What is the definition of relative differential forms of a family $\pi: X \to B$ of (nodal) curves, where $B$ is the base space.



What are the relative differential forms of a family of (nodal) curves?
I learnt the following from the paper *On the relative de Rham sequence* ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=681850) [JSTOR](http://www.jstor.org/stable/2043718)) by Nick Buchdahl. It is more general than the OP situation, but should easily specialise to it.




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Let $f:X \to B$ be a smooth map between manifolds. Then the sheaf $\Omega^1\_f$ of germs of *relative 1-forms* on $X$ is defined by the following exact sequence
$$
f^\*\Omega^1\_B \to \Omega^1\_X \to \Omega^1\_f \to 0
$$


Then the relative differential $k$-forms are defined by
$$
\Omega^k\_f = \Lambda^k \Omega^1\_f
$$
as usual. Differentiating along the fibres gives a differential $d\_f: \Omega^k\_f \to \Omega^{k+1}\_f$. and $(\Omega^\bullet\_f, d\_f)$ is called the *relative de Rham complex*.


When $f$ is a submersion, the relative de Rham complex gives a resolution
$$
0 \to f^{-1}\mathcal{E}\_B \to \Omega^\bullet\_f
$$
of the sheaf of functions of $X$ which are constant on the fibres.


