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]GF⁢[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]GF⁢[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]GF⁢[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]GF⁢[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]GF⁢[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]GF⁢[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.

中文翻译：

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等变共变代数、Steenrod 运算和 Krull 下降定理的相关素理想

令θ:G↪GL⁢(n,F)\theta:G\hookrightarrow\mathrm{GL}(n,\mathbb{F}) 是域𝔽和F⁢[V]上有限群𝐺的表示⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}}}\mathbb{F}[V] 相关的等变共变代数. 本手稿的目的是确定 F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^ {G}}}\mathbb{F}[V] 用于在有限域上定义的表示 𝜃。我们表明，唯一可能的嵌入素数理想是最大理想，它完成了等变共变代数的相关素数理想的描绘，对于最小素数的描述已经在文献中。在本手稿的第一部分中，我们开发了一些特定于 𝔽 是伽罗瓦域的情况的工具，使用伽罗瓦域的 Steenrod 代数 P*\mathcal{P}^{*} 最终形成 W. Krull's Going 的一个版本包含 F⁢[V]↪F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\hookrightarrow\mathbb{F}[V]\otimes_{{\ mathbb{F}[V]^{G}}}\mathbb{F}[V] 的任一个张量因子，然后我们应用这个结果来确定所有 P*\mathcal{P} F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{G}} 中的 ^{*}-不变素理想}\mathbb{F}[V]。由于早就知道 F⁢[V]⊗F⁢[V]GF⁢[V]\mathbb{F}[V]\otimes_{{\mathbb{F}[V]^{ G}}}\mathbb{F}[V] 必须是 P*\mathcal{P}^{*} 不变的，我们的主要结果是一个简单的结果。如上所述，