In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. Among the applications are precise connections between Ringel duality, split quasi-hereditary covers and double centralizer properties, constructions of split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of q-Schur algebras and a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring. They are shown to behave nicely under change of rings techniques.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$-analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$, the $d$-th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$-Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$-Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules.

As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ .

Furthermore, with k as in Maranda’s theorem, we show that if M is R-torsion-free and $H(M)$ is the pure-injective hull of M, then $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

Let $A=\bigoplus _{i\in \mathbb{Z}}A_{i}$ be a finite-dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that if $d\neq 0$ then $A_{-d}$ contains the Higman ideal $H(A)$ and $\dim H(A)\leq \dim A_{0}$, and provide a semisimplicity criterion for $A$ in terms of the centralizer of $A_{0}$.

Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

We prove a Noether–Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.

We prove that if G is a non-trivial finite group, then the integral group ring $\mathbb{Z} G$ possesses a superdecomposable pure-injective module.

The question is addressed of when all pure-projective modules are direct sums of finitely presented modules. It is proved that this is the case over hereditary noetherian rings. Partial results are obtained for uniserial rings. Some of the methods are model-theoretic, and the techniques developed using these may be of interest in their own right.

Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite-dimensional and representation-finite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent:

(1) $A$ admits no outer derivation;

(2) $\Lambda$ admits no outer derivations;

(3) $A$ is simply connected;

(4) $\Lambda$ is strongly simply connected.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

We provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.