Morphism of sheaves induced by a morphism of algebraic varieties

Question

In a book D-module, Perverse Sheaves, and Representation Theory (HTT) page 21, I read that a morphism $f:X\rightarrow Y$ of smooth algebraic varieties canonically induces a $\mathcal{O}_X$-linear morphism $\mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\Omega^1_Y \rightarrow \Omega^1_X$ and its dual $\Theta_X\rightarrow f^*\Theta_Y=\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\Theta^1_Y$. Here $\Omega^1$ and $\Theta$ stand for cotangent sheaf and tangent sheaf, respectively. I wonder how these maps are constructed. It seems to be a very general construction in algebraic geometry but I couldn't find a useful reference yet. Can anyone give an explanation about this, or at least a reference?

Answer 1

$\DeclareMathOperator{\Spec}{Spec}$Let $f\colon X\to Y$ be a morphism of schemes over a scheme $S$. Then it induces a morphism $\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}\Omega^1_{Y/S}\to \Omega^1_{X/S}$ as follows.

Suppose that $S=\Spec A$, $X=\Spec B$ and $Y=\Spec C$. Let $\varphi\colon C\to B$ be the morphism corresponding to $f$. Then we have a morphism of $B$-modules $$ B\otimes_C\Omega^1_{C/A}\to \Omega^1_{B/A};\quad b\otimes dc\mapsto b\cdot d\varphi(c). $$ Informally, this morphism is given by the "pull-back of 1-forms". In a general case, we can construct the morphism by patching these morphisms together.