# Limit preserving and finite presentation

Let A —> B be a ring map. Let R = colim R_i be a filtered colimit of A-algebras. Then there is a canonical map

colim Hom_A(B, R_i) ——> Hom_A(B, R)

By Tag 00QO the following are equivalent

1. A —> B is of finite presentation,
2. the map above is bijective for all R = colim R_i
3. the map above is surjective for all R = colim R_i

Let S be a scheme. Let X be a scheme over S. Let T = lim T_i be a directed limit of affine schemes over S. Then there is a canonical map

colim Mor_S(T_i, X) ——> Mor_S(T, X)

By Tag 01ZC and Tag 0CM0 the following are equivalent

1. X —> S is locally of finite presentation,
2. the map above is bijective for all T = lim T_i
3. the map above is surjective for all T = lim T_i

The same thing is true if X and S are algebraic spaces (Tag 04AK and Tag 0CM6).

I didn’t know you could replace bijectivity by surjectivity in the criterion. But somewhere in the Stacks project we used this fact without proof, so it had better be true, right?

A related result is that to check a morphism f of algebraic stacks is locally of finite presentation, you need only check f is limit preserving on objects (this is the analogue of the above and it says that certain functors are essentially surjective). You can find this in Tag 0CMQ.

Caveat: as this only applies to situations where you already know your functors (or stacks in groupoids) are algebraic spaces (or stacks), it probably won’t be that useful. Often when we try to show a stack is limit preserving, it is part of applying Artin’s criteria and then we don’t yet know our stack is algebraic of course.

Thanks for reading!

[Edit on 6/30/2016: Matthew Emerton just pointed out that this observation was already in Lemma 2.3.15 of his paper with Toby Gee. I must have read it and then forgotten that I had. Apologies to everybody.]

# Up to date again

Worked through your comments online as well as those emailed to me. Please let me know if I forgot to do something. Thanks for all your help!