We extend this result to more general modular rings of invariants where RG0 is quasi-Gorenstein and G0 has a normal, cyclic, p-Sylow subgroup.. We extend this result to modular rings of
Rings of Invariants for Cyclic p-Groups
Our focus in this section is modular actions ofG=Z/p e Z We first explore the eigenspace of a generator of Gas a linear map on V.
Theorem 2.3 Let g ∈G=Z/p e Z be a generator IfV is an n-dimensional representation of G overk, then V is indecomposable if and only if p e−1 < n≤p e In an eigenbasis forV, g acts on a basis via the Jordan block
More generally, let V = V1 ⊕ V2 ⊕ ⊕ Vℓ be an n‑dimensional representation of G, and set ni = dim Vi There exists a basis adapted to this direct sum decomposition such that the action of g on each Vi is given by the corresponding Jordan block Ji, and consequently g admits a Jordan block decomposition with blocks Ji.
, where at least one of the V i has p e−1 < n i ≤p e and each V i is indecomposable.
Proof This follows immediately from the Jordan Normal Form theorem Let g ∈ Z/p^e Z be a generator Recall, a representation of G is a k-vector space V together with a homomorphism π: G → GL(V) ∼= GL_n(k) We have g^{p^e} = 1, that is, g^{p^e} − 1 = (g − 1)^{p^e} = 0 Consequently, the action π(g) satisfies (π(g))^{p^e} = I, so the characteristic polynomial of π(g) is a factor of x^{p^e} − 1; hence all eigenvalues of π(g) are p^e‑th roots of unity, and by the Jordan Normal Form, π(g) is conjugate to a block diagonal matrix with Jordan blocks corresponding to these eigenvalues.
Because T^{p^e−1} = (T−1)^{p^e} ∈ k[T] but not (T−1)^{p^e−1}, the order of π(g) would be p^{e−1} if the latter held This forces p^{e−1} < n ≤ p^e, i.e., p and e bound the dimension of the representation V The linear map π(g) has a unique eigenvalue of 1, and since 1 is the only eigenvalue, each eigenvector yields a distinct fixed subspace of V, i.e., a subrepresentation.
Moreover, each Jordan block in π(g) gives rise to a subrepresentation As the Jordan canonical form of a matrix is unique up to permutation of the Jordan blocks, it follows that
V is indecomposable if and only if π(g) consists of precisely one Jordan block and in particular is represented by J as above.
In the general setting, we again use the Jordan normal form to analyze representations of the cyclic group generated by g, where g^p = e Since π is a representation, the operator π(g) satisfies π(g)^p = I, so its spectrum consists of p-th roots of unity in the base field If π(g) is indecomposable, its Jordan form consists of a single Jordan block, so π(g) is similar to a Jordan block J(λ) of size m with eigenvalue λ, where λ^p = 1 Equivalently, π(g) is of the form λ(I + N) with N nilpotent and N^m = 0; the nilpotent part encodes the indecomposable structure The constraints π(g)^p = I force λ^p = 1, and the size of the block m corresponds to the dimension of the indecomposable module Thus, in the indecomposable case the representation is completely determined up to similarity by a choice of a p-th root of unity λ and a Jordan block size m, giving a concise description of all indecomposable representations of the cyclic group generated by g.
To conclude the argument, it suffices to show that there exists an index i with p^{e−1} < n_i ≤ p^e and that n_i ≤ p^e for every i If some n_i > p^e, then the corresponding Jordan block J_i satisfies J_i^{p^e} ≠ I, which implies π(g)^{p^e} ≠ I and yields a contradiction Conversely, if all n_i ≤ p^{e−1}, then π(g)^{p^{e−1}} = I, again a contradiction Thus the required bounds hold, and the result follows.
Definition 2.3 If Ghas representation V with V indecomposable, then we say that G acts by the indecomposable action Otherwise we say that G acts by adecomposable action.
In the decomposable case, at least one Jordan block J_i must satisfy p^{e-1} < n_i ≤ p^e By a change of basis, i.e., an automorphism of R, we may assume without loss of generality that J_1 is the block meeting this condition, that is p^{e-1} < n_1 ≤ p^e.
Despite the seemingly simple representation theory of the invariant ring for G = (Z/pZ)^e, many questions remain about its structure Noether’s theorem guarantees finite generation, but an explicit generating set is not known in general When the action of G is indecomposable on a representation V with dim V ≤ 5, explicit generating sets are known; in particular, for dim V = 2 or 3 there exist explicit minimal generating sets For dim V > 5, finding an explicit generating set remains open Our study of invariant rings does not depend on these explicit sets, since Chapter 4 develops a technique to investigate the invariant ring RG without requiring them Nevertheless, explicit generators can make the ring more concrete, so we also explore algorithms to compute generators in small examples, which will be helpful for Example 3.5 and Corollary 3.18.
Remark 1 It is not difficult to see how coefficients behave under an indecomposable group action For example, take G = Z/2Z acting on the polynomial ring R = k[x, y] with char k = 2, and let f ∈ R be homogeneous of degree n Writing f = ∑_{i+j=n} a_{i,j} x^i y^j, there is a concrete relation linking the coefficient of the monomial x^2 y^{n−2} in f to the corresponding coefficient in the transformed polynomial g·f, i.e., the coefficient a_{2,n−2} in f determines the coefficient of x^2 y^{n−2} in g·f through a simple, explicit connection.
2 a 0,n where the (n−2)+i i comes from the fact that y (n−2)+i 7→(y+x) (n−2)+i and we must choose i binomials in the product (y+x) (n−2)+i to givex when expanding A similar pattern holds if we increase the number of variables.
Lemma 2.4 Consider G=Z/p e Z acting by the indecomposable action on
R=k[x 1 , , x n ] with p e−1 ≤n ≤p e If g ∈G is a generator, then for 0≤` ≤e−1 g p ` (x i )
Proof We use induction on ` If `= 0 this is by defintion of the indecomposable action. Fix 0≤m≤e−2 and suppose (1) holds for 0≤`≤m By direct calculation, for each x i we have g p m+1 (x i ) = g pp m (x i ) = (g p m ) p (x i ).
We now consider three cases i≤p m , p m < i≤p m+1 , or p m+1 < i.
Case 1: If i≤p m , then by the induction hypothesis g p m (xi) =xi and therefore
Case 2: Suppose p m < i≤p m+1 We make the following two observations.
In the j-th iteration of g_p^m acting on x_i, denoted (g_p^m)^j(x_i), the coefficient of x_{i - s p^m} is obtained by adding the coefficients of x_{i - (s-1) p^m} and x_{i - s p^m} from the (j−1)-th iteration, according to the indecomposable action If we denote the coefficient of x_q in (g_p^m)^j(x_i) by a_{q,j}, then a_{q,j} = a_{q-1,j-1} + a_{q,j-1} Hence, by induction, these coefficients yield the binomial coefficients seen in Pascal’s Triangle.
When i−(p−1)p m ≤p m+1 −(p−1)p m =p m , we have that g p m (x i−(p−1)p m ) =x i−(p−1)p m by the induction hypothesis.
Ifp m < i≤p m+1 , then by the induction hypothesis
(g p m ) p (x i ) = (g p m ) p−j j 0 x i + j 1 xi−p m +ã ã ã+ j j xi−jp m and for j =p,
Case 3 If p m+1 < i, then since i−(p−1)p m > p m+1 −(p−1)p m =p m , the induction hypothesis and the computation in Case 2 gives
With these facts in mind, we compute the generators in two illustrative cases Our first case analyzes a G-action on a two-dimensional indecomposable representation V (so dim V = 2), i.e., a G-module with an indecomposable structure Although this example is classical in representation theory, we spell out the details here to show how the generators arise and to provide a concrete reference for the method.
Example 2.1 analyzes the indecomposable action of G = Z/2Z on the polynomial ring R = k[x, y] in characteristic 2 By Theorem 2.3, with n = 2 and p^{e−1} < n ≤ p^e, we obtain e = 1 for all p > 0, hence the action is given by x ↦ x and y ↦ x + y Consequently the invariant ring R^G is k[x, xy + y^2] To see this, let f ∈ R be written as f = ∑_{i, j} a_{i, j} x^i y^j (as in (2)).
By relating the coefficients in (2) and (3), we obtain constraints on a_{i,j}: for example, a_{0,1}=0, a_{1,1}=a_{0,2}, and 3a_{0,3}=0, which implies a_{0,3}=0 in characteristic not equal to 3, while a_{2,1}=a_{1,2} These relations show that the polynomials x, xy+y^2, and x^2y+y^3 = x(xy+y^2) generate a k-linear subspace, namely k[x, xy+y^2] ⊆ R_G It suffices to prove that any f ∈ R_G of degree at least 3 can be written using the generating set {1, x, xy+y^2} Since the indecomposable action is homogeneous, we may assume f is homogeneous and proceed by induction on deg f ∈ R_G to establish the claim.
Base case: if f is homogeneous of degree three, then f has the form f = c(x^2 y + x y^2) = c x(x y + y^2) with a scalar c in k If f ∈ R_G can be written using the generators 1, x, and xy + y^2 whenever f is homogeneous of degree t, then any f ∈ R_G that is homogeneous of degree t+1 corresponds to the case where t is either even or odd.
Ift is even, then the relation a 1,t =a 1,t + (t+ 1)a 0,t+1 , gives a0,t+1 = 0 Thus ift is even, then sincef is homogeneous we can writef =xh and apply the induction hypothesis.
Ift is odd, then 2|t+ 1, ie,t+ 1 = 2n for some n∈N Moreover, we have t+ 1 = 2 α 1 + 2 α 2 +ã ã ã+ 2 α ` , where α i ∈Z ≥1 for 1≤i≤` Write f =y 2n +g =y 2 α 1 +2 α 2 +ããã+2 α` +g with g a polynomial such that each term has at least onex, ie, g =xg 0 and degg 0 =t.
By direct calculation f =y 2 α 1 ããã2 α` +xg 0
= (y 2 α 1 +ããã+2 α` + (xy) 2 α` −1 y 2 α 1 +ããã+2 α`−1 ) +xg 0 + (xy) 2 α` −1 y 2 α 1 +ããã+2 α`−1
=y 2 α 1 +ããã+2 α`−1 (y 2 +xy) 2 α` −1 +xh+ 2(xy) 2 α 1 −1 +ããã+2 α`−1 −1 (y 2 +xy) 2 α` −1
= (y 2 +xy) 2 α 1 −1 +ããã+2 α` −1 +xh+ (xy) 2 α 1 −1 +ããã+2 α`−1 −1 (y 2 +xy) 2 α` −1 where in the forth equality, xh=xg 0 + (xy) 2 α` −1 y 2 α 1 +ããã+2 α`−1 and as such degh=t We have (y 2 +xy) 2 α 1 −1 +ããã+2 α` −1 ∈R G and therefore xh+ (xy) 2 α 1 −1 +ããã+2 α`−1 −1 (y 2 +xy) 2 α` −1 ∈R G (4)
Since the degree of the expression in equation (4) is at most t, the induction hypothesis applies, allowing us to express it in terms of the required generators Consequently, f can be written using the generators 1, x, and xy + y^2.
We found the ring of invariants in this example by explicitly relating polynomial coefficients before and after applying the group action As the number of variables increases, the complexity of this process becomes significantly more difficult Alternatively, we can use an algorithm developed by Kemper For more details and a proof of what follows see Algorithms 7 and 8 in [20] For any gradedk-algebra R with dimR =n and
Depth and Cohen-Macaulay Rings of Invariants
One question of interest in invariant theory is when a ring of invariants is Cohen–Macaulay Recall that for a Noetherian local ring (S, n) and a finitely generated S-module M ≠ 0, the depth of M is the infimum of integers t such that Ext^t_S(S/n, M) ≠ 0 The S-module M is Cohen–Macaulay when its depth equals the Krull dimension of M, a condition that provides a practical criterion for Cohen–Macaulayness in invariant-theoretic problems.
Cohen-Macaulay if depthM = dimM and the ringS is Cohen-Macaulay if it is
Viewed as a module over itself, RG is Cohen–Macaulay To study its locality, we localize RG at the homogeneous maximal ideal m, ensuring RG is local in this graded setting When the G-action on R is non-modular (i.e., the characteristic does not divide the order of G), one can invoke the celebrated theorem of Eagon and Hochster, which provides a pivotal Cohen–Macaulay result in this context.
Theorem 3.1 [14, Eagon, Hochster] If G is a finite subgroup ofGL(V) and #G is not divisible by the characteristic of k, then k[V] G is Cohen-Macaulay.
When the action of a finite group G on a ring R is modular, the question of whether the skew group ring RG is Cohen-Macaulay becomes less straightforward In this modular context, the p-Sylow subgroups play a central role in determining the Cohen-Macaulay property of RG, providing the criteria and mechanisms that indicate when RG preserves Cohen-Macaulayness.
Lemma 3.2 [18, Jeffries] Let G be a finite subgroup of GL(V) with chark|#G Let
P ≤Gbe a p-Sylow subgroup If k[V] P is Cohen-Macaulay, then k[V] G is Cohen-Macaulay.
This reduces the question of whether R G is Cohen-Macaulay when the action ofG on
R is modular to the case of considering p-Sylow subgroups In this section, we collect known facts and summarize when R G is Cohen-Macaulay or quasi-Gorenstein for
G=Z/p e Z This builds on the work of Kemper in [21] which uses bireflections.
Definition 3.1 LetG be a subgroup of GL(V) We say that g ∈Gis a pseudo-reflection if rank(g−id) = 1 We say thatg ∈G is a bireflection if rank(g −id) ≤2.
Theorem 3.3 [21, Kemper] Let G be a group of order p e and R =k[x 1 , , x n ] If R G isCohen-Macaulay, then G is generated by bireflections.
Thus to determine if R G is Cohen-Macualay, we study the bireflections In particular, Theorem 3.3 tells us that when G is not generated by bireflections, R G is not
Corollary 3.4 Let G=Z/p e Z act on R =k[x 1 , , x n ] by the indecomposable action If n >3, then R G is not Cohen Macaulay.
Proof If g ∈G is a generator, then since n > 3 we have rank(g−id)>2 by definition of the indecomposable action By Theorem 3.3, R G is not Cohen-Macaulay.
Many invariant rings RG are not Cohen–Macaulay, so their depth and Krull dimension do not coincide We want to understand how far apart these two invariants can be Since R is integral over RG, we have dim(RG) = dim(R) = n There are known results for computing the depth under the action of a group on R, which provide methods to determine or bound depth(RG).
Let G = Z/pZ act on a ring R The formula we present here is well known and is proved in [29] using spectral sequences; however, we will avoid such methods in our treatment For cyclic p-groups acting on rings of characteristic p, the well-known results of Ellingsrud and Skjelbred provide the key ingredients for our discussion.
[9, 20] give that depth(R G ) = min{n, n−m+ 2} wherem is the dimension of the k-vector space generated by
We use this to give an elementary proof of the depth when G=Z/p e Z.
1 If G acts on R by the indecomposable action with n = 1,2, then depth(R G ) = n.
2 If G acts on R by the indecomposable action with n ≥3, then depth(R G ) = 3.
3 Let G act on R with representation V 1 ⊕ ã ã ã ⊕V ` We have depth(R G ) = min{n, `+ 2}.
Proof Letg ∈G be a generator Let V denote the vector space generated by the set defined in (6).
1 It is well known that when n= 1,2, R G is a polynomial ring (see Example 2.2).
Let m = dim V = n − 1 For i = 1, g^d(x_i) − x_i = 0 for all d = 1, , p^e − 1 For i > 1, g(x_i) − x_i = x_i − 1 By the definition of the indecomposable action, g(x_j) − x_j = x_j − 1 cannot contain a monomial term x_n for all 1 ≤ j ≤ n, since g(x_n) − x_n = x_n − 1 does not have the term x_n If we consider g^d(x_i) − x_i for any i = 1, , n and d = 1, , p^e − 1, then by the definition of the indecomposable action it is a linear combination of the x_j Since the x_i are linearly independent, it follows that x_1, , x_{n−1} form an F_p-vector space basis for V Thus depth(RG) = min{n, n − m + 2} = min{n, 3} = 3.
To determine the dimension of V, note that each g_d in G acts independently on each subspace V_i Because the G-action is independent on the V_i, a basis for V is obtained by taking the union of the bases of the V_i in the prescribed form By the result from part (2), V_i contributes i−1 basis vectors to a basis of V Therefore the dimension of V is the sum over i from 1 to ℓ of (i−1), i.e dim V = ∑_{i=1}^{ℓ} (i−1) = ℓ(ℓ−1)/2.
Using depth as a guiding invariant, we find that when G acts by an indecomposable representation of size n = 1, 2, or 3, the group algebra RG is Cohen–Macaulay The more interesting case occurs when G acts through a decomposable representation; here, if the decomposition contains enough 1×1 Jordan blocks, RG may become Cohen–Macaulay because these blocks raise the depth of RG without increasing its dimension In other words, adding trivial Jordan blocks can improve the homological depth while keeping the dimension fixed, linking the Jordan decomposition of the G-action to the Cohen–Macaulay property of RG.
Example 3.1 Consider G acting on R with the following Jordan block decomposition of its representation.
According to Theorem 3.5, depth(R G ) = min{4,3 + 2}= 4 and therefore R G is
Cohen-Macaulay since depth(R G ) = dim(R G ) On the other hand, suppose the action ofG onR has the following Jordan block decomposition.
Again by Theorem 3.5, depth(R G ) = min{5,2 + 2}= 4 and therefore R G is not
This allows us to give a characterization of when any action of G=Z/p e Zon R is Cohen-Macaulay.
Corollary 3.6 Let G=Z/p e Z act on R =k[x 1 , , x n ] with representation V 1 ⊕ ã ã ã ⊕V ` Set n i = dimV i
1 If n > `+ 2, then R G is not Cohen-Macualay.
2 If n ≤`+ 2, then R G is Cohen-Macualay when one of the following conditions hold.
(a) If p= 2, then either e= 1 andn i = 2 for one V i , e= 1 and n i = 2 for two V i , or e= 2 and n i = 3 for one V i ; in each case all other V j has n j = 1.
(b) If p≥3, then e= 1 and either ni = 2 for one Vi, ni = 2 for two Vi, or ni = 3 for one V i ; in each case all other V j have n j = 1.
By Theorem 3.5, part (1) shows that if n exceeds ℓ + 2 then dim(RG) > depth(RG) For part (2), Theorem 3.5 forces each ni to be at most 3 If there exists an index i with ni = 3, that index is unique and all other Vi are trivial, which implies n = ℓ + 2 If ni = 2 can occur for at most two indices and all other Vj are trivial, then n equals ℓ + 1 or ℓ + 2 respectively The bounds on e follow from Theorem 2.3.
We have established a formula for the depth of the invariant ring RG By Theorem 3.5, when n ≥ 3, any regular sequence in RG has length at most 3 Our next goal is to identify explicit regular sequences within the invariant ring to demonstrate this bound and to illuminate the structure of RG.
1 If G acts by the indecomposable action, then x 1 , x p−1 1 x 2 −x p 2 is a regular sequence in
2 If n ≥3, G acts by the indecomposable action, and g ∈G is a generator, then x 1 , x p−1 1 x 2 −x p 2 , p
3 If G acts by a decomposable action and depth(R G ) =`+ 2> n with J 1 ,ã ã ã , J ` the Jordan blocks in the Jordan block decomposition of the representation of G and n i ≥3 for some 1≤i≤`, then x1, x p−1 1 x2−x p 2 , p
Y d=1 g d (x3), xn 1 +1, xn 1 +n 2 +1,ã ã ã, xn 1 +ããã+n `−1 +1 is a regular sequence.
4 Let G act by a decomposable action with representation V 1 ⊕ ã ã ã ⊕V ` and suppose R G is Cohen-Macaulay Set S ={S 1 ,ã ã ã , S ` } where S i is a set of primary invariants for
V i The set S is a regular sequence.
Proof Letg ∈G be a generator Denote f 1 =x 1 , f 2 =x p−1 1 x 2 −x p 2 , and f 3 =Qp d=1g d (x 3 ).
1 Since R G is a domain, it is clear that f1 is a regular element in R G Moreover, it is clear thatf 2 is not a zero-divisor inR G /x 1 R G hence is a regular element of R G /x 1 R G
2 Note that R G /(x 1 , f 2 )R G is a subring ofR/(x 1 , x p 2 )R If f 3 is a zero-divisor in
R G /(x 1 , f 2 )R G then it is a zero-divisor in R/(x 1 , x p 2 )R Thus it suffices to prove that f 3 is a regular element ofR/(x 1 , x p 2 )R InR/(x 1 , x p 2 )R f3 =a1x3x p−1 2 +a2x 2 3 x p−2 2 +ã ã ã+ap−1x p−1 3 x2+x p 3 where each a i ∈k and it is clear f 3 is a regular element ofR/(x 1 , x p 2 )R.
3 This follows from (2), the conventions in Definition 2.3, and the fact that it is clear x n 1 +ããã+n i +1 is a regular element of
4 We give the set S in each of the three cases described in Corollary 3.6 using
Remark 2 If n 1 = 2 andn i = 1 for i6= 1, then
These objects form regular sequences, which is easy to see For the first two cases, apply Part 1 and the argument from Part 3 For the third case, apply Part 2 and the argument from Part 3.
Remark 3 Theorem 3.5 implies depth_R G = 3 under the hypotheses of part (2) of Theorem 3.7 However, the proof of part (2) does not justify why the technique used cannot be extended to show that, for example, the relation involving x1, x_{p−1}, x2, and x_p (with the parameter p) could yield a similar conclusion.
Y d=1 g d (x 4 ) is a regular sequence Set I = x 1 , x p−1 1 x 2 −x p 2 ,Qp d=1g d (x 3 )
R^G, the ring of G-invariants, fails to support the standard proof technique here because R^G / I R^G no longer injects into R / I R For a concrete illustration, consider G = Z/3Z acting on k[x1, x2, x3] via the indecomposable action with char(k) = 3 It is well known that in this setting the natural map loses injectivity, which explains why the usual approach cannot be applied.
Quasi-Gorenstein Rings of Invariants
We have established that many of the rings of invariants of interest to us are not
Cohen-Macaulayness is a property whose absence prompts the question of how badly the structure of RG can degenerate Recall that when a ring T is of finite type over a field k, i.e T = k[x1, , xn]/I = S/I, and T is equidimensional with dim T = d, we define the canonical module of T to be the T-module ω_T := Ext^{n-d}_S(T,S).
If T is Cohen-Macaulay and T ∼=ω T , we say that T is Gorenstein If T is not
Let T be Cohen-Macaulay and isomorphic to its canonical module ω_T; in this case we say that T is quasi-Gorenstein The quasi-Gorenstein property is useful for several reasons For example, when a local ring (T, m) admits a canonical module, one may apply local duality to investigate the local cohomology modules.
H m i (T), of T, i.e., we may translate questions regarding local cohomology to questions regarding ω T WhenT is quasi-Gorenstein, this technique becomes especially useful since if
T is not Cohen-Macaulay, then it may have several non-zero local cohomology modules which are in general difficult to compute.
Recall that Noether showed RG is of finite type over a field, which implies RG has a canonical module ω_RG Since RG is normal, this canonical module is unique up to isomorphism and can be realized as an unmixed ideal of height one in RG; in other words, ω_RG corresponds to a divisor on Spec(RG) Our aim is to determine the explicit form of this canonical module.
Because presenting the canonical module directly is typically hard, we adopt an abstract, representation-theoretic viewpoint to describe the structure of ω R G rather than supplying an explicit representation In particular, we determine which of our groups admit pseudo-reflections and then invoke the following theorem to draw conclusions about the module’s structure.
Let G be a finite group acting on the polynomial ring R = k[x1, , xn] over a field k Then the invariant ring RG is a unique factorization domain if and only if there is no nontrivial group homomorphism G → k× that sends every pseudo-reflection to 1.
Recall that in a unique factorization domain T, the divisor class group is trivial, and if ω_{R^G} can be identified with a divisor on Spec(R^G), then R^G being a unique factorization domain implies that R^G is quasi-Gorenstein Our aim is to apply Theorem 3.8 to the rings of invariants we study To prepare, we first provide examples of the kinds of pseudo-reflections that can occur when G ≅ Z/p^e Z, and we note that π(g) denotes the representation of a generator g in G.
Example 3.2 If G=Z/4Zacts on R =k[x, y, z] by the indecomposable action with chark= 2, then we have π(g) 2
and therefore rank(π(g) 2 −id) = 1 Thus π(g) 2 is a pseudo-reflection.
Example 3.3 Let G=Z/8Zand R =k[x 1 , , x n ] with chark = 2 Consider the action of G onR represented by V 1 ⊕V 2 with dimV 1 = 5, dimV 2 = 2, and the following Jordan block decomposition π(g)
and therefore rank(π(g) 4 −id) = 1 Thus π(g) 4 is a pseudo-reflection.
Before giving a general characterization of which actions do not have pseudo-reflections, we invoke Lucas's theorem, a well-known result from 1878 that will guide the proof Theorem (Lucas, 1878) states that for non-negative integers u and n and a prime p, the binomial coefficient C(u, n) modulo p is determined by the base-p digits of u and n: if u = u_0 + u_1 p + u_2 p^2 + and n = n_0 + n_1 p + n_2 p^2 + , then C(u, n) ≡ ∏_i C(u_i, n_i) (mod p) This digit-wise decomposition provides a powerful tool for analyzing congruences and will be used to derive the required characterization of actions without pseudo-reflections in the ensuing proof.
Y i=0 u i n i modp where u=u 0 +u 1 p 1 +u 2 p 2 +ã ã ã+u s p s and n=n0+n1p 1 +n2p 2 +ã ã ã+nsp s are the base p expansions of u and n respectively.
We now demonstrate which actions of G=Z/p e Z do not have pseudo-reflections in both the decomposable and indecomposable case.
Theorem 3.10: Let G = Z/pZ act on R = k[x_1, , x_n] with char k = p by an indecomposable action, and assume n > 2 The representation G ⊆ GL_n(k) has a pseudo-reflection if and only if n = p^{e-1} + 1 More generally, if G acts on R with representation V = V_1 ⊕ ⊕ V_r and dimensions n_i for each summand, then G has a pseudo-reflection if and only if V_1 is the unique summand with p^{e-1} < n_i ≤ p^e (i.e., n_1 = p^{e-1} + 1 and n_i ≤ p^{e-1} for all i > 1).
Proof We first deal with the indecomposable action Let g ∈G be a generator Recall that, g m
(7) and by Theorem 2.3, p e−1 + 1≤n≤p e , that is, p e−1 ≤n−1≤p e −1 It is an observation from (7) that if g m is a pseudo-reflection, i.e rank(g m −id) = 1, then m t
6= 0 We will use Theorem 3.9 to show first that m=cp e−1 for some 1≤c≤p−1 and then to show that we must have n=p e−1 + 1 If t=p j with j < e−1, then by Theorem 3.9
0 ã ã ã m t 1 ã ã ã m 0 0 modp where mi is the ith digit in the base p representation of m and similarly for t But this equation holds if and only if m 1 t
≡0 mod p, that is, m t = 0 since 0≤m t < p Thus m=cp e−1 with 1≤c≤p−1 or m=p e If m=p e , then g m = id is not a pseudo-reflection Hence we must have m=cp e−1 , that is,
6≡0 mod pif and only if
((n−1)e, ,(n−1)0) = (0, q,0,ã ã ã ,0) for some 0< q ≤c Moreover, if q6= 1, then (q−1)p m e−1
6≡0 mod p, whence q= 1 Thus n−1 = p e−1 and therefore n =p e−1 + 1 as desired Conversely, if n=p e−1 + 1, then
Theorem 3.9 and the formula above for g m immediately gives g p e−1 is a pseudo-reflection. For the more general case, with a representation ofG given by V1⊕ ã ã ã ⊕V` and associated Jordan block decomposition π(g)
From Theorem 2.3, a Jordan block contributes a pseudo-reflection precisely if it is the unique Jordan block with dimension greater than p^{e-1}, and in particular when that dimension is p^{e-1}+1.
We make special note of the following corollary regarding the subgroup H ≤Gof pseudo-reflections which we will use in the proof of Theorem 4.10 regarding the structure of
Corollary 3.11 Let G=Z/p e Z act on R =k[x 1 , , x n ] with chark=p Any pseudo-reflection of G is of the form g cp e−1 where 1≤c≤p−1 Moreover, if H ≤G is the subgroup of G generated by pseudo-reflections, then H =hg p e−1 i.
Proof The first claim follows from the proof of Theorem 3.10 The second claim follows from the fact that any generator of H has the form g c p e^{-1} g p e^{-1} c with 1 ≤ c ≤ p−1; in other words, any pseudo-reflection may be written as g0 g p e^{-1} for some g0 ∈ G.
Applying Theorem 3.8, we get the following corollary to Theorem 3.10 regarding R G
Corollary 3.12 If G=Z/p e Z acts on R=k[x 1 , , x n ] with n > 2, then R G is a unique factorization domain.
Proof If G contains no pseudo-reflections, the result follows immediately from Theorem 3.8 Consequently, we may assume that G has pseudo-reflections, i.e., n = p^{e-1} + 1 If g ∈ G is a generator, then by Corollary 3.11, g^{p^{e-1}} is a pseudo-reflection, and the subgroup of pseudo-reflections is H = ⟨g^{p^{e-1}}⟩.
According to Theorem 3.8, RG is a unique factorization domain if and only if there are no non-trivial homomorphisms ϕ: G/H → k× Suppose such a non-trivial homomorphism exists Then G/H ≅ Z/p^{e−1}Z via the map g ↦ 1 Consequently, any non-trivial homomorphism ϕ: G/H → k× yields a non-trivial one-dimensional representation of G/H ≅ Z/p^{e−1}Z, which contradicts the relationship between p^{e−1} and the dimension of the representation required by Theorem 2.3. -**Support Pollinations.AI:**🌸 **Ad** 🌸 Easily create and optimize SEO-friendly math content with Pollinations.AI text APIs—[support our mission](https://pollinations.ai/redirect/kofi) for accessible AI tools!
Remark 4 We note here that this gives a large collections of rings which are examples of unique factorization domains that are not Cohen-Macaulay.
Remark 5 The canonical module is isomorphic to an unmixed ideal of height 1 and can be identified with a divisor on Spec(R_G) From Corollary 3.12 we obtain R_G ≅ ω_{R_G} by noting that R_G is a unique factorization domain, hence has a trivial divisor class group as mentioned earlier This implies that the canonical module for R_G is isomorphic to R_G itself, confirming that R_G is a Gorenstein ring.
R G has order 1, i.e., that R G is quasi-Gorenstein We give another proof with the next lemma utilizing duality.
Set S to be the set of primary invariants when G=Z/p e Z acts onR =k[x 1 , , x n ].
Recall, if G acts by the indecomposable action, then
IfG is represented byV 1 ⊕V 2 ⊕ ã ã ã ⊕V ` , then a set of primary invariants for R G is given by
Let S = {S1, , S′} where each Si is the set of primary invariants for Vi as described, since each Vi is indecomposable In either case, k[S] ≅ R; in particular, k[S] is Gorenstein, i.e., k[S] ≅ ω_{k[S]} Since both R_G and k[S] are normal domains, k[S] is normal as well.
Gorenstein, it follows that ω R G ∼= Hom k[S] (R G , ω k[S] )∼= Hom k[S] (R G , k[S])
We now will use the fact that R G is a unique factorization domain to show explicitly that
Lemma 3.13 Let G=Z/p e Z act on R=k[x 1 , , x n ] Let S denote the set of primary invariants for R G As R G -modules, R G ∼= Hom k[S] (R G , k[S]).
Proof Denote by (−) ∨ := Homk[S](−, k[S]), and consider (R G ) ∨ Ask[S] and R G are both domains, let K ⊆L be the corresponding fields of fractions LetR G L :=R G ⊗ R GL and
R K G :=R G ⊗ k[S] K We have (R G ) ∨ is reflexive as ank[S]-module Moreover rank k[S] (R G ) = dim K (R G K ) = [L:K] To see this, let b 1 , b n be a basis in R G for R G L over
Let a1, , am be a K-basis for L RGK, viewed as a K-vector space, has a basis consisting of b1, , bm modulo the K-span of a1, , am, so dim_K(RGK) = |{b1, , bm} / ⟨a1, , am⟩| = [L:K] Moreover, since the RGK-basis over K yields a corresponding dual basis in (RGK)∨ over K, we have rank_k[S](RG) = rank_k[S]((RG)∨).
F -Singularities of Cyclic Rings of Invariants
Having established that R G is often not Cohen-Macaulay but always quasi-Gorenstein when G=Z/p e Zacts on R=k[x 1 , , x n ], we record some of the known restrictions this puts on theirF-singularities.
Definition 3.2 Fix a ringR with char(R) = p >0 AFrobenius operator on an R-module
M is a map ϕ: M →M satisfying ϕ(rm) = r p e ϕ(m) for somee >0.
Equivalently, giving an R{F^e}-module structure to M means equipping M with the action of the non-commutative ring generated over R by a Frobenius operator F^e that satisfies F^e r = r^{p^e} F^e for all r in R Such a Frobenius operator is said to have degree e Denoting F^e(M) the set of all degree‑e Frobenius operators on M, we can construct the graded ring F(M) with the degree e component equal to F^e(M).
The graded ring F(M) is called the ring of Frobenius operators For more information on rings of Frobenius operators and their importance, see [26].
It is natural to ask when F(M) is finitely generated over F_0(M) Of particular interest are modules such as the local cohomology module H_m^{dim R}(R) and the injective hull of the residue field E_R(R/m) = E We provide a simple example of this phenomenon by taking R to be a power series ring.
Example 3.4 If R=k[[x 1 , , x n ]], then F(E) is finitely generated over F 0 (E) and the ring of Frobenius operators of E =E R (k) is given by
Let R be Gorenstein, complete, normal, and local According to Proposition 4.1 in [19], F(E) is a finitely generated ring extension of F_0(E) In terms of Frobenius complexity, cxF(R) = −∞ The canonical module of R is ω_R ≅ (x_1 \cdots x_n)R, and by Theorem 3.3 of [19], this yields further structural consequences for the ring R.
We have that ω (1−p e ) = (x 1 ã ã ãx n ) 1−p e R Thus
(x 1 ã ã ãx n ) pq−1 F e+1 (z), which yields the desired result.
Notice, the key to the computation in Example 3.4 was thatR was quasi-Gorenstein.
Let (R, m) be a complete local ring with dimension d > 0 that satisfies Serre’s S2 condition; then F(H_m^d(R)) ∼= R{F}, as shown in Example 3.7 of [26] Moreover, the process of computing the ring of invariants commutes with completion at the homogeneous maximal ideal.
For any finite group G, Rc(G) ≅ Rb(G), so we may regard R_G as complete It remains to show that R_b(G) satisfies S2, and it suffices to prove that R_b(G) is normal, which follows by a argument similar to the proof of Theorem 2.1 Consequently, this yields a corollary of Theorem 3.14.
Corollary 3.15 If G=Z/p e Z acts on R=k[x 1 , , x n ], then the ring of Frobenius operators of E is cyclic.
We now consider other examples of singularities for rings of characteristic p > 0 We first give definitions for F-regularity and F-rationality.
Definition 3.3 Let R be a ring of characteristic p > 0 Let F^e: R → R denote the e-th iterate of the Frobenius endomorphism, sending r to r^{p^e} For an R-module M, denote by F^e_*(M) the R-module obtained by restriction of scalars along F^e Thus F^e_*(M) has the same underlying abelian group as M, and for r ∈ R and m ∈ M we have r ·_{F^e} m = F^e(r) · m = r^{p^e} m.
1 We say R isF-finite if F ∗ e (R) is finitely generated over R.
2 Suppose R is an F-finite domain If for every non-zero element f ∈R there exists e∈N and φ∈Hom R (F ∗ e (R), R) such that φ(F ∗ e (f)) = 1, then we say R isstrongly
Let R be a normal, Cohen–Macaulay ring and let φ_R: F^e_*(ω_R) → ω_R denote the canonical dual of Frobenius We say that R has F-rational singularities if there are no nonzero proper submodules M ⊂ ω_R for which φ_R(F^e_*(M)) ⊆ M Equivalently, the only ω_R-submodules invariant in this Frobenius-dual sense are the zero submodule and ω_R itself.
Remark 7 states that in part (2) of this definition we call R strongly F-regular; there is also a notion of weakly F-regular, and it is conjectured that weak F-regularity implies strong F-regularity When R is N-graded, Lyubeznik and Smith proved this in [25], while the general case remains open Since our rings of invariants are all N-graded, the distinction is unnecessary, and we will call R F-regular rather than distinguishing between weakly or strongly F-regular.
Here we present definitions that coincide with the classical ones for F-rational and F-regular singularities within Hochster and Huneke’s theory of tight closure, a framework we introduce and develop further in this work Alongside these definitions, we establish a key tight closure result that we will rely on, originally proved by Smith, which underpins the arguments that follow.
Definition 3.4 Suppose that R is an F-finite domain and I ⊆R is an ideal The tight closure of I is defined to be the set
Let R be an F-finite domain and I an ideal of R If R is F-regular, then I = I^* for every ideal I (i.e., all ideals are tightly closed) Moreover, when R is local, R is F-rational if and only if I = I^* for all parameter ideals I, meaning every parameter ideal is tightly closed.
2 [6, Proposition 10.1.5] If R⊆S is a finite extension of rings, then (IS)∩R ⊆I ∗ for all ideal I ⊆R.
It is normal to ask when R G is either F-regular or F-rational In his dissertation, Jeffries characterized the case whene = 1.
Theorem 3.16 [18, Jeffries] Let G=Z/pZ act on R=k[x 1 , , x n ] The ring of invariants R G isF-regular if and only if R G is F-rational if and only if n= 2 or G acts with representation V 1 ⊕ ã ã ã ⊕V ` where n 1 = 2 and n i = 1 for 2≤i≤`.
Since we have a complete characterization of the Cohen-Macaulay property for our rings of invariants we will use the following theorem to help determine F-regularity when e >1.
Theorem 3.17 [15, Hochster, Huneke] Let R be a ring of characteristic p > 0 If R is strongly F-regular, then R is Cohen-Macaulay and normal.
It is clear that our rings of invariants are F-finite Moreover, given the definition of
F-rational singularities, our characterization of the Cohen-Macaulay property ofR G when
G=Z/p e Zallows us to determine when R G is notF-rational Using Theorem 3.17, it is a straightforward application of Corollary 3.6 to see when R G is neither F-regular nor
F-rational To give a complete characterization of these two types of singularities we need to consider when R G is Cohen-Macaulay By Corollary 3.6 we need only determine if R G is
F-rational orF-regular when G=Z/4Z and n = 3.
Example 3.5 Let G=Z/4Zact on R =k[x, y, z] by the indecomposable action where chark= 2 The ring of invariants R G is neither F-rational nor F-regular.
To prove the claim, it suffices to consider the indecomposable action, since when G acts on R with a representation V = V1 ⊕ … ⊕ Vℓ and n1 = 3 while RG is Cohen–Macaulay, we have ni = 1 for all 2 ≤ i ≤ ℓ Moreover, RG is Gorenstein, and hence RG is F-regular if and only if RG is F-rational We will show that RG is not F-regular Recall from Example 2.3 that
R G =k[x, xy+y 2 , z 4 +z 2 x 2 +zyx 2 +z 2 xy+z 2 y 2 +zy 2 x, xy 2 +y 3 +x 2 z+xz 2 ].
Set I = (x, xy+y 2 , z 4 +z 2 x 2 +zyx 2 +z 2 xy+z 2 y 2 +zy 2 x)⊆R G Notice xy 2 +y 3 +x 2 z+xz 2 6∈IR G
To prove the claim, it suffices to show that any f in I_RG with deg f = 3 is divisible by x Suppose f ∈ I_RG has degree 3 and can be written as f = g0 x + g1 (xy + y^2) with g0, g1 ∈ RG Since deg f = 3, deg g1 = 1, and the only degree-one element of RG is x, hence g1 = x Therefore x divides f, as required Next, consider the ideal IR = (x, y^2, z^4 − y^3 z) RG A direct calculation yields xy^2 + y^3 + x^2 z + x z^2 ≡ y^3 (mod xR).
Thusxy 2 +y 3 +x 2 z+xz 2 ∈IR Sincexy 2 +y 3 +x 2 z+xz 2 ∈R G with xy 2 +y 3 +x 2 z+xz 2 6∈IR G and xy 2 +y 3 +x 2 z+xz 2 ∈IR, it follows from Definition 3.4 part (2) thatI 6=I ∗ and therefore by Definition 3.4 part (1), R G is not F-regular.
Applying Corollary 3.6, Theorem 3.17, Example 3.5, and Theorem 3.16 result we get the following.
Corollary 3.18 Let G=Z/p e Z act on R =k[x 1 , , x n ] The ring of invariants R G is
F-rational if and only if R G is F-regular if and only ifn = 2 or G acts by representation
Proof: For e = 1, the result follows from Theorem 3.16 If e > 1, Theorem 3.17 shows that it suffices to determine whether R_G is not Cohen–Macaulay Moreover, Corollary 3.6 reduces the remaining check to the case p = 2 and e = 2, which is Example 3.5.
Recall, if R⊆S is a split inclusion of rings and S is F-regular or F-rational then R is
F-regular or F-rational respectively In his dissertation, Chan showed that for a group G with P ≤Ga normal p-Sylow subgroup, the inclusion R G ⊆R P is a split inclusion resulting in the following.
Theorem 3.19 [8, Chan] Let G≤GL n (k) act on R=k[x 1 , , x n ] where chark=p Let
1 If R H is F-regular, then R G is F-regular.
2 If R H is F-rational, then R G is F-rational.
Using this along with Corollary 3.18 we immediate get the following characterization of
F-regularity and F-rationality for certain rings of invariants.
Corollary 3.20 Let G≤GL n (k) act on R=k[x 1 , , x n ] where chark =p Suppose
H =Z/p e Z is a normal p-Sylow subgroup of G Ifn = 2 or H acts by representation
V 1 ⊕ ã ã ã ⊕V ` with n 1 = 2 and n i = 1 for 2≤i≤`, then R G is F-regular and F-rational. Proof Under the conditions of the hypothesis, R H is F-rational and F-regular by
Corollary 3.18 so we can apply Theorem 3.19.
4 Rings of Invariants of Subgroups of Z/p e Z
Throughout the section, set R=k[x 1 , , x n ] with chark=p, G=Z/p e Z, g ∈G a generator, and R G the ring of invariants If H ≤Gis a subgroup, then H acts naturally on
R and induces an inclusion of rings R G ⊆R H Recall the Jordan-H¨older filtration of G given by
0 =Ne ≤Ne−1 ≤ ã ã ã ≤N1 ≤N0 =G where N i =hg p i i This yields a chain of subrings of R which are rings of invariants, i.e.,
Our goal is to study R G by studying these intermediate subrings.
Graded Duality and the a-invariant
In this section, we recall an invariant which is useful when determining when a ring has certain F-singularities For a local ring S and an ideal I, denote byH I n (S) the nth local cohomology module of S with respect toI For details on local cohomology see [17].
Definition 4.1 LetS be a positively graded k-algebra with dimS=d where k is a field.
We define the a-invariant of S to be a(S) := max{t |(H n d (S)) t 6= 0} where n is the homogeneous maximal ideal of S.
Remark 8 Suppose thatS admits a canonical module, ω S Let E be the injective hull of the residue field of S and d= dimS By local duality,
H n d (S)∼= HomS(Ext d−d S (S, ωS), E)∼= HomS(HomS(S, ωS), E)∼= HomS(ωS, E).
Elements in (H n d (S)) t are in correspondence with S-linear maps (ω S ) −t →E Thus
(H n d (S)) t 6= 0 if and only if there exists a non-zero map (ω S ) −t →E which happens if and only if (ω S )−t 6= 0 Thus a(S) = max{t|(H n d (S)) t 6= 0}= max{t|(ω S )−t 6= 0}.
We use this formulation when giving bounds on the a-invariant for certain rings of invariants.
The a-invariant has relationships to F-singularities For example, if S is F-rational, then a(S) 0 If
0 =N e ≤Ne−1 ≤ ã ã ã ≤N 1 ≤N 0 =H is a composition series of subgroups acting naturally on R, then a(R G )≤a(R H )≤a(R N 1 )≤ ã ã ã ≤a(R N e−1 )≤a(R).
Proof We only need to prove the inequalitya(R G )≤a(R H ) as the other inequalities follow from Corollary 4.8 Define (−) ∨ = Hom R G(−, R G ) SinceR G is quasi-Gorenstein,
(R G ) ∨ ∼=ω R G Since dimR G = dimR H and R H is a finitely generated R G -module, it follows that ω R H ∼= Ext dim R G R G −dim R H (R H , ω R G )∼= Ext 0 R G(R H , ω R G )∼= (R H ) ∨
Using the same Galois theory argument as in the proof of Theorem 4.6, Tr G H 6= 0 The rest of the proof follows exactly as in the proof of Theorem 4.6.
Example 4.5 Let G=Z/9Zact on R =k[x, y, z, w] by the indecomposable action with chark= 3 Let g ∈G be a generator andH =hg 3 i ≤G By direct computation, π(g 3 )
We have R H =k[x, y, z, x 2 w−w 3 ] which is isomorphic to a polynomial ring Moreover a(R H ) = 4 and by Corollary 4.8, a(R G )≤4.
Remark 9 shows that when k is a perfect field, the results concerning the a-invariant in this section form a concrete instance of the more general Theorem 1.1(1) in [23] This theorem states that for an integral extension A ⊆ B of positively graded Noetherian domains over k, with A regular in codimension 1 and with frac(A) ⊆ frac(B), one has a(A) ≤ a(B) When we consider R_H ⊆ R_N as in Theorem 4.7, the normality of R_H satisfies the first hypothesis, and the second follows from the fact that frac(R_N)/frac(R_H) is a Galois extension with Galois group (G/H)/(G/N) While the proof in [23] is technically involved and uses the module of Kahler differentials to derive an inclusion ω_A → ω_B, our approach provides a simpler proof based on the representation theory of G, and crucially does not require k to be perfect.
A Structure Theorem for R H with H ≤ G
In this section we use changes of basis to reveal an explicit filtration of R = k[V] by representations associated with subgroups H ≤ G We begin with an explicit example to illustrate the kinds of change of basis we will employ For clarity, throughout this subsection when we write V_i we mean the vector space of dimension i, that is, dim_k V_i = i.
Example 4.6 Let G = Z/4Z act on R = k[x, y, z] by the indecomposable action with char k = 2, and let H ≤ G be the subgroup generated by pseudo-reflections; for g ∈ G a generator, H = ⟨h⟩ with h = g^2, so H = {h, id} Consider the map φ: V^3 → V^3 defined by φ(e1) = e1, φ(e2) = e2 + e1, φ(e3) = e3 + e1, where e1, e2, e3 is a basis for V^3 It is clear that φ is an automorphism of V^3 We claim that this change of basis gives an equivalence between the action of H on R in the natural representation and the action of H on R with the representation obtained after conjugation by φ.
V 2 ⊕V 1 The representation for ϕ(h) is given by ϕ(h) = ϕ
We check that this change of basis preserves the group action By direct calculation ϕ(h(e 1 )) = ϕ(e 1 ) = e 1 =h(e 1 ) =h(ϕ(e 1 )), ϕ(h(e2)) = ϕ(e2) = e2+e1 =h(e2+e1) = h(ϕ(e2)) and ϕ(h(e 3 )) = ϕ(e 3 +e 1 ) = (e 3 +e 1 ) +e 1 =h(e 3 +e 1 ) =h(ϕ(e 3 )).
Throughout this section we use changes of bases in the same manner as Example 4.6.
Since the procedure for verifying the equivalence of group actions is the same, we omit the computations when they are clear We present the result for H ≤ G, the subgroup consisting of all pseudo-reflections.
Theorem 4.10 Let G=Z/p e Z act on R=k[x 1 , , x n ] such that the representation of G contains a pseudo-reflection If H ≤G is the subgroup of G generated by pseudo-reflections, then R H is a polynomial ring.
Proof We start with the case whereG acts on R by the indecomposable representation.
By Theorem 3.10, we must haven =p e−1 + 1 Moreover, h=g p e−1 is a pseudo-reflection and a generator for H by Corollary 3.11 The induced action on R by π(h) is given by x 1 7→x 1 x 2 7→x 2 ã ã ã xn−1 7→xn−1 x n 7→x n +x 1
We claim the natural action ofH on R is equivalent toH acting on R with representation
By a targeted change of basis, sending x2 to x2 + x1 and xn to xn + (p−1)x1 while keeping xi fixed for i ≠ 2, n, the action of H on R is preserved and the required equivalence follows This yields R^H = k[x1, , x_{n−1}, x_p^n − x_{p−1} x_n], which we claim is isomorphic to a polynomial ring Consequently, dim R^H = n and R^H has n homogeneous generators.
Let S = {x1, , x_{n−1}, x p n − x p−1 1 x n} be a set of primary invariants Thus RH is its own Noether normalization, and the elements of S are linearly independent over k Consider the map φ: k[y1, , yn] → RH defined by yi ↦ xi for i = 1, , n−1 and yn ↦ x p n − x p−1 1 x n It is a surjective map of k-algebras, so it suffices to show that ker φ = 0, which follows from the fact that dimk[y1, , yn] = dim RH.
Consider G acting on R by a decomposable representation R = V_{j1} ⊕ V_{j2} ⊕ … ⊕ V_{jℓ} Theorem 3.10 shows that, after an appropriate change of basis, j1 = p e^{-1} + 1 and j_i ≤ p e^{-1} for i = 2, …, ℓ Define J_{j_i} to be the Jordan block associated to V_{j_i} in the Jordan form of the representation for the generator g ∈ G If x ∈ G is any pseudo-reflection, Corollary 3.11 implies x = g c p e^{-1} A direct calculation then relates these Jordan blocks via the conjugating action of c p e^{-1}.
= id j i for i= 2, , ` The result now follows from the indecomposable case.
Broer’s broader formulation extends the Chevalley–Shephard–Todd theorem by treating G as an irreducible group generated by pseudo-reflections acting on a ring of dimension n; in this setting the invariant ring RG is generated by n algebraically independent elements if and only if there exists a surjective homomorphism ϕ: k[V] → k[V]G (i.e., onto the invariants k[V]^G) By focusing on the pseudo-reflection subgroup of a cyclic p-group and leveraging the explicit representations of these pseudo-reflections, one can show that RG is a polynomial ring, avoiding Kemper and Malle’s classification of non-coregular invariant rings for irreducible pseudo-reflection groups.
Here are two illustrative examples.
Example 4.7 Let p= 3, e= 2, and n= 4 so that G=Z/9Z and let chark = 3 We have π(g)
LetH ≤G be the subgroup of G generated by pseudo-reflections Note, #H = 3 andH is generated byh=g 3 with representation π(h)
We apply the change of basis, ϕ, described in the proof of Theorem 4.10 which is given by x 1 7→x 1 x 2 7→x 2 +x 1 x 3 7→x 3 x 4 7→x 4 + 2x 1
from which we observe R H =k[x, y 3 −x 2 y, z, w], that is,R H is isomorphic to a polynomial ring.
Example 4.8 Let p= 3 and e= 2 so that G=Z/9Z and let chark= 3 Let g ∈G be a generator with representation as given in Example 4.4, i.e G is represented by
V 4 ⊕V 2 ⊕V 2 If H ≤G is the subgroup generated by pseudo-reflections, then g 3 =h 1 ∈H is a generator Applying the change of basis ϕ, given by x i 7→x i if i6= 2,4 and x 2 7→x 2 +x 1 , x 4 7→x 4 + 2x 1 we get ϕ(h 1 )
This gives R H =k[x 1 , x 3 2 −x 2 1 x 2 , x 3 , , x 8 ] which is isomorphic to a polynomial ring. Notice, the change of basis in this case was the identity map on elements of J i with i6= 1.
We would like to determine the structure of all the subgroups of G in a manner similar to Theorem 4.10 ForH ≤G any subgroup, this will allows us to say, for example, whether
R H is Cohen-Macaulay We begin by taking H ≤ G with |H| = p If g ∈ G is a generator, then h = g^{p^{e-1}} ∈ H is a generator More generally, if G ≅ Z/p^eZ with e > 2, we can consider H ≤ G with |H| = p^d where 1 ≤ d < e If |H| = p^d, then given a generator g ∈ G, we have h = g^{p^{e-d}} ∈ H as a generator Thus to obtain similar results for larger subgroups of G we can apply the same techniques as in the case d = 1. -**Support Pollinations.AI:**🌸 **Ad** 🌸 Optimize your mathematical articles with Pollinations.AI's free text APIs—perfect for group theory and Cohen-Macaulay content; [Support our mission](https://pollinations.ai/redirect/kofi) to keep AI accessible for everyone.
Theorem 4.11 Let G act on R =k[x 1 , , x n ] by the indecomposable action Let H ≤G with #H =p act naturally on R.
1 If n =p e−1 + 2, then R H is Cohen-Macaulay.
2 If p e−1 + 2< n≤2p e−1 , then the natural action of H on R is equivalent to H acting on R with representation V 2 n−p e−1 ⊕V 1 2p e−1 −n
3 If mp e−1 < n≤(m+ 1)p e−1 with 2≤m ≤p−1, then the natural action of H on R is equivalent to H acting on R with representation
Proof Throughout the proof letg ∈G be a generator so thath=g p e−1 ∈H is a generator.
1 If p= 2 and e= 2, then n=p 2−1 + 2 = 4, i.e., we need only consider n≥4 The generator h=g p e−1 ∈H has representation given by π(h)
When n >5, after the change of basis x i 7→x i for i6= 2,4, n−1, n and x2 7→x2+x1, x4 7→x4+x3, xn−1 7→xn−1+ (p−1)x1, xn7→xn+ (p−1)x2,
R H is isomorphic to the ring of invariants of H acting on R with representation
V 2 ⊕V 2 ⊕V 1 n−4 which is Cohen-Macaulay by Corollary 3.6.
For n = 4, apply the change of basis x1 → x1, x2 → x2 + x1, x3 → x3 + (p−1)x1, x4 → x4 + x3 + (p−1)x2; this shows R^H is isomorphic to the ring of invariants of H acting on R with the representation V2 ⊕ V2 For n = 5, apply the change of basis x1 → x1, x2 → x2 + x1, x3 → x3, x4 → x4 + x3 + (p−1)x1, x5 → x5 + (p−1)x2; this shows R^H is isomorphic to the ring of invariants of H acting on R with the representation V2 ⊕ V2 ⊕ V1 In both cases, Corollary 3.6 implies that R^H is Cohen-Macaulay.
2 The action of h onR is given by h(x i )
Set s=n−p e−1 We want to show there is a change of basis that witnesses the natural action of H onR as equivalent toH acting on R with representation G 0 acting on V 2 n−p e−1 ⊕V 1 2p e−1 −n Either 2s < p e−1 + 1 or 2s≥p e−1 + 1 Suppose
2s < p e−1 + 1 The desired equivalence holds after change of basis given by x i 7→
Suppose 2s≥p e−1 + 1 The desired equivalence holds after change of basis given by x i 7→
x i , iodd, i < p e−1 + 1, x i +xi−1, ieven, i < p e−1 + 1, x i + (p−1)x i−p e−1 , iodd, i≥p e−1 + 1, x i +x i−1 + (p−1)x i−p e−1 , ieven, p e−1 + 1≤i≤2s, x i + (p−1)x i−p e−1 , ieven i >2s.
In both cases, after the change of basis theH has representation given by a direct sum of s copies of V 2 and n−2s =n−2(n−p e−1 ) = 2p e−1 −n copies ofV 1 These changes of basis are demonstrated in Example 4.9.
3 The action of h onR is given by h(x i )
Set s=n−mp e−1 We again show there is a change of basis witnessing H acting on
R as equivalent to G 0 =Z/pZacting on R with action induced by the action of G 0 on
V m+1 n−mp e−1 ⊕Vm (m+1)p e−1 −n Consider the change of basis maps φ: V n →V n and ψ: V n →V n which induce the following maps onx i φ(x i )
The change of basis ϕ=ψ◦φ gives the desired equivalence In particular after the change of basis we have π(h)7→
where n i =m+ 1 if i < sand n i =m if i≥s Thus after the change of basis ϕwe haveG 0 acting on the direct sum of s copies ofV m+1 and n−(m+ 1)s m = m 2 p e−1 +mp e−1 −mn m = (m+ 1)p e−1 −n copies of V m
We now consider an example to illustrate the changes of variable described in the proof of Theorem 4.11.
Example 4.9 Let G=Z/49Z act by the indecomposable action and chark = 7 Ifg ∈G is a generator, than h=g 7 is a generator for H ≤G with #H = 7 We first set n = 10 so that a representation for h is given by π(h)
In this cases = 10−7 = 3 and therefore 2s = 6 p e−1 + 2, then R H is not Cohen-Macaulay.
As mentioned before we would like to generalize Theorem 4.11 to the case where
H ≤G=Z/p e Zwith #H =p d for 2≤d≤e To help generalize and illustrate the technique we will use, we look at some examples when d= 2.
Example 4.10 Let G=Z/2 e Z and g ∈G be a generator We first consider e= 3 Set h=g 2 and hhi=H≤G so #H = 4 If n= 5, then a representation of h is given by π(h)
Consider the elementary matrix operations giving the following equivalences
Applying the change of basis T, defined by x1' = x1, x2' = x2 + x1, x3' = x3 + x2 + x1, x4' = x4 + x2, and x5' = x5 + x4 + x3 + x2 + x1, reorganizes π(h) so that the columns indexed by 1, 3, and 5 (those congruent to 1 mod 2) form one Jordan block, while the columns indexed by 2 and 4 (those congruent to 0 mod 2) form a separate Jordan block In this new basis, π(h) is viewed as the Jordan form h0, revealing its block structure and simplifying the analysis of its eigenstructure.
Thus the natural action of H on R is equivalent to the action of H on R with the representation V3 ⊕ V2 If n = 7, then the representations for h and h′, where h′ is the image of π(h) after a change of basis, are given by π(h) [the matrix that follows].
Thus the natural action of H onR is equivalent to the action of H onR with representation V 4 ⊕V 3
We now consider e= 4 Set h=g 4 and hhi=H ≤G If n= 9, then a representation for h is given by π(h)
and after an appropriate change of basis, we may view π(h) as h 0
Thus the natural action of H onR is equivalent to the action of H onR with representation V 3 ⊕V 2 3
Example 4.11 Let p= 3, e= 3 andg ∈G=Z/27Z be a generator Seth=g 3 and hhi=H ≤G If n= 10, a representation for h is given by π(h)
and after an appropriate change of basis, we may view π(h) as π(h)
Hence the natural action of H on R is equivalent to the action of H on R with representation V 4 ⊕V 3 2
With these examples in mind, we now prove the following lemma regarding when
Theorem 4.13 Let G=Z/p e Z act on R=k[x 1 , , x n ] by the indecomposable action and let H ≤G a subgroup with #H=p d , 0< d < e, act naturally on R Let g ∈G be a generator so that H =hg p e−d i If p e−1 +mp e−d < n≤p e−1 + (m+ 1)p e−d with
0≤m≤p d −p d−1 −1, then natural action of H on R is equivalent to the action of H on
Let g_{p^{e-d}} satisfy ≡ 0 mod p A representation π(g_{p^{e-d}}) is given by the n×n matrix (a_{j,i}) with 1’s on the main diagonal and with 1’s at the entries (p^{e-d+i+1}, i) for i = 0, …, n − p^{e-d}, and zeros elsewhere We want to show there exists a change of basis witnessing the natural action of H on R as equivalent to the action of H on R with the representation decomposing as the direct sum of two standard H-modules, V_p(n−p^{d−1}+m+1)_{e−1} ⊕ V_p(p^{d−1})_{e−1+m+(m+1)p^{e−d}−n} The action of H is described by the induced action on these components.
There exists a basis change that transfers the given representation of g p e^{-d} into a form where the Jordan blocks are built from the columns corresponding to the equivalence classes of p e^{-d} In particular, each Jordan block J_j is formed by gathering all columns whose indices lie in the same equivalence class under the relation induced by p e^{-d}.
Modulo p^(e−d) (see Example 4.10) yields p^(e−d) Jordan blocks, each giving an indecomposable representation To determine their sizes, write n = a p^(e−d) + b with 0 ≤ b < p^(e−d); each Jordan block has rank either a or a+1, so exactly b blocks have rank a+1 By hypothesis, n = p^(e−1) + m p^(e−d) + c for some c in {1, , p^(e−d)} Equating a p^(e−d) + b with this expression gives a = p^(d−1) + m, and hence b = n − a p^(e−d) = n − (p^(d−1) + m) p^(e−d) = n − p^(e−1) − m p^(e−d).
Jordan blocks of rank a+ 1 =p d−1 +m+ 1 and p e−d −b=p e−d −(n−p e−1 −mp e−d ) =p e−1 + (m+ 1)p e−d −n
Jordan blocks of rank a=p d−1 +m This gives the desired equivalence.
Recall the filtration of G given by
=G which yields a chain of subrings of R
Using Theorem 4.13 we can now give a more descriptive picture of this chain of subrings.
For 1≤i≤e, by Theorem 4.13, there is an induced map ϕ: k[V]
E such that the following diagram commutes.
Moreover, this gives an injective mapk[V c a i−1 i−1 +1 ⊕Vc b i−1 i−1 ]
1≤i≤e This yields the following corollary to Theorem 4.13.
Corollary 4.14 Let G=Z/p e Z act on R =k[x 1 , , x n ] by the indecomposable action. Let g ∈G be a generator For p e−1 +m i p i < n≤p e + (m i + 1)p i with
5 Noether Numbers, Multiplicity, and p-Sylow Subgroups
Noether Numbers for Modular Ring of Invariants
We open this chapter by introducing a result due to Benson which is analogous to
Noether’s bound on the top degree of a homogeneous generating set for the ring of invariants We start by giving some definitions from invariant theory which will be useful.
Definition 5.1 Let G act on a graded ring R, and let m denote the homogeneous maximal ideal of the invariant subring R^G The Hilbert ideal H of R is defined as the extension mR, i.e., the ideal of R generated by the homogeneous invariants of positive degree The ring of coinvariants is the quotient R/H.
RG is a module over the group ring kG, and its Noether number, denoted β(V) or β(RG), is the smallest integer d such that RG can be minimally generated by homogeneous elements of degree at most d In other words, β(RG) measures the minimal degree bound needed for a minimal generating set of RG within the category of kG-modules This invariant reflects the complexity of RG's module structure under the group action and serves as a valuable metric in invariant theory and the study of polynomial representations Understanding β(RG) helps researchers gauge how generator degrees grow and provides insight into the algebraic properties of the group action on RG.
Let td(R_G) denote the top degree of the graded ring R_G, i.e., the largest degree with a nonzero homogeneous component Since R_G is finite-dimensional as a vector space over k, td(R_G) is finite (td(R_G) < ∞) See [22] for a standard reference.
Definition 5.2 LetG be a group acting on a ring R For an idealI ⊆R we define the invariants of I to beI G :={r ∈I |g(r) =r for all g ∈G} We say I is G-stable provided ga∈I for all g ∈G and a∈I.
Example 5.1 notes that B_n(k) is a subset of GL_n(k), the Borel subgroup of GL_n(k), consisting of all upper triangular matrices By Theorem 2.3, the image π(g) of any element g in G = Z/p^eZ lies in the Borel subgroup, that is, π(g) ∈ B_n(k) Consequently, any ideal that is Borel-fixed is also fixed by the group action, providing a class of examples of G-stable ideals.
Lemma 5.1 [29, Benson, Lemma 2.3.1] Let S be a commutative ring with identity and π: G→Aut(S) be a representation of G by automorphisms of S If #G is invertible in S and I ⊂S is a G-stable ideal then I #G ⊂I G ãS.
Note that Lemma 5.1 requires the action of G on A to be non-modular, since a crucial step in the proof involves dividing by the order of G, |G| In the modular setting, this division is not generally valid, and the following example shows that we do not obtain the same result in the modular case.
Example 5.2 Let G=Z/4Zact on R =k[x, y, z] by the indecomposable action with chark= 2 Let m⊆R be the homogeneous maximal ideal We want to show m 4 6⊆m G R. From Example 2.3,
R G =k[x, xy+y 2 , z 4 +z 2 x 2 +zyx 2 +z 2 xy+z 2 y 2 +zy 2 x, xy 2 +y 3 +x 2 z+xz 2 ].
It is clear that mis G-stable By direct calculationm G R= (x, y 2 , z 4 ) and therefore yz 3 ∈m 4 but yz 3 6∈m G R.
Let R be a graded ring of characteristic p > 0 with R0 = k, and let G be a group acting on R by degree-preserving k-algebra homomorphisms, with p dividing |G| If P ≤ G is a normal p-Sylow subgroup, then the index [G:P] is invertible in R In this p-torsion setting one recalls the relative transfer map, which encapsulates a trace-like construction connecting the P-action to the G-action and underpins the transfer of structure from R to its G-invariants.
Let Tr_{G/P}(r) denote the transfer from P to G, defined by Tr_{G/P}(r) = ∑_{g ∈ G/P} g(r), where the sum runs over a set of representatives for the distinct cosets of P in G (i.e., over the distinct equivalence classes of G/P) Consequently, the image of the relative transfer map lies in RG By combining these observations, we obtain an extension of Lemma 5.1.
Lemma 5.2 Let R be a ring with charR=p >0 and G a group such that p|#G Let
P G be a normal p-Sylow subgroup acting naturally on R If I ⊆R P is G-stable, then
Proof Set s= [G:P] Since P is normal, it is the unique p-Sylow subgroup and therefore
#G/P =s∈R × Fix g 1 , , g s ∈G/P a complete set of distinct equivalence classes and setT ={g 1 , , g s } Choose s elements of I and index them byT, that is, choose
{f g i |i= 1, , s} ⊆I For any h∈G/P, since T is a complete set of representatives for G/P, it follows that
Indeed, there exists g i ∈T such that g i =h −1 and thereforehg i f g i −f g i = 0 Summing over the distinct equivalence classes h∈G/P and expanding yields
∈I G ãR p It suffices to show, this term is, up to a unit, a generic element of I s The term on the right of (9) corresponding toS =∅is ±s
All other terms on the right hand side of (9) are in I G ãR P since I is G-stable and
Q g i ∈Shg i f g i is invariant In particular, if S ={g α 1 , , g α
Since I is G-stable, the product g α 1 f g α
Remark 11 identifies a key step in the proof of Lemma 5.2: we must divide by the index [G:P] In Benson's original proof, there is an analogous step that requires division by a comparable index, highlighting the parallel structure of the argument.
#Gwhich is the obstruction to a similar lemma when G is a p-group.
As a consequence of this lemma, we get the following well-known analogue of Noether’s bound on the top degree of a homogeneous generating set for certain modular rings of invariants.
Corollary 5.3 Let R be a ring with charR =p > 0 and G a group such that p|#G If
P G is a normal p-Sylow subgroup acting naturally on R, then β(R G )≤[G:P]β(R P ).
Proof Denoting n the homogeneous maximal ideal for R P and similarly formand R G , it is clear thatn G =m Moreover,n isG-stable and thereforen [G:P] ⊆n G ãR P We want to show that n G ãR P is generated as an ideal by G-invariants of degree at most [G:P] Considerf a monomial in the generators of R P with degf =s≥[G:P] We have f is a product of s elements of n and therefore f ∈n G ãR P If s >[G:P], then we can writef =gh with degg = [G:P] and degh >1, i.e.,f is not part of a minimal generating set for n G ãR P
Recall the relative transfer map Tr G P :R P →R G defined by Tr G P (r) = [G:P 1 ] P g∈G/P g(r). Since mãR P =n G ãR P , it follows that Tr G P | n G ãR P :n G ãR P →R G is a surjection Moreover, the relative transfer gives a R G -module homomorphism and the generators for n G ãR P are mapped to generators for m which generateR G as an algebra This gives that
R G = (R P ) (G/P ) is generated by elements of degree at most [G:P] when considered as the rings of invariants forG/P acting on R P and the result now follows.
Remark 12: The bound that the generators of the Hilbert ideal RG in RP have degree at most [G:P] was first established in [11] via a rather technical linear-algebra argument Here we avoid that route by applying Lemma 5.2 to obtain the same degree bound and then invoking the surjectivity of the restricted relative transfer map Notably, Fleischmann’s work supplies an algorithm for expressing arbitrary degree G-invariants in terms of invariants of degree at most [G:P]β(RP), without requiring an explicit generating set for RG.
Example 5.3 We claim this bound is sharp LetG=Z/2Z×Z/3Z Let R=F 3 [x, y] and
P =h(0,1)i Since G is abelian, any subgroup ofG is normal, i.e P is a normal 3-Sylow subgroup Suppose G acts by the representation π((1,0)) "
We have [G:P] = 2 and the representation of P is the indecomposable representation of
Z/3Z Thus we have seen R P =F3[x, y 3 −x 2 y], and β(R P ) = 3 This gives β(R G )≤2β(R P ) = 6.
Using the algorithm outlined in Chapter 2, we can computeR G directly In particular,
S ={x 2 , y 6 +x 2 y 4 +x 4 y 2 } forms a set of primary invariants forR G and applying the algorithm yields
Since we can bound the Noether number for a modular ring of invariants by the
Consider the Noether number of a normal p-Sylow subgroup For a p-group G acting on R = k[x1, , xn], a natural question is whether there exists an explicit value or bound for β(RG) In the cyclic case, G ≅ Z/p^eZ, this longstanding question has been answered when e = 1 by the results that follow For the remainder of this chapter we again use the convention that V_i denotes the vector space of dimension i, i.e., dim_k V_i = i.
Consider the action of the cyclic p-group G = Z/pZ on the polynomial ring k[V], with the base field k of characteristic p If k[V] is a reduced finite-dimensional kG-module (where kG denotes the usual group ring), then define s to be the number of non-trivial indecomposable Jordan blocks that appear in the representation of G on k[V].
1 If the representation of G contains a summand isomorphic to V i with i >3, then β(V) = (p−1)s+p−2.
2 If G has representation mV 2 ⊕`V 3 with ` >0, then β(V) = (p−1)s+ 1.
Note, it is well known that β(V 2 ) =β(V 3 ) = β(2βV 2 ) = p It follows from [7] and [31] that β(tV 2 ) = t(p−1) for t >2 Applying these facts and Theorem 5.4 gives the following refinement of Corollary 5.3 when G=Z/pZ.
Corollary 5.5 Let G be a group acting on k[V] with chark =p >0 such that p|#G If
Z/pZ=P G is a normal p-Sylow subgroup acting naturally on R, then setting s to be the number of non-trivial indecomposable Jordan blocks in the representation of P, the following hold.
1 If P has representation V 2 or 2V 2 , then β(R G )≤[G:P]p.
2 If P has representation sV 2 with s >2, then β(R G )≤[G:P]s(p−1).
3 If the representation of P contains a summand isomorphic to V i with i >3, then β(R G )≤[G:P]((p−1)s+p−2).
4 If P has representation mV 2 ⊕`V 3 with ` >0, then β(R G )≤[G:P]((p−1)s+ 1).
Notice that Example 5.3 is a consequence of part (1) of this corollary More generally, if G= (Z/pZ) × ×Z/pZ acts onFp[x, y], then by part (1), β(R G )≤p(p−1).