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I have been trying to really understand the sheaf of differentials defined on a scheme but am having a lot of trouble. Let $X$ be a scheme separated over a field $k$. Suppose we have an open affine cover $\{ U_{i} = \operatorname{spec}A_{i} \}$ of $X$. On each of these $U_{i}$ we would like to define the quasi-coherent sheaf $\mathcal{F}_{i}$ given by $\widetilde{\Omega_{A_{i}/k}}$. The goal then is to glue all the $\mathcal{F}_{i}$.

The main goal is to define isomorphisms, $$ \mathcal{F_{i}}|_{U_{i} \cap U_{j}} {\longrightarrow} \mathcal{F_{j}}|_{U_{i} \cap U_{j}} $$ and this is the part I am having trouble with. Having defined each $\mathcal{F}_{i}$ to be quasi-coherent, and having judiciously chosen $X$ to be separated, we have that intersections are given by $U_{i} \cap U_{j} = \operatorname{spec}(A_{i} \otimes_{k} A_{j})$ and so

\begin{align} \mathcal{F}_{i}|_{U_{i} \cap U_{j}} &\simeq \widetilde{\Omega_{A_{i}/k}}|_{U_{i} \cap U_{j}} \\ &\simeq \big( \Omega_{A_{i}/k} \otimes_{A_{i}} (A_{i} \otimes_{k} A_{j}) \big)^{\sim} \\ &\simeq \widetilde{\Omega_{(A_{i} \otimes_{k} A_{j})/A_{j}}} \end{align} Likewise we have $$ \mathcal{F}_{j}|_{U_{i} \cap U_{j}} \simeq \widetilde{\Omega_{(A_{i} \otimes_{k} A_{j})/A_{i}}}. $$ But now I am at a loss as to how to show there is a canonical (I want it to be canonical so it is easy to see that the cocycle condition holds) isomorphism, $$ \Omega_{(A_{i} \otimes_{k} A_{j})/A_{j}} \longrightarrow \Omega_{(A_{i} \otimes_{k}. A_{j})/A_{i}} $$ I tried to show this using the universal property and Yoneda's lemma, but I'm not even sure how to work with derivations from tensor products of rings. Is this actually the right approach to glue the sheaf of differentials? I am mostly learning from Ravi Vakil's notes, and he seems to glue things by showing that they restrict nicely and then using universal properties to show those restrictions are canonically isomorphic. That is what I have tried to do here, but I'm not sure if it will work.

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  • $\begingroup$ There's a lot going on here. First, have you checked Vakil's definition/construction of $\Omega$? This could probably help a bit. Next, if you want to proceed this way, your tensor products are wrong: they should not be over $k$, because that's going to give you $U_i\times U_j$ instead of $U_i\cap U_j$. To proceed, it's probably better to use proposition 5.3.1, that the intersection of affine schemes is covered by simultaneously-distinguished open sets: this gives you a lot more control over what's going on here, since arbitrary affine open subschemes can be rather funny. $\endgroup$
    – KReiser
    Oct 21 '20 at 6:39
  • $\begingroup$ It isn’t true that $U_i \cap U_j$ is the spectrum of $A_i \otimes_k A_j$, look for instance at the projective space. What’s true is that $U_i \cap U_j$ is the spectrum of a quotient of $A_i \otimes_k A_j$. $\endgroup$
    – Mindlack
    Oct 21 '20 at 6:41
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I do not think you have to use glueing for the following reason. Let $p,q:X\times_S X \rightarrow X$ be the projection morphisms and let $U:=Spec(A) \subseteq X$ be an affine open subscheme mapping to the open subscheme $V:=Spec(R)\subseteq S$. Let $\mathcal{I}\subseteq \mathcal{O}_{X\times_S X}$ be the ideal of the diagonal and define $\Omega:=\Omega^1_{X/S}:=p_*(\mathcal{I}/\mathcal{I}^2)$. It follows

$$\Omega^1(U):=(\mathcal{I}/\mathcal{I}^2)(p^{-1}(U)):= (\mathcal{I}/\mathcal{I}^2)(U\times_S X)\cong (\mathcal{I}/\mathcal{I}^2)(U\times_V U)$$

since $\mathcal{I}/\mathcal{I}^2$ is supported on the diagonal.

It follows

$$\mathcal{I}/\mathcal{I}^2(U\times_V U):=I/I^2$$

where $I\subseteq A\otimes_R A$ is the ideal of the diagonal. Hence for any open subscheme $U$ it follows $\Omega^1(U)$ is the left $A$-module $\Omega^1_{A/R}$.

Note that for $a\in A$ and $\omega:=\sum_i u_i\otimes v_i \in \Omega^1_{A/R}$ it follows

$aw:=\sum_i (au_i)\otimes v_i =\sum_i u_i\otimes (v_ia)=\omega a$.

Hence the left and right $\mathcal{O}_X$-module structure on $\Omega^1_{X/S}$ agree.

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