There's a theory of algebraic geometry over $\mathbb{Z}\_2$-graded commutative rings, often called "[algebraic supergeometry](https://arxiv.org/abs/2008.00700)" or the theory of [superschemes](https://ncatlab.org/nlab/show/super-scheme). 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](https://en.wikipedia.org/wiki/Cohomology_ring) of any space.



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> **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?
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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.)



Applications of $\mathbb{Z}$-graded algebraic geometry to algebraic topology
Lars Hesselholt and Piotr Pstrągowski have since posted a paper to the arXiv doing *exactly this*!



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> Hesselholt–Pstrągowski, *Dirac geometry I: Commutative algebra*. [[arXiv]](https://arxiv.org/abs/2207.09256)
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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](https://arxiv.org/pdf/2207.09256.pdf#page=9) there).


Here's the abstract from the arXiv:



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> **Abstract.** The homotopy groups of a commutative algebra in spectra form a commutative algebra in the symmetric monoidal category of graded abelian groups. The grading and the Koszul sign rule are remnants of the structure encoded by anima as opposed to sets. The purpose of this paper and its sequel is to develop the geometry built from such algebras. We name this geometry Dirac geometry, since the grading exhibits the hallmarks of spin. Indeed, it is a reflection of the internal structure encoded by anima, and it distinguishes symmetric and anti-symmetric behavior, as does spin. Moreover, the coherent cohomology, which we develop in the sequel admits half-integer Serre twists.
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**Edit:** There's now also a sequel paper:



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> Hesselholt–Pstrągowski, *Dirac geometry II: Coherent cohomology* [[arXiv]](https://arxiv.org/abs/2303.13444)
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