Lemma 10.131.6. In diagram (10.131.4.1), suppose that $S \to S'$ is surjective with kernel $I \subset S$. Then $\Omega _{S/R} \to \Omega _{S'/R'}$ is surjective with kernel generated as an $S$-module by the elements $\text{d}a$, where $a \in S$ is such that $\varphi (a) \in \beta (R')$. (This includes in particular the elements $\text{d}(i)$, $i \in I$.)

**First proof.**
Consider the map of presentations (10.131.4.2). Clearly the right vertical map of free modules is surjective. Thus the map is surjective. Suppose that some element $\eta $ of $\Omega _{S/R}$ maps to zero in $\Omega _{S'/R'}$. Write $\eta $ as the image of $\sum s_ i[a_ i]$ for some $s_ i, a_ i \in S$. Then we see that $\sum \varphi (s_ i)[\varphi (a_ i)]$ is the image of an element

in the upper left corner of the diagram. Since $\varphi $ is surjective, the terms $s_ j'[a_ j', b_ j']$ and $s_ k'[f_ k', g_ k']$ are in the image of elements in the lower right corner. Thus, modifying $\eta $ and $\theta $ by subtracting the images of these elements, we may assume $\theta = \sum s_ l'[r_ l']$. In other words, we see $\sum \varphi (s_ i)[\varphi (a_ i)]$ is of the form $\sum s'_ l [\beta (r'_ l)]$. Next, we may assume that we have some $a' \in S'$ such that $a' = \varphi (a_ i)$ for all $i$ and $a' = \beta (r_ l')$ for all $l$. This is clear from the direct sum decomposition of the upper right corner of the diagram. Choose $a \in S$ with $\varphi (a) = a'$. Then we can write $a_ i = a + x_ i$ for some $x_ i \in I$. Thus we may assume that all $a_ i$ are equal to $a$ by using the relations that are allowed. But then we may assume our element is of the form $s[a]$. We still know that $\varphi (s)[a'] = \sum \varphi (s_ l')[\beta (r_ l')]$. Hence either $\varphi (s) = 0$ and we're done, or $a' = \varphi (a)$ is in the image of $\beta $ and we're done as well. $\square$

**Second proof.**
We will use the universal property of modules of differentials given in Lemma 10.131.3 without further mention.

In (10.131.4.1) let $R'' = S \times _{S'} R'$. Then we have following diagram:

Let $M$ be an $S$-module. It follows immediately from the definitions that an $R$-derivation $D : S \to M$ is an $R''$-derivation if and only if it annihilates the elements in the image of $R'' \to S$. The universal property translates this into the statement that the natural map $\Omega _{S/R} \to \Omega _{S/R''}$ is surjective with kernel generated as an $S$-module by the image of $R''$.

From the previous paragraph we see that it suffices to show that $\Omega _{S/R} \to \Omega _{S'/R'}$ is an isomorphism when $S \to S'$ is surjective and $R = S \times _{S'} R'$. Let $M'$ be an $S'$-module. Observe that any $R'$-derivation $D' : S' \to M'$ gives an $R$-derivation by precomposing with $S \to S'$. Conversely, suppose $M$ is an $S$-module and $D : S \to M$ is an $R$-derivation. If $i \in I$, then there exist an $a \in R$ with $\alpha (a) = i$ (as $R = S \times _{S'} R'$). It follows that $D(i) = 0$ and hence $0 = D(is) = iD(s)$ for all $s \in S$. Thus the image of $D$ is contained in the submodule $M' \subset M$ of elements annihilated by $I$ and moreover the induced map $S \to M'$ factors through an $R'$-derivation $S' \to M'$. It is an exercise to use the universal property to see that this means $\Omega _{S/R} \to \Omega _{S'/R'}$ is an isomorphism; details omitted. $\square$

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