Because Kähler differentials are compatible with localization, they may be constructed on a general scheme by performing either of the two definitions above on affine open subschemes and gluing. However, the second definition has a geometric interpretation that globalizes immediately. In this interpretation, represents the "ideal defining the diagonal" in the fiber product of with itself over . This construction therefore has a more geometric flavor, in the sense that the notion of "first infinitesimal neighbourhood" of the diagonal is thereby captured, via functions vanishing modulo functions vanishing at least to second order (see cotangent space for related notions). Moreover, it extends to a general morphism of schemes formula_27 by setting formula_28 to be the ideal of the diagonal in the fiber product formula_29. The "cotangent sheaf" formula_30, together with the derivation formula_31 defined analogously to before, is universal among formula_32-linear derivations of formula_33-modules. If is an open affine subscheme of whose image in is contained in an open affine subscheme , then the cotangent sheaf restricts to a sheaf on which is similarly universal. It is therefore the sheaf associated to the module of Kähler differentials for the rings underlying and .