We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$. We also establish an asymptotic formula for $1\le x,y,z\le H$ such that $x^2+y^2+z^2+k$, $x^2+y^2+z^2+k+1$ are square-free. The method we used in this paper is due to Tolev.

If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group ${}_f\subseteq G*H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_f{=p|}_f{:}_f\to G$. A right inverse (section) $G{\to }_f$ of $p_f$ is called a coaction on G. In this paper we study ${}_f$ and the sections of $p_f$. We consider the following topics: the structure of ${}_f$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

An $R$-module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$-module of countable infinite rank $R^{\left(\omega \right)}$. For every natural number $k>1$, we construct arbitrarily large separable ${\aleph }_k$-free $R$-modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

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 $i$.

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not...

Let $U$ be a dg-$A$-module, $B$ the endomorphism dg-algebra of $U$. We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃(C,d)$ of $C$, $𝐃(B,d)$ of $B$ and $𝐃(A,d)$ of $A$. We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).

We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$ $\epsilon ,$ $k$ and $\beta .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $$\sum _{1\le h\le H}\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{n\in {𝒬}_k}}e\left(\vartheta hn\right).$$

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it...

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

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 $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤=LieG$. Let $(e,h,f)$ be an $𝔰{𝔩}_2$-triple in $𝔤$ with $e$ being a long root vector in $𝔤$. Let $(·,·)$ be the $G$-invariant bilinear form on $𝔤$ with $(e,f)=1$ and let $\chi \in {𝔤}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in 𝔤$. Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...