Definition of locally closed subscheme

Question

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.

Answer 1

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 $f$.