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.
Thanks in advance.
Best Answer
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.