Schema/Syzygienbündel/Geometrische Realisierung/en/Bemerkung

For a surjective morphism

$\varphi \colon {\mathcal {O}}_{X}^{n}\longrightarrow {\mathcal {O}}_{X}$

on a scheme ${}X$ given by elements ${}f_{1},\ldots ,f_{n}\in \Gamma (X,{\mathcal {O}}_{X})$ (the surjectivity means that these elements generate locally the unit ideal) we can realize the corresponding locally free kernel sheaf in the following natural way. We can directly look at the corresponding surjection of geometric vector bundles

$\varphi \colon {{\mathbb {A} }_{X}^{n}}\longrightarrow {\mathbb {A} }_{X}^{1},\,(v_{1},\ldots ,v_{n})\longmapsto \sum _{i=1}^{n}f_{i}v_{i},$

and the kernel consists for every base point ${}x\in X$ in the solution set

${\left\{(v_{1},\ldots ,v_{n})\in (\kappa (x))^{n}\mid \sum _{i=1}^{n}f_{i}(x)v_{i}=0\right\}}$

to this linear equation over the residue class field ${}\kappa (x)$. So fiberwise this syzygy bundle is a very simple object, but of course the solution space varies with the basis. If ${}X=\operatorname {Spec} {\left(R\right)}$ is affine, then one can also describe the syzygy bundle as the spectrum of the ${}R$-algebra

$R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}\right)}.$