# Algebraic spaces

Given an algebraic space X we obtain a ringed topos (Sh(X_{etale}), O_X) of sheaves on the small etale site of X endowed with the structure sheaf. This is a locally ringed topos (as in SGA4, Expose IV, Exercise 13.9). Moreover, a morphism X —> Y of algebraic spaces induces a morphism of ringed topoi in the same direction. In fact it is a morphism of locally ringed topoi (see reference above for definition). In fact I think that

Mor(X, Y) —> Mor( (Sh(X_{etale}), O_X), (Sh(Y_{etale}), O_Y) )

is a bijection, i.e., the category of algebraic spaces is a full subcategory of the category of locally ringed topoi. This is sooooo cool!

Allow me to get excited even if you already knew this ages ago. Namely, it means we can describe algebraic spaces as certain locally ringed topoi. This could be helpful for example when we think about thickenings of algebraic spaces: it will allow us to use the same underlying topos of sheaves and just change the structure sheaf (as we do in the case of schemes).

But it goes farther than that. It also means that we can forget about an algebraic space as just a functor on the category of schemes, and consider it as a geometric object it in its own right. Moreover, one of the things that is currently bogging down the stacks project a bit is writing the interface between schemes theory and the theory of algebraic spaces, where in the schemes language we often use points and locality and on the side of algebraic spaces we constantly worry about all scheme valued points of X. It is conceivable that this can be clarified a bit by using the idea above.

Of course we are not going to rewrite the whole thing from scratch, but I hope to add the observation above to the stacks project and then use it whenever I can!

## 4 thoughts on “Algebraic spaces”

1. Pingback: Update « Stacks Project Blog

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3. When working with algebraic spaces, it suffices to look at all affine-valued points. We can show that the category of étale sheaves on the category of schemes can be identified with the category of étale sheaves on the affine étale site. They’re equivalent in the topos perspective, and checking things on affines is quite a bit easier (when using this approach). Toën-Vezzosi show in HAG II that algebraic spaces are precisely those sheaves who admit an atlas of affine schemes whose structure morphisms are relative algebraic spaces with affine diagonal such that the pullback of any object of the cover by an affine admits an affine cover of relative affines, which when composed through the pullback give an étale morphism of affines (i.e. corresponding to an étale ring map).