• Honam Mathematical Journal
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• v.2 no.1
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• pp.13-18
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• 1980

모든 기호(記號)는 G.E Bredon의 저(著) Sheaf Theory의 기호(記號)를 따른다. A가 torsion free이며 elementary sheaf이라 하자. 그리고 L을 injective L-module이라 하자 $dim_{\varphi}X<{\infty}$이라면 support의 $family{\varphi}$와 locally subset z에 대하여 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{\varphi}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-p}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ 이며 support의 family c와 compact subset z에 대하여도 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{c}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-y}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ A가 elementary이면 locally closed z와 z에서 closed인 $z^{\prime}$ 그리고 $z^{\prime\prime}=z-z^{\prime}$에 대하여 exact sequence ⋯⋯${\rightarrow}H^{z^{\prime}}_{p}\;(X:A){\rightarrow}H^{z}_{p}(X:A){\rightarrow}H^{z^{\prime\prime}}_{p}\;(X:A){\rightarrow}$⋯⋯ 가 존재(存在)한다.

• Journal of Integrative Natural Science
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• v.4 no.4
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• pp.278-281
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• 2011

We discuss an application of 'restrictions on the entries of the maps in the minimal free resolution' and '$SC_r$-condition of modules', and give an alternative proof of the following result of Foxby: Let M be a finitely generated module of dimension over a Noetherian local ring (A,m). Suppose that $\hat{A}$ has no embedded primes. If A is not Gorenstein, then ${\mu}_i(m,A){\geq}2$ for all i ${\geq}$ dimA.

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.239-250
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• 2014

Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$_RH_I^i(M){\leq}k$ for ${\forall}i$ < t, then $$\displaystyle\bigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{\geq}k}=\displaystyle\bigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$$ for ${\forall}j{\leq}t$ and ${\forall}n$ >0. This shows that${\bigcup}_{n>0}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$ is a finite set for ${\forall}i{\leq}t$. Also, we prove that $\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{\ldots},x_i^{n_i})M)_{{\geq}k}=\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{\ldots},x_i)M)_{{\geq}k}$ if $x_1,x_2,{\ldots},x_r$ is M-sequences in dimension > k and $n_1,n_2,{\ldots},n_r$ are some positive integers. Here, for a subset T of Spec(R), set $T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}$.