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1. CJM 2005 (vol 57 pp. 1178)
Asymptotic Behavior of the Length of Local Cohomology Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring,
and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in
$R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show
that
$$
\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d}
=\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d}
$$
always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals
$I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity
of $I$. But we find that this limit may not be rational in general. We give an example
for which the limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.
Keywords:powers of ideals, local cohomology, Hilbert function, linear growth Categories:13D40, 14B15, 13D45 |
2. CJM 1998 (vol 50 pp. 863)
Smooth formal embeddings and the residue complex Let $\pi\colon X \ar S$ be a finite type morphism of noetherian schemes.
A {\it smooth formal embedding\/} of $X$ (over $S$) is a bijective closed
immersion $X \subset \mfrak{X}$, where $\mfrak{X}$ is a noetherian
formal scheme, formally smooth over $S$. An example of such an embedding
is the formal completion $\mfrak{X} = Y_{/ X}$ where $X \subset Y$
is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De~Rham (co)homology.
Our main application is an explicit construction of the Grothendieck
residue complex when $S$ is a regular scheme. By definition the residue
complex is the Cousin complex of $\pi^{!} \mcal{O}_{S}$, as in \cite{RD}.
We start with I-C.~Huang's theory of pseudofunctors on modules with
$0$-dimensional support, which provides a graded sheaf $\bigoplus_{q}
\mcal{K}^{q}_{\,X / S}$. We then use smooth formal embeddings to obtain
the coboundary operator $\delta \colon\mcal{K}^{q}_{X / S} \ar
\mcal{K}^{q + 1}_{\,X / S}$. We exhibit a canonical isomorphism between
the complex $(\mcal{K}^{\bdot}_{\,X / S}, \delta)$ and the residue complex
of \cite{RD}. When $\pi$ is equidimensional of dimension $n$ and
generically smooth we show that $\mrm{H}^{-n} \mcal{K}^{\bdot}_{\,X/S}$
is canonically isomorphic to to the sheaf of regular differentials of
Kunz-Waldi \cite{KW}.
Another issue we discuss is Grothendieck Duality on a noetherian formal
scheme $\mfrak{X}$. Our results on duality are used in the construction
of $\mcal{K}^{\bdot}_{\,X / S}$.
Categories:14B20, 14F10, 14B15, 14F20 |