Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that ${}^{k\left(\right)}$ kills the general local cohomology module $H_{{\Phi }_{}}^j\left(M_{}\right)$ for every integer j less than a fixed integer n, where ${\Phi }_{}:={}_{}:\in \Phi$, then there exists an integer k such that ${}^k{H}_{\Phi }^j\left(M\right)=0$ for every j < n.

Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$-modules. In certain cases, we give some bounds under inclusion for the annihilators of ${\mathrm{Ext}}_R^t(M,N)$ and ${\mathrm{H}}_{𝔞}^t\left(M\right)$ in terms of minimal primary decomposition of the zero submodule of $M$, which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${\left(H_I^i\left(M\right)\right)}_{𝔪}$ is a minimax $R_{𝔪}$-module for all $𝔪\in \mathrm{Max}\left(R\right)$ and for all $i<n$, then the set ${\mathrm{Ass}}_R\left(H_I^n\left(M\right)\right)$ is finite. Also, if $H_I^i\left(M\right)$ is...

We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$-module with $A={\mathrm{End}}_{R}T$. If ${}_{R}T$ is selforthogonal, then we show that $\mathrm{rid}\left(T_A\right)\le \mathrm{findim}\left(A_A\right)\le \mathrm{findim}{(}_{R}T)+\mathrm{rid}\left(T_A\right)$. Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$-module with finite injective dimension, then $\mathrm{rid}\left(T_A\right)\le \mathrm{findim}\left(A_A\right)\le \mathrm{fin}.\mathrm{inj}.\dim {(}_{R}R)+\mathrm{rid}\left(T_A\right)$. Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles...

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s $K_0$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁(G/B)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the...

Let $n,d$ be two non-negative integers. A left $R$-module $M$ is called $(n,d)$-injective, if ${\mathrm{Ext}}^{d+1}(N,M)=0$ for every $n$-presented left $R$-module $N$. A right $R$-module $V$ is called $(n,d)$-flat, if ${\mathrm{Tor}}_{d+1}(V,N)=0$ for every $n$-presented left $R$-module $N$. A left $R$-module $M$ is called weakly $n$-$FP$-injective, if ${\mathrm{Ext}}^n(N,M)=0$ for every $(n+1)$-presented left $R$-module $N$. A right $R$-module $V$ is called weakly $n$-flat, if ${\mathrm{Tor}}_n(V,N)=0$ for every $(n+1)$-presented left $R$-module $N$. In this paper, we give some characterizations and properties of $(n,d)$-injective modules and $(n,d)$-flat modules in...

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$. In the second part, we study the orbifold intersection theory of ${\overline{ℳ}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter...

Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition $C_{}$, defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class ${}_{}$ consisting of all modules satisfying $C_{}$. If and are ideals of R, we get a necessary and sufficient condition for to satisfy $C_{}$ and $C_{}$ simultaneously. We also...

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Let $f$ be a modular eigenform of even weight $k\ge 2$ and new at a prime $p$ dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to $f$ a monodromy module $D_f^{FM}$ and an $ℒ$-invariant ${ℒ}_f^{FM}$. The first goal of this paper is building a suitable $p$-adic integration theory that allows us to construct a new monodromy module $D_f$ and $ℒ$-invariant ${ℒ}_f$, in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $ℒ$-invariants...

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega \left(𝒟\right)$ be the Burnside ring of $𝒟$, $\Delta \left(𝒟\right)$ be the augmentation ideal of $\Omega \left(𝒟\right)$. Denote by ${\Delta }^n\left(𝒟\right)$ and $Q_n\left(𝒟\right)$ the $n$th power of $\Delta \left(𝒟\right)$ and the $n$th consecutive quotient group ${\Delta }^n\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for ${\Delta }^n\left(𝒟\right)$ and determines the isomorphism class of $Q_n\left(𝒟\right)$ for each positive integer $n$.

For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, let ${\pi }_{\varphi }$ be the unitary representation associated to $\varphi$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,{\pi }_{\varphi })$ and reduced 1-cohomology ${\overline{H}}^1(G,{\pi }_{\varphi })$. For example, ${\overline{H}}^1(G,{\pi }_{\varphi })=0$ if and only if either $\text{Hom}(G,ℂ)=0$ or ${\mu }_{\varphi }\left(1_G\right)=0$, where $1_G$ is the trivial character of $G$ and ${\mu }_{\varphi }$ is the probability measure on the Pontryagin dual $\widehat{G}$ associated to $\varphi$ by Bochner’s Theorem. This streamlines an argument of Guichardet...