Lemma 48.31.2. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $j' : U \to X'$ be a compactification of $U$ over $X$ (see proof) and denote $f : X' \to X$ the structure morphism. Then we have a canonical isomorphism $Rj_! \to Rf_* \circ R(j')_!$ of functors $D^ b_{\textit{Coh}}(\mathcal{O}_ U) \to \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ where $Rj_!$ and $Rj'_!$ are as in Remark 48.30.5.
Proof. The fact that $X'$ is a compactification of $U$ over $X$ means precisely that $f : X' \to X$ is proper, that $j'$ is an open immersion, and $j = f \circ j'$. See More on Flatness, Section 38.32. If $j'(U) = f^{-1}(j(U))$, then the lemma follows immediately from Lemma 48.31.1. If $j'(U) \not= f^{-1}(j(U))$, then denote $X'' \subset X'$ the scheme theoretic closure of $j' : U \to X'$ and denote $j'' : U \to X''$ the corresponding open immersion. Picture
\[ \xymatrix{ & & X'' \ar[d]^{f'} \\ & & X' \ar[d]^ f \\ U \ar[rr]^ j \ar[rru]^{j'} \ar[rruu]^{j''} & & X } \]
By More on Flatness, Lemma 38.32.1 part (c) and the discussion above we have isomorphisms $Rf'_* \circ Rj''_! = Rj'_!$ and $R(f \circ f')_* \circ Rj''_! = Rj_!$. Since $R(f \circ f')_* = Rf_* \circ Rf'_*$ we conclude. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)