalent to the full module category where and is the Bongartz complement of modulo its -trace. Moreover, there is a ring epimorphism . We characterize the case when is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225–3237], there exist mutually inverse isomorphisms and between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for and to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when is a Dedekind domain, and if is finitely presented and is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of and contains all injectives. [Copyright &y& Elsevier]

Abstract: Let be an Artin–Tits system, and be the standard parabolic subgroup of generated by a subset of . Under the hypothesis that the Deligne complex has a geometric realization, we prove that the normalizer and the commensurator of in are equal. Furthermore, if is of spherical type, these subgroups are the product of with the quasi-centralizer of in . For two-dimensional Artin–Tits groups, the result still holds without any sphericality hypothesis on . We explicitly describe the elements of this quasi-centralizer. [Copyright &y& Elsevier]