For a sequence of positive integers $a\_0, \ldots, a\_n$ and a ring $R$, there is a graded ring $R[x\_0,\ldots, x\_n]$ where $x\_i$ is in degree $a\_i$. There is a corresponding $\mathbb{G}\_m$-action on $Spec R[x\_0,\dots, x\_n] = \mathbb{A}^{n+1}\_R$ induced by


$R[x\_0,\dots, x\_n] \to R[x\_0,\dots, x\_n] \otimes \mathbb{Z}[x,x^{-1}], x\_i \mapsto x\_i \otimes t^{a\_i}$


The stack quotient $\mathcal{P}\_R(a\_0,\dots, a\_n) = [\mathbb{A}^{n+1}\_R-\{0\}/\mathbb{G}\_m]$ is a called a *weighted projective stack*. For $a\_0 = \cdots = a\_n = 1$, one gets back the usual projective space. 


This seems to be an important example of an algebraic stack, but I am unable to find a reference where basic facts about weighted projective stacks are proven. For example: Is there a reference where it is proven that $\mathcal{P}\_R(a\_0,\dots, a\_n)$ is proper and smooth over $Spec R$?



Reference for Weighted Projective Stacks
I don't know a reference, but I think you can make the following argument. 


The weighted projective stack $X$ is a $\mathbb{G}\_m$-quotient of a scheme smooth over $R$, so it must be smooth over $R$ too: smoothness descends under flat morphisms, in the sense that if $Z \to W$ is faithfully flat and $Z$ is smooth, then so is $W$. (Here the map is a $\mathbb{G}\_m$-torsor, so one doesn't need this stronger statement.)


For properness, you can use the valuative criterion (in the stacky version, as in Deligne-Mumford's paper on [moduli of curves](http://www.math.harvard.edu/~gaitsgde/grad_2009/Deligne-Mumford.pdf)). Namely, assume that $R$ is a complete discrete valuation ring with residue field $K$. Then there are two things to show:


* The restriction from $R$-points of $X$ to $K$-points of $X$ is fully faithful (as a functor of groupoids).
* Given a $K$-point of $X$, after a ramified base extension to $K'$, it extends to an $R'$-point (where $R'$ is the integral closure in $K'$).


Over a local ring $R$, the groupoid of $X$-valued points is obtained by taking the set of tuples $\{x\_0, \dots, x\_n\}$ (at least one of which is invertible) and quotienting by the action of the 
group $R^\*$ (with the weighted action). There is no further sheafification needed since the Picard group is trivial. It follows that given two $R$-valued points, an isomorphism between the associated $K$-valued points necessarily comes from an isomorphism of $R$-valued points: the $K$-scalar inducing the isomorphism must be a unit. 


Conversely, given a $K$-valued point $\{x\_0, \dots, x\_n\}$, then it is isomorphic to an $R$-valued point if the valuation of each $x\_i$ is divisible by $a\_i$, and after a ramified base change you can assume this. (This part, which uses the "stacky" version of the valuative criterion, is not necessary to run the argument for ordinary projective space.)


