Birational morphisms of integral finite-type schemes

Let $k$ be a field and let $X$ and $Y$ be integral $k$-schemes of finite type (Integral schemes, Locally finite type and finite type morphisms). Let ${\eta }_X$ and ${\eta }_Y$ be their generic points (Generic points of irreducible closed subsets) and let $K(X)={\mathcal{O}}_{X,{\eta }_X},K(Y)={\mathcal{O}}_{Y,{\eta }_Y}$ be their function fields; by Function field of an integral finite-type scheme these stalks are fields and are canonically identified with the fraction fields of the coordinate rings of every nonempty affine open of $X$, respectively $Y$.

A morphism of $k$-schemes $f:X\to Y$ is birational when

1. $f({\eta }_X)={\eta }_Y$, and
2. the map on stalks ${\mathcal{O}}_{Y,{\eta }_Y}\to {\mathcal{O}}_{X,{\eta }_X}$ induced by $f$ (Morphisms of schemes) is an isomorphism.

Conditions (1) and (2) are the form taken by the definition of a birational morphism of schemes (Stacks Project, Definition 29.51.1, tag 01RO) for integral schemes: for such $X$ and $Y$ the generic points ${\eta }_X$ and ${\eta }_Y$ are the generic points of the unique irreducible components, and the local rings at them are the function fields. Thus a birational morphism identifies the function field of $Y$ with the function field of $X$.