# Groups acting on schemes: the quotient scheme doesn't always exist.

Preliminary notion: Consider the action of a group $G$ on an object $X$ of some category $\mathcal C$. We have a group homomorphism $\rho:G\longrightarrow\operatorname{Aut}(X)$ which sends $g$ in $\rho_g$. The quotient space is an object $Y$ of $\mathcal C$ with a morphism $p:X\longrightarrow Y$ which satisfies the following universal property:

For every $g\in G$ $p\circ\rho_g=p$ and moreover if $f:X\longrightarrow Z$ is another morphism with this property then there exists a unique morphism $\theta:Y\longrightarrow Z$ such that $f=\theta\circ p$.

Now it is not difficoult to prove (Q.Liu's book exercise 2.14) that given a locally ringed space $(X,\mathcal O_X)$, one can define the quotient locally ringed space $(Y=X/G,\mathcal O_Y)$. In the exercise 3.21 Liu says that for schemes the quotient space is not always well defined (for example it exists when $G$ is finite, the action is free and we are talking about affine schemes). I don't understand where is the obstruction: a scheme is " a particular" locally ringed space because it has a particular decomposition in affine schemes; why this "tiny" difference implies the non-existence of the quotient space?

Edit: Liu also says that sometimes the quotient scheme exists but it is not equal to the quotient space as locally ringed space. This sentece astonished me.

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Liu probably means that the quotient locally ringed space always exists, but it does not have to be a scheme. –  Mikhail Borovoi Feb 25 at 20:02
The point is that the quotient is universal with respect to different categories. Just like the product of schemes is not the same as the product of topological spaces... –  Zhen Lin Feb 26 at 8:30

Well, the category of schemes is a completely different category than the category of locally ringed spaces. It doesn't have colimits (including quotients by group actions), limits, whereas the category of locally ringed spaces has these. Don't forget the forgetful functor here. If you have a group acting on a scheme, then the quotient of the underlying locally ringed space exists, yes, but this doesn't have to be a scheme, and therefore it won't be the quotient in the category of schemes. The exercise mentions an example of this form.

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He also says that sometimes the quotient scheme exists but it is not equal to the quotient space as locally ringed space. This sentece astonished me. –  Dubious Feb 25 at 20:13
See also my answer in the MO thread "common false beliefs" mathoverflow.net/questions/23478 - "(Co)limits may be computed in full subcategories." –  Martin Brandenburg Feb 25 at 21:45

For a finite group acting on a scheme, if each orbit is contained in an open affine, then one can form the quotient scheme, by reducing to the affine case (where one just takes invariants). For example, since any finite set of points in a projective space are contained in the complement of a(n appropriately chosen) hyperplane, we see that for finite groups acting on quasi-projective varieties, the quotient scheme exists.

If there are orbits not contained in open affines, then the quotient need not exist as a scheme. This was one of the factors leading to Michael Artin's invention of algebraic spaces. A particular example is obtained by taking Hironaka's non-projective proper threefold, and taking (or attempting to take) the quotient by an action of the cyclic group of order two. (There is some discussion of this is one of the appendices of Hartshorne.)

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