• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.265-275
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• 1992

The characteristic free representation theory of the general linear group has found a wide range of applications, ranging from the theory of free resolutions to the symmetric function theory. Representation theory is used to facilitate the calculation of explicit free resolutions of large classes of ideals (and modules). Recently, K. Akin and D. A. Buchsbaum [2] realized the Jacobi-Trudi identity for a Schur function as a resolution of GL$_{n}$-modules. Over a field of characteristic zero, it was observed by A. Lascoux [6]. T.Jozefiak and J.Weyman [5] used the Koszul complex to realize a formula of D.E. Littlewood as a resolution of schur modules. This leads us to further study resolutions of Schur modules of a particular form. In this article we will describe some new classes of finite free resolutions associated to the Pieri type skew Young diagrams. As a special case of these finite free resolutions we obtain the generalized Koszul complex constructed in [1]. In section 2 we review some of the basic difinitions and properties of Schur modules that we shall use. In section 3 we describe certain exact complexes associated to the Pieri type skew partitions. Throughout this article, unless otherwise specified, R is a commutative ring with an identity element and a mudule F is a finitely generated free R-module.e.

• 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 $T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}$.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.961-976
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• 2019

Let R be any commutative ring and S be any multiplicative closed set. We introduce an S-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as R is noetherian if and only if $R_S$ is noetherian and every $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension of R, and prove that $R_S$ is noetherian if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is zero; $R_S$ is coherent if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is at most one.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.121-126
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• 1985

D.K. Harrison [5] has shown that if R and S are fields of characteristic different from 2, then two Witt rings W(R) and W(S) are isomorphic if and only if W(R)/I(R)$^{3}$ and W(S)/I(S)$^{3}$ are isomorphic where I(R) and I(S) denote the fundamental ideals of W(R) and W(S) respectively. In [1], J.K. Arason and A. Pfister proved a corresponding result when the characteristics of R and S are 2, and, in [9], K.I. Mandelberg proved the result when R and S are commutative semi-local rings having 2 a unit. In this paper, we prove the result when R and S are 2-fold full rings. Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is nondegenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$ (V, R) induced by B is an isomorphism), and with a quadratic mapping .phi.:V.rarw.R such that B(x,y)=(.phi.(x+y)-.phi.(x)-.phi.(y))/2 and .phi.(rx)= $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U(R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$, .., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2, we reserve the notation [ $a_{11}$, $a_{22}$] for the space.the space.e.e.e.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.167-177
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• 2020

Let I be an ideal of Noetherian local ring (R, m) and M an artinian R-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension r^I (M) of M and the artinianness dimension a^I (M) of M relative to I by r^I (M) = inf{i ∈ ℕ₀ : H^I_i (M) is not representable}, and a^I (M) = inf{i ∈ ℕ₀ : H^I_i (M) is not artinian} and we will prove that i) a^I (M) = r^I (M) = inf{r^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)} ≥ inf{a^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)}, ii) inf{i ∈ ℕ₀ : H^I_i (M) is not minimax} = inf{r^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}. Also, we define the upper representation dimension R^I (M) of M relative to I by R^I (M) = sup{i ∈ ℕ₀ : H^I_i (M) is not representable}, and we will show that i) sup{i ∈ ℕ₀ : H^I_i (M) ≠ 0} = sup{i ∈ ℕ₀ : H^I_i (M) is not artinian} = sup{R^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)}, ii) sup{i ∈ ℕ₀ : H^I_i (M) is not finitely generated} = sup{i ∈ ℕ₀ : H^I_i (M) is not minimax} = sup{R^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}.