What about schemes built up out of graded rings? Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in $C$. For $C=\mathsf{Ab}$ we get the usual category of schemes, where affine schemes correspond to commutative rings. For $C=\mathsf{Set}$ one gets one of the various definitions of schemes over $\mathbb{F}_1$. One of the drawbacks of this quite general theory is that schemes are defined via their functors on commutative monoid objects, without any geometric incarnation.
Question. What happens when $C$ is the category of graded abelian groups (equipped either with the usual symmetry, or with the twisted symmetry)? Here affine schemes correspond to (graded) commutative rings. Is there any connection with the usual Proj construction? Is there any more geometric interpretation of these schemes? For example one might hope for a fully faithful functor into the category of locally ringed spaces. Is this category of schemes something new at all?
 A: It is my impression (without being very careful about it) is that the C-geometry (for C the category of graded abelian groups) is the same as geometry over $BG_m$, i.e., geometry of schemes with 
an action of the multiplicative group. This should follow from the identification of $C$ with representations of $G_m$. (In particular there is a close relation with Proj - except we don't throw away the "irrelevant ideal", we just consider the stacky quotient of a homogeneous variety in affine space by $G_m$.)
More generally, suppose your category $C$ is identified by a generalized Tannakian reconstruction theorem with $QCoh(X)$, the (complete cocomplete closed symmetric monoidal) category of quasicoherent sheaves on $X$, a quasicompact stack with affine diagonal. Then $C$-schemes should be identified with schemes over $X$, so that $C$-geometry just means geometry relative to the base $X$. Thus for example $C$ could be $R-mod$ for a commutative ring $R$ ($X=Spec(R)$), or $Rep(G)$ for an affine group scheme $G$ ($X=BG$).
