Abstract view
Smooth formal embeddings and the residue complex


Published:19980801
Printed: Aug 1998
Abstract
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 IC.~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
KunzWaldi \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 $\mcal{K}^{\bdot}_{\,X / S}$.