Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_C$-injective module $G$, the character module $G^{+}$ is $G_C$-flat, then the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$...

Let $\mathbb{Z}{̂}_p$ be the ring of p-adic integers, $U\left(\mathbb{Z}{̂}_p\right)$ the unit group of $\mathbb{Z}{̂}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $\mathbb{Z}{̂}_p^{\lambda }G$ the twisted group algebra of G over $\mathbb{Z}{̂}_p$ with a 2-cocycle $\lambda \in Z²(G,U\left(\mathbb{Z}{̂}_p\right))$. We give necessary and sufficient conditions for $\mathbb{Z}{̂}_p^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $\mathbb{Z}{̂}_p^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $\mathbb{Z}{̂}_p^{\lambda }{G}_p$-module V and an irreducible $\mathbb{Z}{̂}_p^{\lambda }B$-module W.

Let $K$ be a finite extension of ${\mathbf{Q}}_p$. The field of norms of a $p$-adic Lie extension $K_{\infty }/K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm{Gal}(K_{\infty }/K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi ,\Gamma )$-modules, and give a condition for the existence of certain types of lifts.

Let $R$ be a graded ring and $n\ge 1$ be an integer. We introduce and study the notions of Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules by using the notion of special finitely presented graded modules. On $n$-gr-coherent rings, we investigate the relationships between Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules. Among other results, we prove that any graded module in $R$-gr (or gr-$R$) admits a Gorenstein $n$-FP-gr-injective (or Gorenstein $n$-gr-flat) cover and preenvelope,...

Let $\Sigma$ be a compact connected oriented surface with one boundary component, and let $\pi$ be the fundamental group of $\Sigma$. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $\Sigma$, whose $k$-th term consists of the self-homeomorphisms of $\Sigma$ that act trivially at the level of the $k$-th nilpotent quotient of $\pi$. Morita defined a homomorphism from the $k$-th term of the Johnson filtration to the third homology group of the $k$-th nilpotent quotient of $\pi$. In this paper, we...

An $n\times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$. Let ${𝒟}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$ $(i=1,\cdots ,r)$ and $(i,i-r+1)$ $(i=r+1,\cdots ,n)$ entries are negative, and zeros elsewhere. We prove that for $r\ge 3$ and $n\ge 4r-2$, the sign pattern ${𝒟}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.

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...

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜=𝕂\left[{𝔸}^n\right]$ is said to be of Kostant type, if its centre $Z\left(𝒜\right)$ is freely generated by homogeneous polynomials $F_1,...,F_r$ such that they give Kostant’s regularity criterion on ${𝔸}^n(d_x{F}_i$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$). If the initial Poisson algebra is of Kostant type and $F_i$ satisfy a certain degree-equality, then the contraction...

Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B=R^{\left[1\right]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{\left[2\right]}$? We give a negative answer to this question. The interest in...

Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

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.

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...

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

An $𝒮$-closed submodule of a module $M$ is a submodule $N$ for which $M/N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$-closed submodule $N$ of $M$ is a direct summand of $M$. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$-modules are projective if and only if all right $R$-modules are GCS-modules.

A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $\left|G\right|$, the order of $G$. Let $P\in {\mathrm{Syl}}_p\left(G\right)$. As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$-subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b|a^{p^{n-1}}=1,b^2=1,b^{-1}ab=a^{1+p^{n-2}}\rangle$, where $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed. ...