# ct.category theory – Applications of \$mathbb{Z}\$-graded algebraic geometry to algebraic topology

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.)