ag.algebraic geometry - Functorial characterization of morphisms of schemes

Check out the paper of Kontsevich-Rosenberg Noncommutative spaces and Noncommutative Grassmannian and related constructions. You will get what you want. i.e. the definition of properness and separatedness of presheaves(as functors, taken as "space") and morphism between presheaves(natural transformations).

Notice that these definitions are general treatment for algebraic geometry in functorial point of view,nothing to do with noncommutative.

Definition for separated morphism and separated presheaves

Let $X$ and $Y$ be presheaves of sets on a category $A$(in particular, $CRings^{op}$). We call a morphism $Xrightarrow Y$ separated if the canonical morphism

$Xrightarrow Xtimes _{Y}X$ is closed immersion We say a presheaf $X$ on $A$ is separated if the diagonal morphism:
$Xrightarrow Xtimes X$ is closed immersion

Definition for strict monomorphism and closed immersion

For a morphism $f$: $Yrightarrow X$ of a category $A$, denote by $Lambda _{f}$ the class of all pairs of morphisms

$u_{1}$,$u_{2}$:$XRightarrow V$ equalizing $f$, then $f$ is called a strict monomorphism if any morphism $g$: $Zrightarrow X$ such that $Lambda_{f}subseteq Lambda_{g}$ has a unique decomposition: $g=fcdot g'$

Now we have come to the definition of closed immersion: Let $F,G$ be presheaves of sets on $A$. A morphism $Frightarrow G$ a closed immersion if it is representable by a strict monomorphism.

Example

Let $A$ be the category $CAff/k$ of commutative affine schemes over $Spec(k)$, then strict monomorphisms are exactly closed immersion(classcial sense)of affine schemes. Let $X,Y$ be arbitrary schemes identified with the correspondence sheaves of sets on the category $CAff/k$. Then a morphism $Xrightarrow Y$ is a closed immersion iff it is a closed immersion in classical sense(Hartshorne or EGA)

Definition for proper morphism just follows the classical definition: i.e. universal closed and separated. You can also find the definition of universal closed morphism in functorial flavor in the paper I mentioned.

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