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Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

For a fixed CM field K with maximal totally real subfield F, we consider isomorphism classes of dihedral Artin representations of F which are induced from K, distinguishing between those which are "canonically" induced from K and those which are "noncanonically" induced from K. The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones. [ABSTRACT FROM AUTHOR]

For a higher hereditary algebra, we calculate its upper (lower) Serre dimension, the entropy and polynomial entropy of Serre functor, and the Hochschild (co)homology entropy of Serre quasi-functor. These invariants are determined by its Calabi-Yau dimension for a higher representation-finite algebra, and by its global dimension and the spectral radius and polynomial growth rate of its Coxeter matrix for a higher representation-infinite algebra. For this, we will prove the Yomdin type inequality on Hochschild homology entropy for a finite dimensional elementary algebra of finite global dimension. Our calculations imply that the Kikuta and Ouchi's question on relations between entropy and Hochschild (co)homology entropy has positive answer, and the Gromov-Yomdin type equalities on entropy and Hochschild (co)homology entropy hold, for the Serre functor on perfect derived category and the Serre quasi-functor on perfect dg module category of an indecomposable elementary higher hereditary algebra. [ABSTRACT FROM AUTHOR]

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has a radical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremum being finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J³ = 0 and without considering that rad² (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case. [ABSTRACT FROM AUTHOR]

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: 'Classification of singularities of curves can be interpreted in dual terms as a description of "co-artin" subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)'. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let K be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group Aut K (A) is explicitly described. The automorphism group of the algebra A is finite if and only if the algebra A is not isomorphic to a monomial algebra, and in this case | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definitions of relative injective envelopes and relative projective covers of modules on ring extensions, and study their basic properties. In particular, we give their equivalent characterizations in terms of relative essential monomorphisms and relative superfluous epimorphisms, and prove that relative injective envelopes and relative projective covers on ring extensions are unique up to isomorphism whenever they exist. Moreover, for an extension of Artin algebras, we show that every finitely generated module has both a relative injective envelope and a relative projective cover. In addition, we compare relative injective envelopes and relative projective covers on two ring extensions linked by surjective homomorphisms of rings respectively. [ABSTRACT FROM AUTHOR]

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or D Λ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if rad C ∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C -approximations and right C -approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C. [ABSTRACT FROM AUTHOR]

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2. When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a , b of a two-dimensional Artin group of hyperbolic type, there exists an integer n ≥ 1 such that a n and b n either commute or generate a non-abelian free subgroup. [ABSTRACT FROM AUTHOR]

Given an odd prime number p and a p-stabilized Artin representation \rho over \mathbb {Q}, we introduce a family of p-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the p-part of the Tamagawa number conjecture for Artin motives at s=0 and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of p-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a p-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 (T , −) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 (T , −) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 (T , −) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i (T , T (J)) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang's paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations. [ABSTRACT FROM AUTHOR]

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin's primitive root conjecture that the order is 'almost always almost maximal' assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments. [ABSTRACT FROM AUTHOR]

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study the behavior of the extensions dimensions of algebras under different equivalences. We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the length of the tilting complex associated with the derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences. [ABSTRACT FROM AUTHOR]

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Let n be an even natural number. We compute the periods of any n 2 -dimensional complete intersection algebraic cycle inside an n -dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting. [ABSTRACT FROM AUTHOR]

We interpret the two old and still unsettled conjectures of Ankeny-Artin-Chowla and of Mordell, concerning the unit of the real quadratic field Q (p) , in terms of its p -rationality. [ABSTRACT FROM AUTHOR]

In this paper, we succeed in proving that a connected Artin algebra whose Jacobson cubic radical is zero with each simple module and each Gorenstein projective module having the square radical of the cover projective of its first syzygy to be zero and verifying the coincidence covers, is either self-injective or CM-free. [ABSTRACT FROM AUTHOR]

We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements. Let Λ′, Λ, Λ″ be artin algebras such that (mod Λ′, mod Λ, mod Λ′) is a recollement, and let D ′ and D ′ ′ be subcategories of mod Λ′ and mod Λ″ respectively. For any n, m ≥ 0, under some conditions, we get dim Ω k (D) ≤ dim Ω n (D ′) + dim Ω m (D ′ ′) + 1 , where k = max{m, n} and D is the subcategory of mod Λ glued by D ′ and D ′ ′ ; moreover, we give a sufficient condition such that the converse inequality holds true. As applications, some results for Igusa–Todorov subcategories and syzygy finite subcategories are obtained. [ABSTRACT FROM AUTHOR]

We construct a finite-dimensional quiver algebra from the non-simply laced type B Dynkin diagram, which we call type B zigzag algebra. This leads to a faithful categorical action of the type B Artin (braid) group A.B/, acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of A.B/, viewed as mapping class group of the punctured disc – hence, our exposition can be seen as a type B analogue of Khovanov–Seidel’s work. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra over an algebraically closed field. We study sections in the Auslander–Reiten quiver of the categories of complexes of fixed size and in the Auslander–Reiten quiver of the bounded derived category. One of the aims of this work is to show where the indecomposable A -modules can be found in the Auslander–Reiten quiver of the derived category, whenever A is a piecewise hereditary algebra. We also prove that we can obtain a transjective component of the bounded derived category from a section. As an application of the results of sections, we find a bound on the strong global dimension of some piecewise hereditary algebras of tree type. Moreover, we also determine the strong global dimension of some algebras taking into account their ordinary quivers with relations. [ABSTRACT FROM AUTHOR]

We apply the character sums method of Lenstra, Moree, and Stevenhagen to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form ∑ n = 1 ∞ g (n) # G (n) in which g(n) is a multiplicative arithmetic function and { G (n) } is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants. [ABSTRACT FROM AUTHOR]

In this paper, we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions, which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation. [ABSTRACT FROM AUTHOR]

We generalize Grothendieck's generic freeness lemma to modules over semi-constructible extensions of normally bounded domains, extending previous results of Irving, McConnell, Robson, Artin, Small and Zhang, in which the base ring is commutative. [ABSTRACT FROM AUTHOR]

Let k be a number field and G be a finite group. Let F k G (Q) be the family of number fields K with absolute discriminant D K at most Q such that K / k is normal with Galois group isomorphic to G . If G is the symmetric group S n or any transitive group of prime degree, then we unconditionally prove that for all K ∈ F k G (Q) with at most O ε (Q ε) exceptions, the L -functions associated to the faithful Artin representations of Gal (K / k) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that: there exist infinitely many degree n S n -fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke; for a prime p , the periodic torus orbits attached to the ideal classes of almost all totally real degree p fields F over ℚ equidistribute on PGL p (Z) ∖ PGL p (R) with respect to Haar measure; for each ℓ ≥ 2 , the ℓ -torsion subgroups of the ideal class groups of almost all degree p fields over k (resp. almost all degree n S n -fields over k ) are as small as GRH implies; and an effective variant of the Chebotarev density theorem holds for almost all fields in such families. [ABSTRACT FROM AUTHOR]

We prove that every right‐angled Artin group occurs as a finite‐index subgroup of the outer automorphism group of another right‐angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to (Z/2Z)N$(\mathbb {Z}/2\mathbb {Z})^N$ for some N$N$. For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day–Wade and Wade–Brück about when this group is a right‐angled Artin group and when it has finite index. [ABSTRACT FROM AUTHOR]

This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi)$, where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$, can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho)$ for all virtual representations ρ of the corresponding Weil group WE$W_E$. Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. [ABSTRACT FROM AUTHOR]

We show that the perverse t-structure induces a t-structure on the category D A (S , Z ℓ) of Artin ℓ -adic complexes when S is an excellent scheme of dimension less than 2 and provide a counter-example in dimension 3. The heart Perv A (S , Z ℓ) of this t-structure can be described explicitly in terms of representations in the case of 1-dimensional schemes. When S is of finite type over a finite field, we also construct a perverse homotopy t-structure over D A (S , Q ℓ) and show that it is the best possible approximation of the perverse t-structure. We describe the simple objects of its heart Perv A (S , Q ℓ) # and show that the weightless truncation functor ω 0 is t-exact. We also show that the weightless intersection complex E C S = ω 0 I C S is a simple Artin homotopy perverse sheaf. If S is a surface, it is also a perverse sheaf but it need not be simple in the category of perverse sheaves. [ABSTRACT FROM AUTHOR]

Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right-angled Artin groups can be expressed in terms of their defining graph. In this paper, we address the question of understanding the structure of a class of subgroups of right-angled Artin groups in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina–Brady groups or Artin kernels) from a finiteness conditions viewpoint. Unlike the right-angled Artin groups the isomorphism type of Bestvina–Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina–Brady groups can be expressed as an iterated amalgamated product. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient right-angled Artin group. [ABSTRACT FROM AUTHOR]

We study the special value u=1 of Artin-Ihara L-functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest. [ABSTRACT FROM AUTHOR]

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]

This paper aims to initiate a study of the representation theory of representation-finite artin algebras in terms of the nilpotency of the radical of their module category. Firstly, we shall calculate this nilpotency explicitly for hereditary algebras of type A n and for Nakayama algebras. Surprisingly, this nilpotency for a given algebra coincides with its Loewy length if and only if the algebra is a hereditary Nakayama algebra. Secondly, we shall find all artin algebras for which this nilpotency is equal to any given positive integer up to four and describe completely their module category. [ABSTRACT FROM AUTHOR]

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group. [ABSTRACT FROM AUTHOR]

We derive a geometric model for the module category mod k Q of a Dynkin quiver Q via the heart of a total stability condition on the bounded derived category of mod k Q. As an application, we prove Reineke's conjecture that there is a stability function on mod k Q making any indecomposables stable. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q(s τ -tilt A) of A and the tilting quiver Q(tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices. [ABSTRACT FROM AUTHOR]

This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite Q -algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the Q -algebra Q [ V n ] of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set [ n ] : = { 1 , 2 , ... , n } . All results are equally true if we replace Q [ V n ] by K [ V n ] , where K is any perfect field of characteristics ≠ 2 . The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework. [ABSTRACT FROM AUTHOR]

The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction, computational effects, and type abstraction. We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a para- metricity structure. Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof-relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syn- tactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types, by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntac- tic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant. [ABSTRACT FROM AUTHOR]

We consider the conjugation action of a quantum group G over an arbitrary field k. In particular we consider the coordinate algebra k [ G (n) ] of the quantised general linear group G (n) , at an arbitrary non-zero parameter q , as a G (n) -module and give analogues of results of Kostant and Richardson. We also consider the case in which G is a product of quantum general linear groups and the problem of describing the conjugation invariants of k [ G ] for the action of a quantum subgroup. This is approached via finite dimensional sub-coalgebras of k [ G ] and the theory of quasi-hereditary algebras. [ABSTRACT FROM AUTHOR]

Quandles are non-associative algebraic structures with defining axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper gives two approaches to write efficient presentations for a large class of quandles, called Dehn quandles, using presentations of their underlying groups. The first approach gives finite presentations for Dehn quandles of a class of Garside groups and Gaussian groups. The second approach is for general Dehn quandles when the centralisers of generators in their underlying groups are known. Several examples including Dehn quandles of spherical Artin groups, surface groups and mapping class groups of orientable surfaces are given to illustrate the results. [ABSTRACT FROM AUTHOR]

We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4-cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non-simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi-Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of Dn when n is even or in any of the exceptional cases. In Baumeister et al. (J. Algebra 629 (2023), 399-423), we show using different methods that this result holds for type Dn for all ??? 4. [ABSTRACT FROM AUTHOR]

Birational properties of generically finite morphisms X→Y$X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of X. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. [ABSTRACT FROM AUTHOR]

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Borel's rank theorem identifies the ranks of algebraic K-groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin L-functions by considering equivariant algebraic K-groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic K-theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in L-functions. [ABSTRACT FROM AUTHOR]

We study which irreducible morphisms in the category of complexes of fixed size C n (proj A) with n ≥ 2 and A an artin algebra, are irreducible in C [ 0 , n ] (proj A) or either in C n + 1 (proj A). Moreover, we determine which of them are also irreducible in the bounded derived category. [ABSTRACT FROM AUTHOR]

We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the conjecture in a specific case. Along the way, we provide short and almost self-contain algebraical proofs of several classical results on Artin-Tits groups, such as those of Van der Lek on intersection of standard parabolic subgroups. [ABSTRACT FROM AUTHOR]

Let A be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo–Mumford regularity, Tor-regularity, Artin–Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones. [ABSTRACT FROM AUTHOR]

Given a finite group G, we study the monoid algebra RG, over a field K , generated by the monomial characters of G. In particular, we note that the integral closure of RG is contained in the algebra generated by those characters χ for which their associated Artin L-function L(s, χ) is holomorphic at s0 ∈ ℂ \ {1}. Also, we discuss the supercharacter theoretic case. [ABSTRACT FROM AUTHOR]

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group W (D n). We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of W (D n) cannot be isomorphic to the Artin group of type D n. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family. [ABSTRACT FROM AUTHOR]

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions. [ABSTRACT FROM AUTHOR]

We prove a version of a theorem of Auslander for finite group actions or coactions on noetherian polynomial identity Artin-Schelter regular algebra. [ABSTRACT FROM AUTHOR]