Let $(R,𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\mathrm{hd}(𝔞,M)=n$. We determine the annihilator of the top local homology module ${\mathrm{H}}_n^{𝔞}\left(M\right)$. In fact, we prove that $${\mathrm{Ann}}_R\left({\mathrm{H}}_n^{𝔞}\left(M\right)\right)={\mathrm{Ann}}_R\left(N(𝔞,M)\right),$$ where $N(𝔞,M)$ denotes the smallest submodule of $M$ such that $\mathrm{hd}(𝔞,M/N(𝔞,M\left)\right)<n$. As a consequence, it follows that for a complete local ring $(R,𝔪)$ all associated primes of ${\mathrm{H}}_n^{𝔞}\left(M\right)$ are minimal.

Let $M$ be a 1-connected closed manifold of dimension $m$ and $LM$ be the space of free loops on $M$. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_{*}(LM;k)$. When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ and the shifted homology $H_{*+m}(LM;k)$. We also prove...

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f:R\to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi :M\to N$ be an $R$-homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $\left(R{\bowtie }^{f}J\right)$-module called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and denoted by $M{\bowtie }^{\varphi }JN$. We study some homological properties of the $\left(R{\bowtie }^{f}J\right)$-module $M{\bowtie }^{\varphi }JN$. Among...

Let X be a Riemann domain over ${ℂ}^k\times {ℂ}^{\ell }$. If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P\subset {ℂ}^k$ such that every slice $X_a$ of X with a∉ P is a region of holomorphy with respect to the family ${f|}_{X_a}:f\in ℱ$.

For a prime $p\ge 5$, we compute the algebraic $K$-theory modulo $p$ and $v_1$ of the mod $p$ Adams summand, using topological cyclic homology. On the way, we evaluate its modulo $p$ and $v_1$ topological Hochschild homology. Using a localization sequence, we also compute the $K$-theory modulo $p$ and $v_1$ of the first Morava $K$-theory.

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...

Let $R$ be a ring. A subclass $𝒯$ of left $R$-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. Then a left $R$-module $M$ is called $𝒯$-finitely generated if there exists a finitely generated submodule $N$ such that $M/N\in 𝒯$; a left $R$-module $A$ is called $(𝒯,n)$-presented if there exists an exact sequence of left $R$-modules $$0⟶K_{n-1}⟶F_{n-1}⟶\cdots ⟶F_1⟶F_0⟶M⟶0$$ such that $F_0,\cdots ,F_{n-1}$ are finitely generated free and $K_{n-1}$ is $𝒯$-finitely generated;...

Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(𝒯,n)$-injective if ${\mathrm{Ext}}_R^n(C,M)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(𝒯,n)$-flat if ${\mathrm{Tor}}_n^R(M,C)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(𝒯,n)$-projective if ${\mathrm{Ext}}_R^n(M,N)=0$ for each $(𝒯,n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(𝒯,n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(𝒯,n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(𝒯,n)$-projective; the ring $R$ is called...

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...

We introduce two entire functions $f_{{\mathrm{A}}_{\frac{1}{2}\infty }}$ and $f_{{\mathrm{D}}_{\frac{1}{2}\infty }}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps ${\mathbf{C}}^2\to \mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\setminus \{0,1\}$. All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type ${\mathrm{A}}_{\frac{1}{2}\infty }$ and ${\mathrm{D}}_{\frac{1}{2}\infty }$, respectively. Coxeter elements...

Let $k\ge 2$ and let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence defined by $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}$$ for $n\ge 2$ with initial conditions $$P_{-(k-2)}^{\left(k\right)}=P_{-(k-3)}^{\left(k\right)}=\cdots =P_{-1}^{\left(k\right)}=P_0^{\left(k\right)}=0,P_1^{\left(k\right)}=1.$$ In this study, we handle the equation $P_n^{\left(k\right)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{\left(k\right)}=y^m$ with $2\le y\le 1000$ has only one solution given by $P_7^{\left(2\right)}={13}^2.$

Let $(R,𝔪)$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq {(x_1,...,x_n)}^2$ is a graded ideal of a polynomial ring $S=K[x_1,...,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ of the first syzygy module ${\mathrm{Syz}}_1\left(𝔪\right)$ of $𝔪$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

A QTAG-module $M$ is an $\alpha$-module, where $\alpha$ is a limit ordinal, if $M/H_{\beta }\left(M\right)$ is totally projective for every ordinal $\beta <\alpha$. In the present paper $\alpha$-modules are studied with the help of $\alpha$-pure submodules, $\alpha$-basic submodules, and $\alpha$-large submodules. It is found that an $\alpha$-closed $\alpha$-module is an $\alpha$-injective. For any ordinal $\omega \le \alpha \le {\omega }_1$ we prove that an $\alpha$-large submodule $L$ of an ${\omega }_1$-module $M$ is summable if and only if $M$ is summable.

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...