Title:  Smooth Formal Embeddings

Publication status: Canadian Math. J. 50 (1998) 863-896.

Let $\pi : X \ar S$ be a finite type morphism of noetherian schemes. A {\em 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 $\delta : \mcal{K}^{q}_{X / S} \ar \mcal{K}^{q + 1}_{X / S}$. 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 $\mcal{K}^{\bdot}_{X / S}$.

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(updated 23 May 2009)