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

  • Larry Smith EMAIL logo
Published/Copyright: June 23, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 4

Abstract

Let θ:G↪GL⁢(n,F) be a representation of a finite group 𝐺 over the field 𝔽 and F⁢[V]⊗F⁢[V]GF⁢[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] 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* 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] of either of the tensor factors and we then apply this result to determine the height and the coheight of all the P*-invariant prime ideals in F⁢[V]⊗F⁢[V]GF⁢[V]. Since it has long been known that the associated prime ideals in F⁢[V]⊗F⁢[V]GF⁢[V] must be 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] for 𝔽 a Galois field are either minimal or the maximal ideal, meaning the ideal consisting of all forms of strictly positive degree.

Keywords: Invariant theory; extension theore; Going Down; Steenrod operations
MSC 2010: 13A50; 13B02; 55S10; 13B21

A Appendix: A derivation lemma

It is often necessary to verify that elements h1,…,hn in a commutative graded algebra 𝐻 over an arbitrary field 𝔽 are algebraically independent over 𝔽. The following lemma, while elementary in character, seems not to be as well known as it ought to be.

Lemma A.1

Lemma A.1 (Derivation lemma)

Let 𝐻 be a commutative connected graded algebra over the field 𝔽, and let h1,…,hn∈H and ∂1,…,∂n:H→H be derivations. Suppose that

det⁡[∂1⁡(h1)…∂1⁡(hn)⋮⋮⋮∂n⁡(h1)…∂n⁡(hn)]≠0

and is not a zero divisor. Then h1,…,hn are algebraically independent.

Proof

Without loss of generality, we can assume that 𝔽 is perfect by passing to an algebraic closure.

Introduce the polynomial algebra F⁢[X1,…,Xn], where deg⁡(Xi)=deg⁡(hi) for i=1,…,n, and define the map ϕ:F⁢[X1,…,Xn]→H by the requirement that ϕ⁢(Xi)=hi for i=1,…,n. We need to show that 𝜙 is a monomorphism. So suppose it is not, and choose r⁢(X1,…,Xn)∈ker⁡(ϕ) to be a nonzero element of minimal degree. Then

r⁢(h1,…,hn)=0∈H.

If we apply ∂i to this equation, we receive by the chain rule

0=∂i⁡r⁢(h1,…,hn)=∑j=1n∂⁡r∂⁡Xj|(X1,…,Xn)=(h1,…,hn)⁢∂i⁡hj

for i=1,…,n. If we rearrange the terms and write this as one matrix equation instead of 𝑛 separate equations, we obtain the following:

0=[∂1⁡h1⋯∂1⁡hn⋮⋮⋮∂n⁡h1⋯∂n⁡hn]⋅[∂⁡r∂⁡X1⋮∂⁡r∂⁡Xn]|(X1,…,Xn)=(h1,…,hn).

Since det⁡[∂i⁡hj]≠0 is not a zero divisor, we conclude from Cramer’s rule the following equalities:

0=∂⁡r∂⁡X1|(X1,…,Xn)=(h1,…,hn)=⋯=∂⁡r∂⁡Xn|(X1,…,Xn)=(h1,…,hn).

If, for some 1≤i≤n, one has ∂⁡r∂⁡Xj≠0∈F⁢[X1,…,Xn], then, since ∂⁡r∂⁡Xj∈ker⁡(ϕ) and deg⁡(∂⁡r∂⁡Xj)<deg⁡(r), we would have a contradiction to the choice of r∈ker⁡(ϕ) as a nonzero element of minimal degree in ker⁡(ϕ). Therefore,

0=∂⁡r∂⁡X1=⋯=∂⁡r∂⁡Xn∈F⁢[X1,…,Xn].

If 𝔽 has characteristic zero, this says that r=0, whereas if 𝔽 has characteristic p≠0, it says that r=sp. In either case, we again have a contradiction to the choice of r∈ker⁡(ϕ) as a nonzero element of minimal degree. ∎

Remark

The converse of this lemma can fail in characteristic p≠0. For example, if Fp is the Galois field with 𝑝 elements, then xp,yp∈Fp⁢[x,y] are algebraically independent, but their Jacobian matrix is identically zero.

As a special case, one has the P*-derivation lemma that uses J. W. Milnor’s primitive elements PΔi∈P*, i∈N0 (see e.g. [23, Chapter 10] or [16]), from the Steenrod algebra of a Galois field as the derivations.

Lemma A.2

Lemma A.2 (P*-derivation lemma)

Let 𝐻 be an unstable algebra over the Steenrod algebra of the Galois field 𝔽, q=pν with 𝑝 a prime integer, and ν∈N. If

det⁡[PΔi⁢(h1)…PΔi⁢(hn)⋮⋮⋮PΔi+n⁢(h1)…PΔi+n⁢(hn)]≠0

and it is not a zero divisor in 𝐻, then h1,…,hn are algebraically independent.∎

Acknowledgements

The author would like to thank the referee for the extensive list of corrections to the manuscript that have improved its readability.

  1. Communicated by: Frederick R. Cohen

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Received: 2020-11-14
Revised: 2020-12-12
Accepted: 2021-04-25
Published Online: 2021-06-23
Published in Print: 2021-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
https://doi.org/10.1515/forum-2020-0324
Keywords for this article
Invariant theory; extension theore; Going Down; Steenrod operations

Articles in the same Issue

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
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