컨텐츠 시작

학술대회/행사

초록검색

제출번호(No.) 0627
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