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.

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{A}\mathbb{G}\left( R \right)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\,=\,\left( 0 \right)$. Here we show that if $R$ is a reduced ring and the independence number of $\mathbb{A}\mathbb{G}\left( R \right)$ is finite, then the edge chromatic number of $\mathbb{A}\mathbb{G}\left( R \right)$ equals its maximum degree and this number equals ${{2}^{\left| \min \left( R \right) \right|-1}}-\,1$; also, it is proved that the independence number of $\mathbb{A}\mathbb{G}\left( R \right)$ equals ${{2}^{\left| \min \left( R \right) \right|-1}}$, where $\min \left( R \right)$ denotes the set of minimal prime ideals of $R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph $\mathbb{A}\mathbb{G}\left( R \right)$ is not Eulerian, and that it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand.

Let $R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}$ be a graded Noetherian ring with local base ring $\left( {{R}_{0}},{{\text{m}}_{0}} \right)$ and let ${{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}$ . Let $M$ and $N$ be finitely generated graded $R$ -modules and let $\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}$ an ideal of $R$ . We show that $H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)$ and ${H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;$ are Artinian for some $i\text{ s}$ and $j\,\text{s}$ with a specified property, where ${{\mathfrak{b}}_{o}}$ is an ideal of ${{R}_{0}}$ such that ${{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}$ is an ${{\mathfrak{m}}_{0}}$ -primary ideal.

We prove that $r$ independent homogeneous polynomials of the same degree $d$ become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose $(d-1)$- osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called the Weak Lefschetz Property) and a differential geometric notion, concerning varieties that satisfy certain Laplace equations. In the toric case, some relevant examples are classified, and as a byproduct we provide counterexamples to Ilardi's conjecture.

Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $R/\mathfrak{a}\,\le \,1$, then all generalized local cohomology modules $H_{\mathfrak{a}}^{i}(M,\,N)$ are $\mathfrak{a}$-cofinite.

We show that if $R\,=\,{{\oplus }_{n\in \mathbb{N}0}}\,{{R}_{n}}$ is a Noetherian homogeneous ring with local base ring $({{R}_{0}},\,{{m}_{0}})$, irrelevant ideal ${{R}_{+}}$, and $M$ a finitely generated graded $R$-module, then $H_{{{m}_{0}}R}^{j}\,(H_{R+}^{t}\,(M))$ is Artinian for $j\,=\,0,\,1$ where $t\,=\,\inf ${$i\in {{\mathbb{N}}_{0}}:H_{R+}^{i}(M)$ is not finitely generated}. Also, we prove that if $\text{cd(}{{R}_{+}},M)\,=\,2$, then for each $i\,\in \,{{\mathbb{N}}_{0}},\,H_{{{m}_{0}}R}^{i}\,(H_{R+}^{2}\,(M))$ is Artinian if and only if $H_{{{m}_{0}}R}^{i+2}(H_{R+}^{1}(M))$ is Artinian, where $ \text{cd(}{{R}_{+}},\,M)$ is the cohomological dimension of $M$ with respect to ${{R}_{+}}$. This improves some results of R. Sazeedeh.

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a},\,\mathfrak{b},\,\mathfrak{a}\,\cap \,\mathfrak{b}$ and $\mathfrak{a}\,+\,\mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen–Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\,\le \,r\,<\,{{\dim}_{R}}(M)$, any maximal element q of the non-empty set of ideals {$\mathfrak{a}\,:\,\text{H}_{\mathfrak{a}}^{i}(M)$ is not artinian for some $i$, $i\,\ge \,r$} is a prime ideal, and all Bass numbers of $\text{H}_{\mathfrak{q}}^{i}(M)$ are finite for all $i\,\ge \,r$.

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is finitely generated for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ is finitely generated. In this paper it is shown that if $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is Artinian for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$ is Artinian.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.

These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

Let S = k[[x,y]] be a formal power series ring in two variables x, y over a field k and I an (x, y)-primary ideal of S. We show that S/I is selfinjective if Hi(S/I, S/I ⊗k S/I) = 0 for i = 1 and 2.