Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem

Abstract

Abstract Let θ : G ↪ GL ⁢ ( n , F ) \theta:G\hookrightarrow\mathrm{GL}(n,\mathbb{F}) be a representation of a finite group 𝐺 over the field 𝔽 and F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] the associated equivariant coinvariant algebra. The purpose of this manuscript is to determine the associated prime ideals of F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] for representations 𝜃 which are defined over a finite field. We show that the only possible embedded prime ideal is the maximal ideal completing the delineation of the associated prime ideals of an equivariant coinvariant algebra for which descriptions of the minimal primes are already in the literature. In the first part of this manuscript we develop some tools particular to the case where 𝔽 is a Galois field using the Steenrod algebra P * \mathcal{P}^{*} of a Galois field 𝔽 culminating in a version of W. Krull’s Going Down Theorem for the inclusion F ⁢ [ V ] ↪ F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\hookrightarrow\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] of either of the tensor factors and we then apply this result to determine the height and the coheight of all the P * \mathcal{P}^{*} -invariant prime ideals in F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] . Since it has long been known that the associated prime ideals in F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] must be P * \mathcal{P}^{*} -invariant, our main result is an easy consequence. As indicated above, our main result is that the associated prime ideals of F ⁢ [ V ] ⊗ F ⁢ [ V ] G F ⁢ [ V ] \mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] for 𝔽 a Galois field are either minimal or the maximal ideal, meaning the ideal consisting of all forms of strictly positive degree.