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Various Row Invariants on Cohen-Macaulay Rings
Lee, Kisuk The Basic Science Institute Chosun University 2014 조선자연과학논문집 Vol.7 No.4
We define a numerical invariant $row^*_j(A)$ over Cohen-Macaulay local ring A, which is related to the presenting matrices of the j-th syzygy module (with or without free summands). We show that $row_d(A)$=$row_{CM}(A)$ and $row^*_d(A)$=$row^*_{CM}(A)$ for a Cohen-Macaulay local ring A of dimension d.
PRESENTING MATRICES OF MAXIMAL COHEN-MACAULAY MODULES
Lee, Ki-Suk Korean Mathematical Society 2007 대한수학회보 Vol.44 No.4
We define a numerical invariant $row_{CM}(A)$ over Cohen-Macaulay local ring A, which is related to rows of the presenting matrices of maximal Cohen-Macaulay modules without free summands. We show that $row(A)=row_{CM}(A)$ for a Cohen-Macaulay(not necessarily Gorenstein) local ring A.
Some remarks on types of Noetherian local rings
이기석 충청수학회 2014 충청수학회지 Vol.27 No.4
We study some results which concern the types of Noe- therian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either Ap is Cohen-Macaulay, or r(Ap) ≤ depthAp +1 for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.
ON TYPES OF NOETHERIAN LOCAL RINGS AND MODULES
Lee, Ki-Suk Korean Mathematical Society 2007 대한수학회지 Vol.44 No.4
We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${\le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${\le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.
SOME REMARKS ON TYPES OF NOETHERIAN LOCAL RINGS
Lee, Kisuk Chungcheong Mathematical Society 2014 충청수학회지 Vol.27 No.4
We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.
A NOTE ON TYPES OF NOETHERIAN LOCAL RINGS
Lee, Kisuk Korean Mathematical Society 2002 대한수학회보 Vol.39 No.4
In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring A is less than or equal to depth A + 1, and $\hat{A}_p$ is Cohen-Macaulay for every prime $p\neq\hat{m}$, then A is Cohen-Macaulay. (This implies the previously known result: if A satisfies $(S_{n-1})}$, where n is the type of a .ins A, then A is Cohen-Macaulay.)
ON CERTAIN GRADED RINGS WITH MINIMAL MULTIPLICITY
Kim, Mee-Kyoung Korean Mathematical Society 1996 대한수학회논문집 Vol.11 No.4
Let (R,m) be a Cohen-Macaulay local ring with an infinite residue field and let $J = (a_1, \cdots, a_l)$ be a minimal reduction of an equimultiple ideal I of R. In this paper we shall prove that the following conditions are equivalent: (1) $I^2 = JI$. (2) $gr_I(R)/mgr_I(R)$ is Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal N. (3) $N^2 = (a'_1, \cdots, a'_l)N$, where $a'_i$ denotes the images of $a_i$ in I/mI for $i = 1, \cdots, l$.
A NOTE ON COHOMOLOGICAL DIMENSION OVER COHEN-MACAULAY RINGS
Bagheriyeh, Iraj,Bahmanpour, Kamal,Ghasemi, Ghader Korean Mathematical Society 2020 대한수학회보 Vol.57 No.2
Let (R, m) be a Noetherian local Cohen-Macaulay ring and I be a proper ideal of R. Assume that β<sub>R</sub>(I, R) denotes the constant value of depth<sub>R</sub>(R/I<sup>n</sup>) for n ≫ 0. In this paper we introduce the new notion γ<sub>R</sub>(I, R) and then we prove the following inequalities: β<sub>R</sub>(I, R) ≤ γ<sub>R</sub>(I, R) ≤ dim R - cd(I, R) ≤ dim R/I. Also, some applications of these inequalities will be included.
A note on cohomological dimension over Cohen-Macaulay rings
Iraj Bagheriyeh,Kamal Bahmanpour,Ghader Ghasemi 대한수학회 2020 대한수학회보 Vol.57 No.2
Let $(R,\m)$ be a Noetherian local Cohen-Macaulay ring and $I$ be a proper ideal of $R$. Assume that $\beta_R(I,R)$ denotes the constant value of ${\rm depth}_R(R/I^{n})$ for $n\gg0$. In this paper we introduce the new notion $\gamma_R(I,R)$ and then we prove the following inequalities: $$\beta_R(I,R)\leq\gamma_R(I,R)\leq \dim R - \cd(I,R)\leq \dim R/I.$$ Also, some applications of these inequalities will be included.