Condition for a morphism of schemes to factor through a closed subscheme

Suppose $$f\colon Y\to X$$ is a morphism of schemes. Suppose $$Z\hookrightarrow X$$ is a closed subscheme defined by the quasi-coherent ideal $$\mathcal J\subset \mathcal O_X$$. The Stacks Project claims that $$f$$ factors through the inclusion $$Z\hookrightarrow X$$ precisely if the morphism $$f^\ast \mathcal J\to f^\ast \mathcal O_X=\mathcal O_Y$$ is the zero morphism. As far as I can tell, this is equivalent to the condition that $$\mathcal J\hookrightarrow \mathcal O_X\to f_\ast \mathcal O_Y$$ is the zero morphism, since the adjunction should preserve the zero morphisms.

However, this is never stated anywhere, which makes me suspicious, since I find my condition more straightforward than the one on the Stacks Project, so I don't know why it is not mentioned.

I would very much appreciate if anybody is willing to verify (or falsify) my thoughts.

• Indeed, the two conditions are equivalent. Nov 29, 2018 at 9:01
• Do you know the support of an indeal sheaf? Nov 30, 2018 at 22:09
• @Armando j18eos Yes I certainly do. I don't see how this is related to my question, though.
– asdq
Dec 1, 2018 at 4:59

As one knows, a morphism of schemes is a pair $$(f,f^{\sharp}):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X)$$ where $$f:Y\to X$$ is a map of topological spaces and $$f^{\sharp}:\mathcal{O}_X\to f_{*}\mathcal{O}_Y$$ is a morphism of (quasi-coherent) sheaves; let $$\mathcal{I}=\ker f^{\sharp}$$ be, it is a quasi-coherent ideal sheaf and one has the factorization $$\begin{equation*} 0\to\mathcal{I}\hookrightarrow\mathcal{O}_X\underbrace{\xrightarrow{i^{\sharp}}\mathcal{O}_{X\displaystyle/\mathcal{I}}\hookrightarrow}_{f^{\sharp}}f_{*}\mathcal{O}_Y\to0 \end{equation*}$$ where $$i^{\sharp}$$ is a locally surjective morphism of (quasi-coherent) sheaves; I mean, for any $$y\in Y,\,i^{\sharp}_{f(y)}:\mathcal{O}_{X,f(y)}\to\left(\mathcal{O}_{X\displaystyle/\mathcal{I}}\right)_{f(y)}$$ is a surjective morphism of (local) rings.
Applying the $$\mathbf{Spec}$$ functor to previous right-exact sequence of quasi-coherent $$\mathcal{O}_X$$-algebras, one has the sequence of schemes $$\begin{equation*} Y\underbrace{\to\mathbf{Spec}\left(f_{*}\mathcal{O}_Y\right)\to\mathbf{Spec}\left(\mathcal{O}_{X\displaystyle/\mathcal{I}}\right)\cong V(\mathcal{I})\hookrightarrow}_f\mathbf{Spec}(\mathcal{O}_X)=X \end{equation*}$$ where $$V(\mathcal{I})$$ is the support of $$\mathcal{I}$$, and it is a closed subscheme of $$X$$.