Let $mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $mathrm{Supp}_RMsubseteqmathrm{V}(mathfrak{a})$. We show that if $mathrm{dim}_RMleq2$, then $M$ is $mathfrak{a}$-cofinite if and only if $mathrm
{Ext}^i_R(R/mathfrak{a},M)$ are finitely generated for all $ileq 2$, which generalizes one of the main results in [Algebr. Represent. Theory 18 (2015) 369--379]. Some new results concerning cofiniteness of local cohomology modules $mathrm{H}^i_mathfrak{a}(M)$ for any finitely generated $R$-module $M$ are obtained.
Let $mathfrak{a},mathfrak{b}$ be two ideals of a commutative noetherian ring $R$ and $M$ a finitely generated $R$-module.~We continue to study $textrm{f}textrm{-}mathrm{grad}_R(mathfrak{a},mathfrak{b},M)$ which was introduced in [Bull. Malays. Math.
Sci. Soc. 38 (2015) 467--482], some computations and bounds of $textrm{f}textrm{-}mathrm{grad}_R(mathfrak{a},mathfrak{b},M)$ are provided.~We also give the structure of $(mathfrak{a},mathfrak{b})$-$mathrm{f}$-modules,~various properties which are analogous to those of Cohen Macaulay modules are discovered.
We introduce the notions of Koszul $N$-complex, $check{mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $mathbf{D}_N(R)$ of $N$-complexes using the $check{mathrm{C}
}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the category of cohomologically $mathfrak{a}$-torsion $N$-complexes and the category of cohomologically $mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.
The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosuppo
rt and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $mathrm{Spec}R$, called the big cosupport.
Segmentation is a prerequisite yet challenging task for medical image analysis. In this paper, we introduce a novel deeply supervised active learning approach for finger bones segmentation. The proposed architecture is fine-tuned in an iterative and
incremental learning manner. In each step, the deep supervision mechanism guides the learning process of hidden layers and selects samples to be labeled. Extensive experiments demonstrated that our method achieves competitive segmentation results using less labeled samples as compared with full annotation.
