# Definition of locally closed subscheme

I'm wondering how to define a locally closed subscheme formally. My attempt is to define it as a morphism $$f:X\rightarrow Y$$ which can factor as $$f = g\circ i$$ where $$g$$ is an open immersion and $$i$$ closed immersion.

My question:

1. Is the factorization $$f = g\circ i$$ unique in some sense? Could it be described by ideal sheaf?

2. For any given locally closed subset, is there a unique reduced scheme structure making it as a locally closed subscheme?

3. Is quasiprojective scheme locally closed subscheme? If it is, when we say $$X$$ is a quasiprojective scheme, is the immersion $$f$$ part of data of $$X$$ or we only need $$X$$ can be immerse into some $$\mathbb P^n$$?

In Hartshorne, quasiprojective morphism is defined as a morphism $$f:X\rightarrow Y$$ is quasiprojective if it factors into an open immersion $$j:X'\rightarrow X$$ followed by a projective morphism $$g:X'\rightarrow Y$$. The order of composition is different from that in immersion.

In some other place, quasiprojective variety might be even defined as: a quasiprojective variety is a open subset of a projective variety. For which I don't know if projective variety is part of data.

1. No. Let $$x\in\Bbb A^1_k$$ be a closed point, and consider $$\{x\}\to U\to \Bbb A^1_k$$ as $$U$$ varies among the open subsets of $$\Bbb A^1_k$$ containing $$x$$.
2. Yes. Suppose $$W\subset X$$ is a locally closed subset. Write $$W=U\cap V$$ with $$U$$ open and $$V$$ closed. Equip $$U$$ with the restriction of the structure sheaf on $$X$$ and $$V$$ with the reduced induced subscheme structure. Then $$W=U\times_X V$$ is a reduced subscheme structure on $$W$$.
3. Hartshorne's definition of a locally closed immersion is famously not quite correct in general. To be more specific, Hartshorne's locally closed immersion is the opposite order from what you've written - an open immersion in to a closed subscheme. One can interchange the order of these when either the source is reduced or the composite morphism is quasi-compact. The latter condition is satisfied when the source is (locally) noetherian, and given Hartshorne's general perspective on noetherian hypotheses, I think the best way to proceed is to assume he's working in the locally noetherian situation. This gives that a scheme $$X$$ quasi-projective over $$S$$ via $$f:X\to S$$ can be written as a locally closed subscheme of $$\Bbb P^n_S$$ for some $$n$$. Generally one interprets this as the second condition you've written - there exists a map, but it's not generally packaged as part of the data of $$f$$.