# 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 '14 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 '14 at 8:30