# Well-definedness of conormal sheaf

In the definition of the conormal sheaf, we are given a locally closed immersion $$X \to Y$$, which factors through some closed subscheme $$Z$$ so that we have $$X \to Z \to Y$$, where the first map is a closed immersion and the second an open immersion.

We then take the ideal sheaf $$\mathcal{I}$$ corresponding to the closed immersion $$f:X\to Z$$, and define the conormal to be $$\mathcal{I}/\mathcal{I^2}$$, viewed as an $$\mathcal{O}_X$$-module (is it correct to say that the conormal is $$f^*(\mathcal{I}/\mathcal{I^2})$$?).

I've tried to show that is independent of the choice of $$Z$$, but haven't really managed to work it out successfully (perhaps factoring through the scheme-theoretic closure might work?). How does one show that the conormal is well-defined?

Oops, this is rather silly. If $$\Delta (X)\subset U,V\subset Y$$ so that $$X\to U\subset Y,X\to V\subset Y$$ are both the same locally closed immersion $$\Delta$$, and also noting that if we ever have a morphism $$f:X\to Y$$ with $$U\subset X$$ maps into $$V\supset f(U)$$, then $$f^*$$ commutes with restriction, we get $$\Delta^*(\mathcal{I}/{\mathcal{I}^2})|_X=\Delta_X^*(\mathcal{I}/\mathcal{I}^2|_V)=\Delta_X^*(\mathcal{I}/\mathcal{I}^2|_U)$$. In particular, there is nothing special about $$\mathcal{I}/\mathcal{I^2}$$, If I'm not mistaken.

• You wrote $X\rightarrow U\subset Y$. Later you wrote $f: X\rightarrow Y$ with $U\subset X$. Is $X$ contained in $U$ or contains $U$? Commented Jun 24, 2023 at 12:20
• I find [noting that if we ever have a morphism $f:X\to Y$ with $U\subset X$ maps into $V\supset f(U)$] confusing. Commented Jun 24, 2023 at 12:23

Factor the locally closed immersion $$i: X\rightarrow Y$$ as below, such that $$i_U: X\rightarrow U$$ is a closed immersion and $$U$$ is an open subscheme of $$X$$.

Let $$\mathscr{I}_U$$ be the ideal sheaf for $$X$$ in $$U$$. The conormal sheaf is defined as:

$$\mathscr{N}_{X/Y}^{\vee} = i_U^*\mathscr{I}_U.$$

To show the definition is independent of the choice of $$U$$, pick another open subscheme $$V$$ of $$Y$$ having $$X$$ as a closed subscheme. Then $$X$$ is a closed subscheme of $$U\cap V$$, and we have the following commutative diagram, where $$i_{U\cap V}$$ is a closed immersion with ideal sheaf $$\mathscr{I}_{U\cap V}=\mathscr{I}_U|_{U\cap V}$$.

Thus $$i_U^*\mathscr{I}_U = i_{U\cap V}^*\left(\mathscr{I}_{U}|_{U\cap V}\right) = i_{U\cap V}^*\mathscr{I}_{U\cap V}.$$ Similarly $$i_V^*\mathscr{I}_V = i_{U\cap V}^*\mathscr{I}_{U\cap V}$$.