컨텐츠 시작
학술대회/행사
초록검색
제출번호(No.) | 0627 |
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분류(Section) | Poster Session |
분과(Session) | Algebra / Representation Theory / Lie Theory (SS-02) |
영문제목 (Title(Eng.)) |
[CANCELED] On the structure of modules in local rings |
저자(Author(s)) |
Pham Hung Quy1 FPT University1 |
초록본문(Abstract) | Let $(R, \frak m)$ be a Noetherian local ring. We always assume that $R$ is the image of a Cohen-Macaulay local ring. Let $M$ be a finitely generated $R$-modules of dimension $d$. Let $\frak a_i(M) = \mathrm{Ann}H^i_{\frak m}(M)$ for all $0 \leq i \leq d-1$ and $\frak a(M) = a_0(M)\cdots a_{d-1}(M)$. Let $\mathfrak{b}(M) = \cap_{\underline{x};i=1}^d \mathrm{Ann}(0:x_i)_{M/(x_1,\ldots,x_{i-1})M},$ where $\underline{x} = x_1,\ldots, x_d$ runs over all systems of parameters of $M$. It is well known that $\sqrt{\frak a(M)} = \sqrt{\frak b(M)}$ and $\dim R/\frak b(M) < d$. We study the splitting of local cohomology modules with respect to a parameter element $x \in \frak b(M)^3$. Recall that the largest submodule of $M$ with dimension less than $d$ is called the unmixed component of $M$ and denoted by $U_M(0)$. We obtain the following theorem. \begin{theorem} Let $R$ and $M$ as above. Let $I$ be an ideal of $M$ and $x \in \frak b(M)^3$ a parameter element of $M$. Let $\overline{M} = M/U_M(0)$ we have $$H^i_I(M/xM) \cong H^i_I(M) \oplus H^{i+1}_I(\overline{M})$$ for all $i < d- \dim R/I$. \end{theorem} Applying Theorem 1 we get the following result. It sheds a light on the structure of non-Cohen-Macaulay modules. It should be noted that $M$ is Cohen-Macaulay if and only if $U_{M/(x_{i+1},\ldots,x_d)M}(0) = 0$ for all $1 \leq i \leq d$ and for some (and hence for all) system of parameters $x_1,\ldots,x_d$ of $M$. \begin{theorem} Let $M$ be a finitely generated $R$-module of dimension $d$, $\underline{x} = x_1,\ldots, x_d$ a system of parameters of $M$ satisfying that $x_i \in \mathfrak{b}(M/(x_{i+1},\ldots,x_d)M)^3$ for all $i \leq d$. Then $U_{M/(x_{i+1},\ldots,x_d)M}(0)$ is independent of the choice of $\underline{x}$ for all $1 \leq i \leq d$ (up to an isomorphism). \end{theorem} |
분류기호 (MSC number(s)) |
13C05, 13D45 |
키워드(Keyword(s)) | local rings, local cohomology |
강연 형태 (Language of Session (Talk)) |
English |