The spectrum of a twisted commutative algebra

A twisted commutative algebra is (for us) a commutative $\mathbf{Q}$-algebra equipped with an action of the infinite general linear group. In such algebras, the “$\mathbf{GL}$-prime” ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct $\mathbf{GL}$-primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish $\mathbf{GL}$-primes. This yields an effective method for analyzing $\mathbf{GL}$-primes.