• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.9-17
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• 2018

Let $R ={\oplus}_{n{\in}Z}R_n$ be a graded commutative Noetherian ring and let I be a graded ideal of R and J be an arbitrary ideal. It is shown that the i-th generalized local cohomology module of graded module M with respect to the (I, J), is graded. Also, the asymptotic behaviour of the homogeneous components of $H^i_{I,J}(M)$ is investigated for some i's with a specified property.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.1077-1097
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• 2019

Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.191-207
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• 2000

Let R be a ommutative ring with unity and F a finite free R-module. For a nonnegative integer r, there exists a natural filtration of$S_r(S_2F)$ such that its associated graded module is isomorphic to $\Sigma_{{\lambda}{\epsilon}{\tau}_r}\;L_{\lambda}F$, where ${\Gamma}_{\gamma}$ set of partitions such that $$\mid${\lambda}$\mid$-2r,{{\widetilde}{\lambda}}-{{\widetilde}{\lambda}}_1},...,{{\widetilde}{\lambda}}_k},\;each\;{{\widetilde}{\lambda}}_t}$,is even. We call such filtrations plethysm formulas. We extend the above plethysm formula to the version of chain complexes. By plethysm formula we mean the composition of universally free functors. $Let{\emptyset}:G->F$ be a morphism of finite free R-modules. We construct the natural decomposition of $S_{r}(S_2{\emptyset})$,up to filtrations, whose associated graded complex is isomorphic to ${\Sigma}_{{\lambda}{\varepsilon}{\tau}}_r}\;L_{\lambda}{\emptyset}$.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.127-135
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n＋l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)＋as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.

• Pediatric Gastroenterology, Hepatology & Nutrition
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• v.3 no.2
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• pp.151-159
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• 2000

Purpose: Recently, a wide application of gastrofiberscopy in the pediatric group have revealed that nodular duodenitis is not an uncommon disease in children and is suspected to be associated with H. pylori infection. The aim of this retrospective study was to investigate the clinical and histopathologic features in children with nodular duodenitis, and to assess the correlations beween both. Methods: During a period of 5 years (Jan. 1995~Dec. 1999), we investigated clinical, endoscopic and histopathologic features of 39 consecutive patients diagnosed as having nodular duodenitis at Pediatric department of Seoul Red Cross Hospital. In 35 children with nodular duodenitis endoscopic biopsy specimens were stained with Hematoxylin & Eosin and Giemsa's stain, and were graded according to the criteria outlined by Triadafilopoulos, Whitehead et al., and Prieto et al.. Statistical analyses were performed with Graph PAD InStat. Results: The prevalence rate of nodular duodenitis was 17.1% and the most frequent chief complaint was abdominal pain (69.2%). Endoscopically grade 1 was the most common (45.7%) and nodular gastritis was coexistent in 28.3%. The most common histology of the duodenum was grade 2 (54.3%), and the most common histologic score of the stomach was 2 (42.9%). H. pylori was found in the duodenum in 37.1%, and in the stomach in 31.4%. The correlation coefficient between the endoscopic grade and the histologic grade of nodular duodenitis was 0.3983 (p=0.0178). And the correlation coefficient between the histologic grade and the grade of H. pylori colonization in the duodenum was 0.5154 (p=0.0018). Conclusion: There was significant correlation between the endoscopic grade and the histologic grade of nodular duodenitis, and was also significant correlation between the histologic grade and the grade of H. pylori colonization in the duodenum. Therfore H. pylori infection should be regarded as an etiologic factor of nodular duodenitis.