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...

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 X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

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...

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\mathrm{Ext}}_R^i(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H_I^i\left(M\right)$ are $I$-cominimax for all $i<t$ and ${\mathrm{Ext}}_R^i(R/I,H_I^t\left(M\right))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\mathrm{Ext}}_R^j(N,H_I^i\left(M\right))$ and ${\mathrm{Tor}}_j^R(N,H_I^i\left(M\right))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i$.

Let $p$ be a prime number. This paper introduces the Roquette category ${ℛ}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In ${ℛ}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of ${ℛ}_p$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

For a finite group $G$ denote by $N\left(G\right)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N\left(G\right)=N\left(L\right)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z\left(G\right)=1$ and $N\left(G\right)=N\left(A_i\right)$ is necessarily isomorphic to $A_i$, where $i\in \{2p,2p+1\}$.

Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $\tau$-semibrick, then the Auslander-Reiten quiver $\Gamma \left(ℱ\right(𝒳\left)\right)$ of the filtration subcategory $ℱ\left(𝒳\right)$ generated by $𝒳$ is $\mathbb{Z}{𝔸}_{\infty }$. If $𝒳={\left\{X_i\right\}}_{i=1}^t$ is a $\tau$-cycle semibrick, then $\Gamma \left(ℱ\right(𝒳\left)\right)$ is $\mathbb{Z}{𝔸}_{\infty }/{\tau }_{𝔸}^t$.

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

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...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q\left(m\right)$, where $Q$ is a polynomial with integral coefficients, then $P\left(x\right)=Q\left(R\right(x\left)\right)$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P\left(x\right)={\left(R\left(x\right)\right)}^q$ for some polynomial $R\left(x\right)$.