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### Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...

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### Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.
I know well the plus-construction of sheafification, which is presented in Artin's paper ...

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### When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor ...

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### Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf ...

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### How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?
Is there any reference for such staff?
My example of functors underlying this ...

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### restriction and pullback of representable etale sheaf along closed immersion

I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my ...