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?

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Aren't these just schemes with a $G_m$ action, or equivalently schemes over $BG_m$? More generally if you have a Tannakian reconstruction identifying $C$ with $QCoh(X)$ for some geometric stack $X$, isn't $C$-geometry just geometry over $X$? –  David Ben-Zvi Mar 14 '13 at 1:10
This sounds quite reasonable, thank you. I would appreciate it if you expand this to an answer. –  Martin Brandenburg Mar 14 '13 at 1:22
If these are schemes with a $\mathbb G_m$-action, then GIT modding out by the $\mathbb G_m$ action should get you the Proj, no? –  Will Sawin Mar 14 '13 at 1:58
You could have a look at Michael Temkin's paper "On local properties of non-Archimedean spaces II" Isr. J. of Math. 140 (2004), 1-27 (see math.huji.ac.il/~temkin/papers/Local_Properties_II.pdf), especially the first section. –  Jérôme Poineau Mar 14 '13 at 7:09
And you're dealing with locally graded-ringed spaces. These are spaces with a sheaf of graded rings so that the stalks are graded local in the sense that they have one homogeneous maximal ideal. –  Dylan Wilson Mar 14 '13 at 12:34

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 $G$ ($X=BG$).
Hi James - excellent point! Idle question: is the problem with the word "scheme"? i.e. if we replace the Zariski topology with the etale (or flat) topology (which I think T-V define in this generality), then things should get better? In other words, shouldn't algebraic spaces or stacks over C again be just spaces over $X$? –  David Ben-Zvi Mar 14 '13 at 14:58