# Another point of view

if you follow the page you quote: Functorial characterization of open subschemes. There are also correspondence notions for separatedness and properness and so on. Let me elaborate a bit.

What you do is to identify a commutative scheme $X$ with $Qcoh_{X}$(Gabriel-Rosenberg reconstruction theorem). Let$f_{*}$=$F$

:$Qcoh_{X}\rightarrow Qcoh_{Y}$ (Assume $X,Y$ are quasi compact and quasi separated).

**Affineness**
$F$ is affine if $f_{*}$ is conservative(faithful in abelian case),having left adjoint functor

$f^{*}$ and having right adjoint functor $f^{!}$

**closed immersion**

Let $C_{X}=Qcoh_{X}$ and $C_{U}=Qcoh_{U}$.(Suppose they are abelian categories). Then $C_{U}\rightarrow C_{X}$ ($u_{*}$) is *closed immersion* if ($u_{*}$) is an categorical equivalence of:

$C_{U}$ and full **topologizing subcategory** $C_{V}$ of $C_{X}$(*topologizing* subcategory is full subcategory which is closed under finite direct sum and subquotient taken in $C_{X}$)

**thickennings**

We call a closed immersion $U\rightarrow T$ a *thickenning*, if the smallest saturated multiplicative system in $HomC_{T}$ containing $(u*)(HomC_{U})$ coincides with $Hom(C_{T})$

**Formally smooth,formally unramified,formally etale**

I will talk about these notions later. They are defined via thickennings.

**separatedness and properness**

Once you have the definition of closed immersion given above, then the definition of separatedness is free(follows the same pattern as EGA) Properness is similar. I will formulated later.

# Notice

The reason to identify space with category of quasi coherent sheaves on it is mainly for noncommutative algebraic geometry. What I wrote here is trivial case of this consideration because we can drop the categorical language in commutative case.

Functor point of view and categorical point of view are not equivalent in general