Given two schemes $X, Y$ with a map $f: X \rightarrow Y$, one should have a map $df: T^*Y \times_Y X \rightarrow T^*X$. How is this map defined (using the language of algebraic geometry, rather than the language of manifolds)?
Definition of differential of map (in algebraic geometry)
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algebraic-geometry
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2One doesn't usually have a cotangent *bundle* so much as a cotangent *sheaf*, in which case it is not appropriate to use fibre product notation for pullback. – 2012-12-18
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1And the cotangent sheaves are relative some basis. – 2012-12-18