There's a theory of algebraic geometry over $\mathbb{Z}_2$-graded commutative rings, often called "algebraic supergeometry" or the theory of superschemes. From what I understand, there's also a variant theory of $\mathbb{Z}$-graded algebraic geometry, for rings whose multiplication is $\mathbb{Z}$-graded commutative, satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$.
Now, many structures arising in algebraic topology are not commutative, but some are instead graded-commutative―for instance, this is the case for the cohomology ring of any space.
Question. Can one use the theory of $\mathbb{Z}$-graded algebraic geometry to say something useful about some of the graded-commutative structures found in algebraic topology, such as e.g. cohomology rings?
One thing I imagine one could do is say take the $\mathrm{Spec}$ of a cohomology ring, and then study it algebro-geometrically as a scheme in the $\mathbb{Z}$-graded setting. Has this sort of strategy ever been successfully carried out?
(Of course there's DAG/SAG, which work wonderfully for the purposes of homotopy theory, but I'm nevertheless curious about this question considered from the point of view of graded-commutative algebraic geometry.)
Best Answer
Lars Hesselholt and Piotr Pstrągowski have since posted a paper to the arXiv doing exactly this!
In their paper, they develop a theory of $\mathbb{Z}$-graded-commutative algebraic geometry in the sense of schemes built from $\mathbb{Z}$-graded rings satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$, which they call Dirac rings.
My (limited) understanding of it is that the main example and motivation for such a theory is that the $\pi_*$ of a commutative algebra in spectra is a Dirac ring (Example 2.2 there).
Here's the abstract from the arXiv:
Edit: There's now also a sequel paper: