Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free resolution of the $i^{-1}\mathcal{O}_Y$-module $\mathcal{O}_X$ called the Koszul resolution. Namely, they say it is $$ 0 \rightarrow K_{n-r} \rightarrow \dots \rightarrow K_1 \rightarrow K_0 = i^{-1}\mathcal{O}_Y \rightarrow \mathcal{O}_X, $$ and choosing a local coordinate system such that on the open subset $U \subseteq Y$ we have $X \cap U = \{ y_{r+1} = \dots = y_n = 0 \}$, we have $$ K_{j} = \bigwedge^j \left( \bigoplus_{s = r+1}^n \; i^{-1}\mathcal{O}_Y dy_{r} \right). $$ I don't understand whether this resolution exists globally or only locally. I guess it exists globally, but I don't understand how to define it. Both answers and references are welcome.
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$\begingroup$ I think one can just define $K_1$ to be the image of $d:i^{-1}\mathcal{I}_X \to i^{-1}\Omega_Y$, and $K_j=\bigwedge^j E_1$. Your local description shows that $K_j$ are locally free $i^{-1}\mathcal{O}_Y$ modules. $\endgroup$– Chan Ki FungCommented Apr 21, 2021 at 15:42
1 Answer
It exists locally, and more generally when $X$ is the zero locus of a global section $s$ of a rank $r$ vector bundle $E$ on $Y$, where $r$ is the codimension of $X$ in $Y$. Then the resolution is given by the celebrated Koszul complex $$0\rightarrow \bigwedge^rE^*\xrightarrow{\ i(s)\ }\bigwedge^{r-1}E^*\rightarrow \cdots \rightarrow E^* \xrightarrow{\ i(s)\ } \mathcal{O}_Y\rightarrow \mathcal{O}_X\rightarrow 0\, ,$$where $i(s)$ denotes the interior product with the section $s$ (googling "Koszul complex" should give you thousands of references). Locally $X$ is given by $r$ equations $f_1=\cdots =f_r=0$, and you can just take $E=\mathcal{O}_Y^r$ and $s=(f_1,\ldots ,f_r)$. Globally such a vector bundle may or may not exist, depending on the geometry of $X$.
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$\begingroup$ Do you know whether it exists globally assuming projective? $\endgroup$ Commented Jul 11, 2020 at 20:18