The question is related to another question asked here a couple of minutes ago: Does vanishing of cohomology of locally free sheaves imply affiness of scheme

In Hartshorne Exercise 4.2,we have the following statement:let $f:X\rightarrow Y$ be a finite surjective morphism of noetherian separated scheme,with $X$ affine,then $Y$ is affine.I wonder whether one is able to formulate purely categorical argument.Let's see what Serre affiness criterion said:If $X$ is quasi compact scheme,if all higher cohomology of quasi coherent sheaves vanish,then $X$ is affine.It is known that one can argue this purely categorical as following: consider category of quasi coherent sheaves on $X$,denoted by $Qcoh(X)$,since all higher cohomology of quasi coherent sheaves vanishing,it means that $Ext^i(O_X,-)=0$ for all $i\geq 1$ which means that $O_X$ is projective generator of this category.(There is some issue here about the definition of sheaf cohomology since we usually define it in category of sheaves of $O_X$-module,but it is not a big deal since for quasi compact scheme $X$,we have adjoint functors between $Qcoh(X)$ and $O_X-mod$,then the injectives transferred,one is able to prove the statement in Exercise 3.6).Then by Gabriel-Popscue theorem,$Qcoh(X)\cong \Gamma(X,O_X)-mod$,then by reconstruction theorem,$X\cong Spec(\Gamma(X,O_X))$,then $X$ is affine.In fact,we can avoid using Gabriel-Pospcue theorem by using Beck theorem(carefully verify the condition and then establish the categories equivalence).

Another similar statement which can be proved categorical is the following well known statement: $f:X\rightarrow Y$ with $f$ affine and $Y$ is affine scheme,then $X$ is affine scheme,the argument is just interpreting $f$ being affine in categorical language($f_*$ has left and right adjoint and it is conservative),then use Beck's theorem and certain routine argument in category theory)

So,I want to formulate a categorical argument to prove Chavelly's theorem.But the problem is the following:How to characterize the surjective morphism of scheme categorically?I noticed that there was a question https://math.stackexchange.com/questions/399892/what-is-the-definition-of-surjective-morphism-of-schemes,but since category of schemes is not abelian category, in general,class of epimorhisms and class of surjective map(or called strict epimorphism in the sense of Kashiwara-Schapira).I am not sure what the "surjective morphism" in the statement of Chavelly's theorem means."Epimorhism" or "strict epimorphism"?

It seems that there are a lot of statements like "this scheme is affine" "this scheme is projective" (to characterize the shape of scheme)can be formulated categorically by essential use of Beck's theorem.(For example, a scheme $X$ with ample line bundle is projective scheme can be proved using Beck's theorem)



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