I have a question on Lei Fu's Etale Cohomology theory, p.75, proposition 2.5.3.
First see the next two linked piles
In this proof, the author argues that since $ \Delta_{X/Y} $ is closed and open immersion and $q$ closed immersion, and $\Delta _{X/S} = q\Delta_{X/Y}$, and $(f \times f)^{*} \ \mathcal{I}$ is the $\mathcal{O}_{X \times_{S} X} $- ideal defining the closed immersion q, it follows that
$ \Omega_{X/S} \cong \Delta^{*}_{X/Y} q^{*}( (f \times f)^{*} \ \mathcal{I}/ (f \times f)^{*} \ \mathcal{I}^{2}) $ $(\star)$
But now I can't understand the ($\star$). By the definition, $ \Omega_{X/S} $ is $\Delta _{X/S} ^{*}$ of the sheaf of ideal defining the closed immersion $\Delta _{X/S} $. So, to show the ($\star$), it suffices to show that $(f \times f)^{*} \ \mathcal{I}$ is also $\mathcal{O}_{X \times_{S} X} $ - ideal defining the closed immersion $\Delta _{X/S} = q\Delta_{X/Y}$ .
Is it true?
My first Attempt : First consider the next reduced case :
$ U \overset{i} \hookrightarrow Y \overset{j} \hookrightarrow Z $ , where i is closed open inclusion ( ; i.e, closed open subscheme), and j is closed inclusion, and $ k:= j \circ i : U \hookrightarrow Z $ is closed immersion.
Then now define $ \mathcal{J} := ker(\mathcal{O}_{Z} \overset {j^{b}} \rightarrow j_{*} \mathcal{O}_{Y} ) $ and $ \mathcal{I} := ker(\mathcal{O}_{Z} \overset {k^{b}} \rightarrow k_{*} \mathcal{O}_{Y}|_{U} ) $ ; i.e., the sheaf of ideals defining $j, k$ respectively.
($j^{b}, k^{b} $ are surjective since j, k are closed immersion)
Q. Then $\mathcal{J} = \mathcal{I}$ ?
To show this, fix an open $V \subseteq Z$ . Then since $ k^{b}_{V} = (j \circ i)^{b}_{V} = i^{b}_{j^{-1}(V)} \circ j^{b}_{V}$, $\mathcal{J}(V) \subseteq \mathcal{I}(V)$ . So it remains to show that $\mathcal{I}(V) \subseteq \mathcal{J}(V)$ . Note that if $i^{b}_{j^{-1}(V)}$ is injective, then $\mathcal{J}(V) \subseteq \mathcal{I}(V)$ and we are done.
But I'm stucked on this step. Can we show that $i^{b}_{j^{-1}(V)}$ is injective? If not, can we show $\mathcal{J} = \mathcal{I}$ or the above $(\star)$ by other method?
Further progress : If we can assume that $X,Y,S$ are affine in the original statement in the linked pile, then $X, X \times _{Y} X , X \times _{S} X $ are also affine schemes and we can consider next reduced situation ; $ U:= Spec((A/a)/(b/a)) \overset{\cong} \rightarrow V(b/a) \subseteq Y:= Spec(A/a) \overset{\cong} \rightarrow V(a) \subseteq Z:=Spec(A) $
, where $a$ is an ideal of $A$ and $b$ is an ideal of $A$ containing $a$ and $j : Y : \rightarrow Z$ closed immersion and $ i : U \rightarrow Y$ is closed open immersion, which are induced from $\varphi _{1} : A \rightarrow A/a $ and $ \varphi_{2} : A/a \rightarrow ((A/a)/(b/a)) $ respectively. But even in this simpler case, I am stucked to show that $i^{b}_{j^{-1}(V)}$ is injective when $V$ is and open set in $Z$, even in the case $V=D(f)$ is principal.
Can anyone help me?