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 $(R,𝔪)$ 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 $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

Given a locale $L$ and a join semilattice $J$ with bottom element $0_J$, a new concept $(\sigma ,J)$ called $L$-slice is defined,where $\sigma$ is as an action of the locale $L$ on the join semilattice $J$. The $L$-slice $(\sigma ,J)$ adopts topological properties of the locale $L$ through the action $\sigma$. It is shown that for each $a\in L$, ${\sigma }_a$ is an interior operator on $(\sigma ,J)$.The collection $M=\{{\sigma }_a;a\in L\}$ is a Priestly space and a subslice of $L$-$Hom(J,J)$. If the locale $L$ is spatial we establish an isomorphism between the $L$-slices $(\sigma ,L)$ and $(\delta ,M)$. We have shown that the fixed...

Given a coarse space $(X,ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X\times X$ to the natural coarse structure on $(ℬ\setminus \{\varnothing \left\}\right)\times (ℬ\setminus \{\varnothing \left\}\right)$ and say that a macro-uniform mapping $f:(ℬ\setminus \{\varnothing \left\}\right)\to X$ (or $f:{\left[X\right]}^2\to X$) is a selector (or 2-selector) of $(X,ℰ)$ if $f\left(A\right)\in A$ for each $A\in ℬ\setminus \left\{\varnothing \right\}$ ($A\in {\left[X\right]}^2$, respectively). We prove that a discrete coarse space $(X,ℰ)$ admits a selector if and only if $(X,ℰ)$ admits a 2-selector if and only if there exists a linear order “$\le$" on $X$ such that the family of intervals $\left\{\right[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $ℬ$.