The topological nilpotence degree of a Noetherian unstable algebra

We investigate the topological nilpotence degree, in the sense of Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$ is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian $p$-subgroups. By replacing centralizers of elementary abelian $p$-subgroups with components of Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn's result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our results apply much more generally, for example, we establish results for $2$-local compact groups in the sense of Broto-Levi-Oliver, for connected $H$-spaces with Noetherian mod $2$ cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold at the prime 2. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson's depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

Here the product is taken over those elementary abelian p-subgroups E of G for which C(G) is strictly contained in E, and the map is the map induced by the inclusions C G (E) ≤ G. He then shows that for any compact Lie group G. Combining these two results gives Theorem 1.2. We make the following remarks about this theorem.
(1) As noted by Kuhn, it suffices in Theorem 1.2 to only consider those E which contain C(G).
(2) By [Kuh13,Theorem 2.30] the central essential ideal CEss(G) is non-zero if and only the cohomology H * G has depth equal to the rank c(G) of the maximal central elementary abelian p-group C(G).
(3) The appearance of − dim(G) in the theorem comes from Symonds' theorem [Sym10] that the Castelnuovo-Mumford regularity Reg(H * G ) (see Appendix B) is less than or equal to − dim(G). Using these three remarks, one could restate Kuhn's theorem in the following way: {e(C G (E)) + Reg(H * CG(E) )}.
We state it in this way, as this is closer to the generalization we prove below.
1.2. Unstable algebras and the central essential ideal. In the previous section we saw that the topological nilpotence degree of H * G is controlled by cohomology of elementary abelian p-subgroups of G. In order to generalize this to an arbitrary unstable Noetherian algebra R we need to explain what plays the role of the centralizer of R. For this, we use Lannes' T -functor [Lan92].
We recall in Section 2.2 that for any pair (E, f ) such that E is an elementary abelian group and f is a finite morphism R → H * E of unstable algebras, we can produce a new unstable algebra T E (R; f ), along with a canonical map ρ = ρ R,(E,f ) : R → T E (R; f ). If R = H * G , and E < G is an elementary abelian p-subgroup, then the fundamental computation of Lannes is that T E (H * G ; res * G,E ) ∼ = H * CG(E) , where res * G,E : H * G → H * E is the induced map, and ρ : H * G → H *

CG(E)
is simply the map induced by the inclusion C G (E) → G. Inspired, by this Dwyer and Wilkerson [DW92] used the components of the T -functor to define centrality in a Noetherian unstable algebra. In particular, we say that Pairs (E, f ) as considered above naturally assemble into a category A R (see Section 2.1). This category has the property that every endomorphism is an isomorphism, and as such the set of isomorphism classes of objects forms a poset, where [(E, f )] ≤ [(V, g)] if and only if Hom AR ((E, f ), (V, g)) = ∅ Using work of Dwyer and Wilkerson, we show in Theorem 3.12 that there exists a unique (up to isomorphism) maximal central element (C, g) ∈ A R . If R = H * G for a finite p-group G with group-theoretic center C(G), then C = C(G), however this does not hold in general for a compact Lie group. Instead, there is a monomorphism C(G) → C, which need not be an isomorphism in general, see Example 3.14 for an example due to Mislin. We refer to a choice of maximal central element as the center of R, and write (E, f ) ⊆ (V, g) if [(E, f )] ≤ [(V, g)]. Inspired by Kuhn's work, we define the central essential ideal of a Noetherian unstable algebra R with center (C, g) as the unstable algebra fitting in the exact sequence where the product is taken over the maps ρ R,(E,f ) . This does not depend on the choice of isomorphism class of the center of R.
For G a finite group, Kuhn has proved that the Krull dimension of CEss(G) is at most the rank of C. The proof uses a result about transfers due to Carlson [Car95] that is not available for a general unstable algebra. We instead use U-technology to prove the following result, which is crucial in the sequel.
Theorem A. (Theorem 4.3). Let R be a connected Noetherian unstable algebra with center (C, g), then the Krull dimension of CEss(R) is at most the rank of C.
1.3. The topological nilpotence degree. We now move on to the calculation of the topological nilpotence degree of unstable algebras, i.e., the calculation of the invariant d 0 . The first step is to calculate d 0 of the central essential ideal of a Noetherian unstable algebra. In order to do this, there is a technical point we must introduce. If R is a Noetherian unstable algebra with center (C, g), the image of g : R → H * C is either a polynomial algebra (when p = 2) or a polynomial tensor an exterior algebra (when p > 2). In particular, there always exists a subalgebra B ⊂ R such that B → Im(g) is an isomorphism. Borrowing terminology from Kuhn, we call such a B a Duflot algebra. We then have the following, which is the analog of (1.4) above, where e(R) denotes the top degree of the finitely generated R-module H * C . Theorem B. (Theorem 4.20 and Theorem 4.22) Let R be a connected Noetherian unstable algebra at the prime p with center (C, g), and suppose that the Duflot algebra B is a polynomial algebra, 1 then if CEss(R) = 0 we have d 0 (CEss(R)) ≤ e(R) + Reg(R).
Moreover, CEss(R) = 0 if and only if depth(R) = rank(C). In this case, CEss(R) is a Cohen-Macaulay R-module of dimension rank(C).
The statement that if depth(R) = rank(C), then CEss(R) = 0 can be considered a form of Carlson's depth conjecture (see [CTVEZ03,Question 12.5.7]) in the case of a Noetherian unstable algebra of minimal depth. Indeed, we always have depth(R) ≥ rank(C) by the author's generalized version of Duflot's theorem [Hea20], see also Corollary B.7 in this paper (Carlson considers the case R = H * G for G a finite group). We next introduce the p-central defect of a Noetherian unstable algebra, which is the difference between the Duflot minimum (the rank of C) and the Krull dimension of the ring. Algebras of p-central defect 0 are precisely those for which R ∼ = CEss(R) (and are the analogs of p-central groups if R = H * G is the mod p cohomology of a compact Lie group). In this case, one can see the above theorem as an estimate of d 0 (R) itself.
We then extend the previous result for connected unstable algebras with a local cohomology theorem in the sense of Greenlees and Lyubeznik [GL00]. We recall that these are those unstable algebras for which there is a spectral sequence E s,t 2 = H s,t m (R) =⇒ (I m ) ν−t−s where I m is the injective hull of F p in the category of R-modules, and ν is an integer, called the shift. In this case, we have the following.
Theorem C. (Theorem 5.9) Let R be a connected Noetherian unstable algebra at the prime p, and suppose that the Duflot algebra B is a polynomial algebra. If R has p-central defect 0 and R has a local cohomology theorem of shift ν, then Reg(R) = ν and d 0 (R) = e(R) + ν.
Using work of Broto and Crespo [BC99], we show that at the prime 2, a H-space with Noetherian mod 2 cohomology ring always has p-central defect 0. Moreover, the cohomology always has a local cohomology theorem, and we thus compute d 0 (H * (X)) in Theorem 5.15.
Theorem C also specializes to the following, generalizing Kuhn's result for compact Lie groups.
We then show that d 0 (R) is controlled by d 0 (CEss(T E (R; f )), analogous to (1.3) above. Putting all these result together, we obtain an estimate for d 0 (R) for an unstable algebra (at the prime 2) in Theorem 5.24.
This leads to the following group theoretic result, where a subgroup E < G is said to be cohomologically p-central if C G (E) → G is a mod p cohomology equivalence. We will see that there is (up to isomorphism) a maximal cohomologically p-central subgroup C p (G), whose rank may be greater than the rank of the usual group-theoretic center of G.
Theorem E. (Theorem 6.8) Assume we are in one of the following cases: (1) G is a compact Lie group.
(2) G is a discrete group for which there exists a mod p acyclic G-CW complex with finitely many G-cells and finite isotropy groups. (3) G is a profinite group such that the continuous mod p cohomology H * G is finitely generated as an F p -algebra. (4) G is a group of finite virtual cohomological dimension such that H * G is finite generated as an F p -algebra. (5) G is a Kac-Moody group. Then, for any prime p we have where c(C G (E)) is the rank of the maximal cohomologically p-central subgroup of G.
Of course, by including additional summands, one can rewrite this as to give a result analogous to Theorem 1.2. At the prime 2, we have similar results in the case of the mod 2 cohomology of 2-local compact groups [BLO07], see Section 6.2.
Example 1.5. In Example 6.11, we compute that 1 ≤ d 0 (H * GL2(Z3) ) ≤ 2 when p = 3. Similarly, in Example 6.12 we compute that d 0 (H * S2 ) = 2 at the prime 3, where S 2 is the Morava stabilizer group which features prominently in the chromatic approach to stable homotopy theory.
Finally, in an appendix, we show that a slight variation of our methods shows the following.
Theorem F. (Theorem A.2) Fix p = 2. Let G be a compact Lie group, and X a manifold, then where C(G, X) is the maximal central elementary abelian p-subgroup of G that acts trivially on X, and e(G, X) denotes the top degree of a generator of the restriction map H * G (X) → H * C(G,X) . Notation. The following is some of the notation used in this paper.

U
The category of unstable modules over the Steenrod algebra (Section 2.1) K The category of unstable algebras over the Steenrod algebra (Section 2.1) R Generic unstable algebra (Section 2.1) E Elementary abelian p-group A R Rector's category associated to a Noetherian unstable algebra R (Section 2.1) Topological nilpotence degree of an unstable module (Section 2.3) CEss(R) The central essential ideal of a Noetherian unstable algebra (Section 4.1) The module of primitives for a comodule (Section 4.2) The space of indecomposables for a B-module M (Section 4.2) F Fusion system associated to a discrete p-toral group S (Section 6.2) Conventions. We will always write H * G (X) for the mod p G-equivariant cohomology of a space X. In particular, taking X to be a point, then H * G denotes the group cohomology of G. For a space X we will always write H * (X) for the mod p cohomology of X; thus H * G = H * (BG). If R is an augmented F p -algebra we will write ǫ R : R → F p for the canonical map; in the case of R = H * (X), we will often abbreviate this to ǫ X , or even ǫ G if X = BG .
We observe that if M ∈ U, then M is trivial in negative degrees. If M 0 ∼ = F p , then we say the M is connected.
The mod p cohomology of a space H * (X) is always an unstable module. In fact, it also has an algebra structure satisfying certain properties, which leads to the following definition.
Definition 2.2. An unstable A-algebra R is an unstable A-module, together with maps µ : R ⊗ R → R and η : F p → R which determine a commutative, unital, F p -algebra structure on R and such that the Cartan formula holds (equivalently, φ is A-linear) and Sq n x = x 2 if p = 2 and n = |x|, P n x = x p if p > 2 and 2n = |x|.
We let K denote the category of unstable algebras over A. This is the category with objects unstable algebras, and morphisms degree preserving maps which are both A-linear and maps of graded algebras.
Finally, we say that R is a Noetherian unstable algebra if R is finitely generated as an algebra.
Example 2.3. The mod-p cohomology of an elementary abelian p-group E of rank n is of fundamental importance in the theory of unstable algebras over the Steenrod algebra. We recall that with |x i | = 1 when p = 2, and where |y i | = 1 and β denotes the Bockstein homomorphism associated to the sequence 0 → Z/p → Z/p 2 → Z/p → 0. In particular, H * E is a Gorenstein ring of dimension n. Its importance comes from the fact that it is an injective object in the category U, see [Car83,Mil84,LZ86].
Given an unstable algebra R, we can also define a category R−U, whose objects are unstable Amodules M together with A-linear structure maps R⊗M → M which make M into an R-module, and whose morphisms are the A-linear maps which are also R-linear. The full subcategory of R − U consisting of the finitely generated R-modules will be denoted R f g − U.
Example 2.4. Let G be a compact Lie group and X a manifold, then the Borel equivariant cohomology H * G (X) is an object of R f g − U for R = H * G , see [Qui71]. The following categories, first studied by Rector [Rec84], will play a crucial role in the sequel.
Definition 2.5. Let R be a Noetherian unstable algebra, then the category V R is the category with objects (E, f ) where E is an elementary abelian p-group, and f : R → H * E is a homomorphism of unstable algebras. A morphism α : necessarily arises form a monomorphism E → V of elementary abelian p-groups. We have the following properties of A R , where we recall that a Noetherian unstable algebra always has finite Krull dimension.
Proposition 2.6. Let R be a Noetherian unstable algebra of Krull dimension d.
(1) The category A R has a finite skeleton.
(2) For each (E, f ) ∈ A R we have rank(E) ≤ d. In fact, Proof. Part (1)  as needed. This leads to the following.
Finally, we note the following.
Lemma 2.14. For any morphism α : where G is a compact Lie group, due to Lannes [Lan86,Lan92]. More specifically, let E < G be an elementary abelian p-subgroup, with induced map res * G,E : . The adjoint to this gives rise to an isomorphism are the maps induced on cohomology by the obvious maps Note that the claim of Corollary 2.12 then follows immediately.
It follows that T E (R; f ) plays the role of the 'centralizer' of the pair (E, f ) ∈ A R . We investigate this analogy further in the following sections.
2.3. The nilpotent filtration of an unstable algebra. In this section, we review Schwartz's nilpotent filtration on the category of unstable modules over the Steenrod algebra, and the associated localization functors of Henn, Lannes, and Schwartz. We recall that in the previous section we introduced the categories U and K of unstable modules and unstable algebras over the Steenrod algebra respectively. As noted in the introduction, Schwartz [Sch88] introduced a natural filtration on U, known as the nilpotent filtration. We take the following from [HLS95].
Definition 2.16. Let M, N be unstable modules.
(1) M is called n-nilpotent if and only if every finitely generated submodule admits a filtration such that each filtration quotient is an n-fold suspension.
(2) The category Nil n is the full subcategory of U that contains all n-nilpotent modules. Definition 2.17. Let M be an unstable A-module, then the topological nilpotence degree of M is We note that if R is Noetherian, and M ∈ R f g − U, then d 0 (M ) is finite [HLS95, Theorem 4.3]. In particular, d 0 (R) itself is finite.
There are a number of alternative characterizations of the number d 0 . For example, the subcategories Nil n are localizing, and the general theory of localization in abelian categories implies there exists a functor L n : U → U, and a natural transformation λ n : 1 U → L n such that L n M is Nil n -closed, and λ n has n-nilpotent kernel and cokernel. In this case, we have Further equivalent characterizations can be found in [Kuh07, Definition 3.11]. One particular result of interest for us is the following, which is a direct consequence of [HLS95, Theorem 4.9].
Proposition 2.18. Let R be a Noetherian unstable algebra, and M ∈ R f g − U, then for n ≥ d 0 (M ) there is a monomorphism in R f g − U: induced by the product of the maps η M,(E,f ) .
Here we write K ≤n for the quotient of a graded module K by all elements of degree greater than n.
We also have the following properties of d 0 , which are a combination of [HLS95, Proposition 3.6] and [Kuh07, Proposition 3.12].
Proposition 2.19. Let R be an unstable module.
(1) If R is concentrated in finitely many degrees, then d 0 (R) ≤ n, where n is the top degree in which R is non-zero.
The topological nilpotence degree of a Noetherian unstable algebra R is related to algebraic nilpotence in the following way, compare [Kuh13, Corollary 2.6].
Lemma 2.20. Let R be a connected Noetherian unstable algebra, and define e to be d 0 (R) for p = 2, or d 0 (R) + dim(R) for p odd. Then e is the maximal integer d such that rad(R) d = 0. In particular, for s > e, the product of any s nilpotent elements in R is zero.
Proof. Let d alg (R) be the maximal d such that rad(R) d = 0, so that our claim is d alg (R) ≤ e. It is clear that It then follows from Proposition 2.18 that Here we have used that rank(E) ≤ dim(R) for each (E, f ) ∈ A R , see Proposition 2.6(2). It follows that d alg (R) ≤ e as claimed.

The center of a Noetherian unstable algebra
In this section, following Dwyer and Wilkerson, we define central objects of a Noetherian unstable algebra with respect to the objects of A R , and show that, up to isomorphism, there is a maximal such element with respect to a natural poset structure. We prove that for each central object (E, f ) ∈ A R , the unstable algebra R naturally obtains the structure of a H * E -comodule. 3.1. Central objects of a Noetherian unstable algebra. Throughout this section we assume that R is a connected Noetherian unstable algebra. These assumptions can be weakened, however they are suitable for all the applications we have in mind. Based on Example 2.15 it is natural to make the following definition.
In light of Example 2.15, given a central elementary abelian p-subgroup E of a compact Lie group G, we see that the pair (E, res * G,E ) is central inside A H * G . We will see later the converse is true if G is a finite p-group, but not in general.
We recall that κ R, gives, respectively, the identity map of R and the map f .
We use this in the following result, which is an algebraic analog of the fact that if E is an elementary abelian p-subgroup of a group G, then E is always a central subgroup of C G (E).
The map h is defined in the proof of Proposition 3.6 of [DMW92]. Namely, the map is an equivalence, and the following diagram commutes: . In order to see that h is central, we use Proposition 3.3 with the map κ R,(E,f ) : A relatively straightforward diagram chase shows that this map has the desired property, see the proof of Proposition 3.8 of [Hea20].   Let (E, f ) and (C, g) be objects of A R , and assume that (C, g) is central. Then there is a unique pair (E ⊕ C, f ⊞ g) ∈ V R which restricts to f (resp. g) along the summand inclusion E → E ⊕ C (resp. C → E ⊕ C).
Remark 3.7. The map f ⊞ g : We observe that it is not necessarily the case that (E ⊕ C, f ⊞ g) ∈ A R , i.e., the map f ⊞ g : R → H * E⊕C is not necessarily finite. However, given any object (V, j) ∈ V R by [DW92, Proposition 4.8] there is the notation of a kernel ker(j) ⊂ V , which is a subgroup of V , such that j : R → H * V extends uniquely to a mapj : jj commutes. Applying this to the construction in Lemma 3.6 leads to the following definition.
Definition 3.8. Let (E, f ) and (C, g) be objects of A R , and assume that (C, g) is central, then As a diagram, we can represent this as 3.2. The poset of central objects. Observe that the category A R has the property that every endomorphism is an isomorphism. Such a category is called an EI-category (see [Lüc89]), and the set of isomorphism classes of objects is partially ordered by the relation Recall that this implies that there exists a monomorphism ι : Consider the full subcategory A C R ⊂ A R consisting of the central objects. This inherits the partial order from A R . We shall show that, with respect to this partial order, A C R has, up to isomorphism, a unique maximal element, i.e., there is, up to isomorphism, a unique maximal central object in A R . To do this, we briefly recall the definition of an under category.
A crucial observation is the following, which is shown in the proof of Proposition 4.10 of [DW92].
Proof. By the previous proposition ι : Theorem 3.12. With respect to the poset structure above, there exists a unique (up to isomorphism) maximal central element (C, g) ∈ A R .
Proof. By Proposition 2.6(1) there are only finitely many isomorphism classes of objects in A R and hence A C R . It follows that there exist maximal isomorphism classes of central objects. We now show that there is a unique such isomorphism class. To that end, suppose we are given two central objects (E, f ) and (V, g) in A R . By Proposition 3.10 the pair (E•V, σ(f, g)) ∈ (E, f ) ↓ A R and by Definition 3.13. Let R be a Noetherian unstable algebra, then the center (C, g) ∈ A R is a choice of isomorphism class of the maximal central object with respect to the poset structure on A R .
Example 3.14 (Mislin). The following example is due to Mislin [Mis92]. Let G = Σ 3 and let p be an odd prime, then the inclusion C 2 → Σ 3 of a 2-Sylow subgroup induces an isomorphism Note that Σ 3 actually has 3 conjugacy classes of elementary abelian subgroups of order 2, and that Σ 3 has trivial group theoretic center.
3.3. Hopf algebras and comodules. One of the key properties of H * G used by Kuhn is that for a central elementary abelian subgroup C, H * G is a H * C -comodule, and moreover the restriction map H * G → H * C is a morphism of H * C -comodules. A similar result occurs for general unstable algebras.
Proposition 3.15. Let R be a connected Noetherian unstable algebra with center (C, g), then R is a H * C -comodule, and g : Here, the H * C -comodule arises as the composite For the second part, we first recall that by Corollary 2.12 we have ρ R, We note that this is consistent with Proposition 3.3. Now to see that g : The left square commutes because R is a H C -comodule via Ψ R,(C,g) , while the right hand square commutes by naturality, and so the diagram is commutative. By the previous paragraph the top composite is g and the bottom is 1 ⊗ g. We deduce that g is a H * C -comodule morphism as claimed. Now suppose we are given (C, g) in A R , and we are given a non-trivial homomorphism α : (C, g) → (E, f ); in particular, there is a monomorphism α : C ֒→ E. As discussed previously, C -comodules, see the discussion (before passing to components) on the bottom of page 30 of [HLS95]. In particular, the following diagram commutes: This leads to the following result.
With the comodule structures as described above, Proof. By definition of the comodules structures, we must show that the diagram commutes. To see that the top and bottom square commute, we use Lemmas 2.11 and 2.14 and the definition of T α (f ) to see that there are isomorphisms Finally, the middle square commutes by the fact that T α (f ) is a morphism of H * C -comodules. Thus, the diagram commutes as claimed.
3.4. Central elements and the nilpotence degree. We offer the following improvement of Proposition 2.18 in the case M = R. In the case R = H * G for a compact Lie group G, this has also been shown by Kuhn (in fact, Kuhn shows more, see [Kuh07, Section 4.4]).
Proposition 3.17. Let R be a Noetherian unstable algebra with center (C, g), then for induced by the product of the maps η R,(E,f ) .
Proof. Given (E, f ) ∈ A R , we recall that we can form the object (C • E, σ(f, g)) ∈ (C, g) ↓ A R . Moreover, by Lemma 2.14, there is a commutative diagram This fits into a commutative diagram is an isomorphism by Proposition 3.10. By commutativity of the diagram,φ R is a monomorphism as required.
4. The topological nilpotence degree of the central essential ideal In this section we introduce the central essential ideal CEss(R) of a connected Noetherian algebra R, following the definition of Kuhn for compact Lie groups. We give an upper bound for d 0 (CEss(R)), and prove that CEss(R) is non-zero if and only if the depth of R is minimal, which is a version of Carlson's depth conjecture in this case.
4.1. The central essential ideal. We recall that in [Kuh13] Kuhn defines the central essential ideal for a compact Lie group G to be the kernel of the map where the product is taken over those elementary abelian p-subgroups of G strictly containing the maximal central subgroup C(G). The analog for a general unstable algebra R replaces H * G with R and H * CG(E) with components of the T -functor. Definition 4.1. Let R be a connected Noetherian unstable algebra with center (C, g) ∈ A R , then the central essential ideal CEss(R) is defined by Note that CEss(R) is independent of the choice of the isomorphism class of the center. Moreover, by replacing A R by a choice of skeleton if necessary, we can assume this product is finite (see Proposition 2.6).
Proof. This is a consequence of Lemma 3.16.
The main result of this section is the following. We refer the reader to Appendix B for a brief discussion on the basic commutative algebra needed in this section, in particular, for the definition of the depth and dimension of an R-module.
Theorem 4.3. Let R be a connected Noetherian unstable algebra with center (C, g) ∈ A R . Let c(R) be the rank of C, then the Krull dimension of the R-module CEss(R) is at most c(R).
The proof will require some preliminary results. We recall the following definitions, due to Henn [Hen96] and Powell [Pow07]. (1) (Henn) The T -support of M is The following result justifies the terminology of the R − U transcendence degree, see [Pow07, Proposition 7.2.6].
The proof relies on the existence of Brown-Gitler modules J R (n) in the category R − U (see [Hen96, Section 1.5]), which represent the functor We now present the proof of Proposition 4.5.
Proof. (Powell) Since Powell's work is not published, we sketch Powell's proof here. To that end, let R = R/ Ann R (M ), which is a Noetherian unstable algebra (note that the annihilator ideal is closed under the action of the Steenrod algebra) such that α : R → R is a morphism of unstable algebras, and let M ∈ R f g − U denote the object obtained by inducing M along the morphism α, (if the reader prefers a published reference, this is also easily deduced from the formulas on page 1756 of [NR10]).
by Proposition 2.6, which gives one inequality.
For the reverse inequality, we recall that the Dickson invariants are defined by D n = (H * (Z/2) n ) GLn(Z/2) for p = 2 and D n = (P n ) GLn(Z/p) for p > 2 where P n is the subalgebra of H * (Z/p) n generated by βH 1 (Z/p) n . As is well known, D n ∼ = F p [c 1 , . . . , c n ]. Then, for s another non-negative integer, one lets D n,s denote the subalgebra of D n whose elements are the p s -th powers of elements D n , which naturally obtains an action of the Steenrod algebra. Specifically, D n,s ∼ = F p [c p s 1 , . . . , c p s n ]. Suppose now that dim(R) = n, then by [BZ97, Appendix A] there exists a natural number s and a monomorphism of unstable algebras ι : D n,s → R for which R is a finitely-generated D n,s -module. We let ω ι denote the image of the top Dickson where each component is non-trivial, and (E i , f i ) ∈ A R , so that, in particular by Proposition 2.6(2), rank(E i ) ≤ n. Using exactness of localizations, there exists an i ∈ I for which We will need the following computation, which is an almost immediate consequence of [Hen96, Lemma 3.6]. The proof is given in [Hea20, Proposition 3.14].
Proposition 4.7. Let R be a Noetherian unstable algebra and M ∈ R f g − U, then Finally, we also need the following result, also due to Powell [Pow07, Proposition 7.3.1]. The proof is also given in [Hea20, Proposition 3.17].
With these preparations, we can now prove Theorem 4.3.
Proof of Theorem 4.3. If CEss(R) = 0 then the result is clear, thus we can assume that CEss(R) = 0. By Proposition 3.17 we can find n large enough so that We factor λ as a product λ = λ >c × λ ′ where By construction, CEss(R) is contained in the kernel of λ >c , and since λ is injective, we deduce that the restriction of λ ′ to CEss(R) ⊂ R is injective. We deduce that where the last inequality uses Proposition 4.7. By Proposition 4.5 we have TrDeg R (CEss(R)) = dim R (CEss(R)) ≤ c(R), as claimed.
We will prove the converse in Theorem 4.22.

Primitives and indecomposables.
Let R be a connected Noetherian unstable algebra with center (C, g). 3 We recall from Proposition 3.15 that g : R → H * C is a morphism of H * Ccomodules. In particular, the image K of g is a Hopf subalgebra of H * C (see the proof of [BH93a, Proposition 1.2]). As noted in [BH93a, Remark 1.3], it follows from the Borel structure theorem [MM65,Theorem 7.11] that there is a basis x 1 , . . . , x c for H 1 C such that (4.10) . , x c ) if p is odd, for some natural numbers j 1 ≥ j 2 ≥ · · · , and where y i = βx i for β the Bockstein homomorphism.
Definition 4.11. Let e(R) denote the maximum degree of a generator for H * C as a R-module, i.e., otherwise.
In order to proceed, we need one more definition, due to Kuhn [Kuh13, Definition 2.15].
Definition 4.12. A Duflot algebra of R is a subalgebra B ⊆ R that maps isomorphically to K = Im(R → H * C ). Since the image K is a free graded-commutative algebra over F p , such Duflot algebras always exist (as the natural epimorphism R → K always splits).
Given a Noetherian unstable algebra R, we fix a Duflot algebra B ⊆ R.
Definition 4.13. If M is a graded B-module, then the space of indecomposables is We let e indec (M ) be its largest nonzero degree, or −∞ if M = 0.
As shown in Lemma 4.2, CEss(R) is a sub H * C -comodule of R. Moreover, it is an unstable module, as it is the kernel of a morphism of unstable modules, and the comodule structure map is a morphism of unstable modules. Comodules with this additional structure are called unstable H * C -comodules in [Kuh13]. Definition 4.14. Let M be an unstable H * C -comodule, then the modules of primitives is We let e prim (CEss(R)) denote the supremum of the degrees in which P C (CEss(R)) is non-zero, with the convention that this is −∞ if CEss(R) = 0.
It is also a module over K = Im(R g − → H * C ), which we have seen is a sub-Hopf algebra of H * C . By assumption, the H * C -comodule structure and the R-module structure are compatible. Applying a lemma of Kuhn [Kuh07, Lemma 5.2] we deduce that gr i M is a free B-module, and the composite The filtration of M given by the M i is separated, and so the fact that each gr i M is a free B-module implies that M is also a free B-module.
Theorem 4.17. Let R be a connected Noetherian unstable algebra with center (C, g), and fix a Duflot algebra B of R.
(3) There is an exact sequence Proof. Everything in (1) and (2) except for the claim that CEss(R) is finitely-generated is a consequence of the previous lemma with M = CEss(R). Now, B has Krull dimension equal to the rank of C, namely c(R). Since we know CEss(R) is a free B-module, it suffices to check that the Krull dimension of CEss(R) is at most c(R), which is Theorem 4.3. For (3), consider the exact sequence Note that these are R-modules and H * C -comodules in a compatible way. We can apply Lemma 4.16 to the images and cokernels of these maps to deduce that they are free B-modules. It follows that the maps split, and we have an exact sequence as claimed.
Proof. In [Kuh13, Lemma 2.11] Kuhn proves that any unstable H * C -comodule with P C M finitedimensional has the property that d 0 M = e prim (M ) (under our conventions this is only true if M = 0). But Theorem 4.17(2) implies that P C CEss(R) is finite-dimensional if and only Q B CEss(R) is finite-dimensional (this is the same argument as [Kuh13, Corollary 2.19]), and that in this case e prim (CEss(R)) ≤ e indec (CEss(R)) < ∞. But it is clear that if CEss(R) is finite-dimensional, then so is Q B CEss(R), and this is a consequence of Theorem 4.17(1).

4.3.
Regularity and e indec (CEss(R)). We now give the following version of [Kuh13,Proposition 2.27]. This proposition is the first point of the paper we need to make some assumptions on the Duflot algebra.
This gives an exact sequence Because H * E is a finitely generated R-module via f so is the quotient ring Q B H * E . It follows from Lemma B.5 that r − c = depth(Q B H * E ) = depth R (Q B H * E ). In particular, by Lemma B.3 there exists elements y i ∈ m = R >0 such that Q B H * E is a finitely generated free module over the graded polynomial subring S ∼ = k[y 1 , . . . , y r−c ] ⊆ R.
Since S has dimension r −c > 0, we can find a non-zero element ℓ with positive degree which is a non-zero divisor on Q B H * E . It follows that the sequence f 1 , . . . , f c , ℓ ∈ R restricts to a regular sequence in H * E . By Proposition 3.4 there exists a h : Theorem 4.20. Let R be a connected Noetherian unstable algebra, and suppose that the Duflot algebra B is a polynomial algebra, then e indec (CEss(R)) ≤ e(R)+Reg(R), and hence if CEss(R) = 0, we have d 0 (CEss(R)) ≤ e(R) + Reg(R).
Proof. The first claim is equivalent to the statement that Q B CEss(R) is concentrated in degrees at most e(R) + Reg(R). By Proposition 4.19 it is equivalent to show that H 0 m (Q B R) is concentrated in degrees at most e(R) + Reg(R).
We now observe (see [Tot14,p. 137]   This stronger result gives the following, which is a version of Carlson's depth conjecture (originally conjectured for finite groups) is the case of a Noetherian unstable algebra of minimal depth.

Theorem 4.22. The central essential ideal CEss(R) is non-zero if and only if the depth of R is equal to the Duflot minimum c(R). Moreover, in this case CEss(R) is a Cohen-Macaulay R-module of dimension c(R).
Proof. The only if direction is Corollary 4.9, so we prove the converse. To this end, suppose that depth(R) = c(R), so that H c(R) m (R) = 0. By Remark 4.21 we have e indec (CEss(R)) ≥ 0, and hence CEss(R) = 0.
For the second claim, observe that we have by Corollary 4.9 and Theorem 4.17.

The topological nilpotence degree of a Noetherian unstable algebra
In this section we introduce the p-central defect of a Noetherian unstable algebra, which is the analog of p-centrality for finite groups. In the case where the p-central defect is 0 or 1, we give a very explicit description of d 0 (R) when R has a local cohomology theorem in the sense of Greenlees and Lyubeznik [GL00]. We finish by giving an upper bound for d 0 (R) when R is a Noetherian unstable algebra at the prime 2.
5.1. The p-central defect of a Noetherian unstable algebra.
Definition 5.1. Let R be a connected Noetherian unstable algebra with center (C, g). Let c(R) be the rank of C, and Remark 5.2. It is clear that c(R) ≤ p(R), thus the p-central defect of R is always greater than or equal to 0. In the case that c(R) = 0, then R ∼ = CEss(R). Moreover, we have The first inequality is Corollary B.7, the second always holds, and the final one is Proposition 2.6(2). Thus, if the p-central defect is 0, then depth(R) = dim(R) = c(R), so that R is a Cohen-Macaulay ring.
Remark 5.3. The Cohen-Macaulay defect of R is defined as dim(R) = p(R) − depth(R). By Duflot's depth theorem (Corollary B.7) depth(R) ≥ c(R), so that the Cohen-Macaulay defect is always less than or equal to the p-central defect of R.
Lemma 5.4. Let R be a connected Noetherian unstable algebra with center (C, g). If (C, g) (E, f ), then the p-central defect of T E (R; f ) is strictly less than that of R.
Proof. We first claim that p(T E (R; f )) ≤ p(R). Indeed, if (V,g) ∈ A TE (R;f ) , then we can precompose with ρ R,(E,f ) : R → T E (R; f ) to get a pair (V, g) ∈ A R for g =g • ρ R,(E,f ) . On the other hand, we recall there exists h : . Combining these two inequalities we see that hence the result.
For algebras of p-central defect 0, we have an immediate estimate for d 0 (R).
Proposition 5.5. Suppose that R = 0 has p-central defect 0, and that p = 2, or more generally the Duflot algebra of R is polynomial, then depth(R) = c(R) and d 0 (R) ≤ e(R) + Reg(R).
Proof. By Remark 5.2 we have depth(R) = c(R), and then the estimate for d 0 (R) is an immediate consequence of the fact that R ∼ = CEss(R), and Theorem 4.20.
We can give a more explicit result when R has a local cohomology theorem in the sense of [GL00]. We recall the definition here, where we write m = R >0 for the maximal ideal of the graded local ring R.
Definition 5.6 (Greenlees-Lyubeznik). Let R be a connected Noetherian unstable algebra. We say that R has a local cohomology theorem with shift ν if there is a spectral sequence Example 5.8. If R = H * G for a compact Lie group G and either p = 2 or the adjoint representation is orientable, then R admits a local cohomology theorem with shift − dim(G) [BG97a].
Theorem 5.9. Let R be a connected Noetherian unstable algebra whose Duflot algebra of R is polynomial. Suppose that R is of p-central defect 0 and has a local theorem with shift ν. Then Reg(R) = ν and d 0 (R) = e(R) + ν.
Proof. Because R ∼ = CEss(R), R is a Cohen-Macaulay ring (see Remark 5.2). Thus, the spectral sequence of (5.7) collapses, and shows that H c(R),−c(R)+k m ∼ = (I m ) ν−k . This is maximal when k = ν, and it follows that Reg(R) = a c(R) (R) + c(R) = ν. By Theorem 4.20 and Remark 4.21 we have d 0 (R) = e prim (R) ≤ e indec (R) = e(R) + ν. Thus, we must show that the topmost class of Q B R is primitive, where B denotes the Duflot algebra of R. Recall that B ∼ = F p [f 1 , . . . , f c ] by assumption and that R is a free B-module. In addition, the collapsing of the spectral sequence (5.7) shows that R is even a Gorenstein ring. By [MS05, Proposition I.1.4] Q B R is a Poincaré duality algebra of formal dimension e(R) + ν. This implies that the topmost class is a one dimensional primitive class, and hence that e prim (R) = e indec (R).
for p odd. For example, the unstable algebra B 1 is the cohomology of S 3 3 , the 3-connected cover of S 3 . The problem of realizing these unstable algebras as the cohomology of spaces has been investigated by Cooke [Coo79], and Aguadé, Broto, and Notbohm [ABN94,ABN97]. Because this unstable algebra is so simple, it is not hard to directly compute that d 0 (B i ) = 2p i + 1. Indeed, there is a short exact sequence of unstable modules . By Proposition 2.19 we then have ). It then suffices to show that d 0 (F p [x]) = 0, which follows, for example, from the injection F p [x] → H * Z/p given by obvious inclusion, and the calculation that d 0 (H * Z/p ) = 0. We now show how to deduce this from the previous theorem. Let f : B i → F p [x] → H * Z/p be the map given by projection onto the polynomial part, followed by the inclusion map. Then, (Z/p, f ) ∈ A Bi . Moreover, this pair is central, see [ABN94, Theorem 3.1(2)] when p is odd (where B i is denoted B i,1 ) and the proof of Theorem 13 of [ABN97] for p = 2. Because B i has Krull dimension 1, it is of p-central defect 0. It also clearly has a local cohomology theorem, as does any Gorenstein ring. The shift is easy to calculate as ν = (1 − 2p i ) + (1 + 2p i ) = 2.
Let s denote the generator H 1 Z/p and t = β(s) ∈ H 2 Z/p (so that t = s 2 if p = 2). Then the Duflot algebra is F 2 [s 2 i+1 ] when p = 2, and F p [t p i ] when p is odd. In particular, it is always polynomial, and e(B i ) = 2p i − 1 for all primes p. Thus, Theorem 5.9 applies to show that Definition 5.13. Let X be a connected H-space with Noetherian mod 2 cohomology, then the Poincaré dimension of H * (X) is s i=1 (|y i | 2 as −1 ). Proposition 5.14. Let X be a connected Noetherian H-space with cohomology as in (5.12).
(1) (Broto- (where the map f is denoted µ X ). That f is central is contained in the proof of Lemma 2.3 of [BC99]. Finally, H * (X) has p-central defect 0 because H * (X) has dimension r.
Theorem 5.15. Let p = 2, and suppose X is a connected H-space with Noetherian mod 2 cohomology. If X has Poincaré dimension d, then d 0 (H * (X)) = d.
Proof. Because H * (X) is Gorenstein it has a local cohomology theorem, with shift 4 given by Because H * (X) has 2-central defect 0, we can use Theorem 5.9 to see that From the description of the image in the previous proposition, we have Im(f ) ∼ = F p [u 2 β 1 1 , . . . , u 2 βr r ], and so  (1) If the polynomial generator is of degree 4, then the minimal possibility of a H-space with a four-dimensional polynomial generator is We calculated in Example 5.10 that d 0 (B 1 ) = 5, and indeed S 3 3 has Poincaré dimension 5.
(2) If the polynomial generator is of degree 8, then there are two possible minimal examples namely We note that X 1 and X 2 correspond to the 3-connected covers of the Lie groups G 2 and SU (3) respectively. By the previous theorem, we see that d 0 (H * (X 1 )) = 20 and d 0 (H * (X 2 )) = 14.

5.2.
Unstable algebras of p-central defect 1. Suppose now that R has p-central defect 1.
In this case, we have the following.
Theorem 5.17. Suppose R is a Noetherian unstable algebra with center (C, g) and p-central Proof. The short exact sequence and Proposition 2.19 show that However, d 0 (T E (R)) = d 0 (R) (Proposition 2.19), and because T E (R; f ) is a summand of T E R, we see that d 0 (T E (R; f )) ≤ d 0 (R). Thus, the previous inequality is actually an equality. By Lemma 5.4 each T E (R; f ) with (E, f ) ∼ = (C, g) has p-central defect 0 and hence Because R has p-central defect 1, the spectral sequence (5.7) only has two non-zero columns and collapses to give a short exact sequence The result then follows because R ∼ = T C (R; g).
Remark 5.18. The previous result can be made slightly more precise. There are two possibilities: either R is Cohen-Macaulay, or the depth and the dimension differ by 1 (so that R is almost Cohen-Macaulay). By Theorem 4.22 CEss(R) = 0 if and only if R is almost Cohen-Macaulay. Thus, if R is almost Cohen-Macaulay, the proof of the previous theorem in fact shows that

Examples.
In this section, we give some examples coming from the cohomology of groups, or homotopical versions of groups, in particular for p-compact groups (see [DW94]) and p-local finite groups (see [BLO03b]). We defer, however, a thorough discussion of homotopical groups for Section 6.2. For the following, we recall that a discrete group is a duality group of dimension d over It is called a Poincaré duality group if I ∼ = F p with some G-action, and it is orientable if the action is trivial. A virtual duality group of virtual dimension d (which we write vdim(G)) is then a group G with a normal subgroup N of finite index which is a duality group of dimension d.
The result then follows by combining Theorems 5.9 and 5.17 once we explain why we can assume the Duflot algebra is polynomial in Cases (i) to (iii). This follows the same argument as in [Kuh07] (see the discussion after Theorem 7.4); G will admit a decomposition G = C 0 × G 1 where C 0 is an elementary abelian p-group, and G 1 has no Z/p-summands. The Duflot algebra then decomposes as B = H * C0 ⊗B 1 where B 1 is a polynomial Duflot subalgebra of H * G1 . Moreover, Q B H * G = Q B1 H * G1 and e(H * G ) = e(H * G1 ). See also the discussion in the proof of Theorem 6.8.
One can also give an estimate for d 0 (H * G ) when H * G has p-central defect 1 using Theorem 5.17, at least in cases (i),(iv), and (v). In cases (ii) and (iii) one would also need to know that the centralizer of an elementary abelian p-group is still an orientable virtual Poincaré duality group or an orientable p-adic Lie group.
Remark 5.20. In Remark 6.7 we will see that in Cases (i)-(iii) that if G is p-central, in the sense that the maximal central elementary abelian p-subgroup of G has rank equal to the p-rank of G, then H * G is of p-central defect 0. Thus, the above theorem is a direct generalization of Kuhn's result [Kuh07, Theorem 2.9] that if G is a p-central finite group, then d 0 (H * G ) = e(H * G ). In Cases (4) and (5) there is also a natural notation of 2-centrality, and in this case G is 2-central if and only if H * G has 2-central defect 0, see Lemma 6.22. Remark 5.21. In cases (i),(ii), and (v) of Theorem 5.19 the regularity of the mod p cohomology has been computed in general, and so this result also follows from Theorem 5.24 below. The strength of the above theorem is that in the case of algebras of p-central defect 0 or p-central defect 1, we can compute the regularity in the presence of a local cohomology theorem, even if we do not know it in general. For example, the regularity of the mod p cohomology of profinite groups or p-compact groups is currently unknown.
5.4. The topological nilpotence degree of an unstable algebra. In this section we prove our main result (Theorem 5.24), which gives an estimate for d 0 (R) for an unstable algebra at the prime 2. We need the following.
Proposition 5.22. For any connected Noetherian unstable algebra R we have Proof. Suppose that R has p-central defect d. The proof will be by induction on d. If d = 0, then the statement of the proposition is clear (in fact, in this case the inequality is even an equality). Inductively, we assume that the proposition holds for all connected Noetherian unstable algebras of p-central defect 0 ≤ k < d. Choose a pair (E, f ) with (C, g) (E, f ), and let (C E ,g E ) denote the center of T E (R; f ). By Proposition 3.4 there exists h : T E (R; f ) → H * E such that (E, h) is central in A TE (R;f ) and the following diagram commutes: By centrality, we have (E, h) ⊆ (C E ,g E ), and hence (by composing with ρ R,(E,f ) ) we have (C, g) (E, f ) ⊆ (C E , g E ), where g E = ρ R,(E,f ) •g E . By Lemma 5.4 the p-central defect of T E (R; f ) is less than that of R, and in particular, the inductive hypothesis applies to show that Let j = ρ R,(E,f ) •j, then by Lemma 5.25 below we have an isomorphism of unstable algebras From the definition of the central essential ideal and Proposition 2.19, we have Combining the previous two equations and observing that (C, g) (C E , g E ) gives the desired result.
Theorem 5.24. Let R be a connected Noetherian unstable algebra with center (C, g), and suppose p = 2, then Proof. Combine Theorems 4.20 and 4.22 and Proposition 5.22.
We still owe the reader the proof of the following. This is a T -functor version of the observation that if G is a group and E and V are elementary abelian p-subgroups of G, with Z(C G (E)) < V < C G (E), then C CG(E) (V ) ∼ = C G (V ), where Z(−) denotes the maximal central elementary abelian p-subgroup of a group.
Lemma 5.25. Let R be a Noetherian unstable algebra with center (C, g). Choose (C, g) ⊆ (E, f ) ∈ A R , and let (C E ,g E ) be the center of the Noetherian unstable algebra Remark 5.26. We follow the notation of Proposition 5.22, and let h : , and so (E, f ) (V, j). We let ι : E → V denote the corresponding morphism of elementary abelian p-groups. Finally, we let m : T V (R; j) → H * V be the central map factoring j : R → H * V . In summary, the situation of the lemma is displayed in the following diagram: Proof. We first observe that by [DW92, Proposition 4.5] (which applies because (E, h) ∈ A TE (R;f ) is central) there is a unique pair (E ⊕ V,j ⊕ h) ∈ V TE (R;f ) (i.e., a mapj ⊕ h : T E (R; f ) → H * E⊕V ) whose composition with the projection maps to H * E and H * V gives h andj, respectively. We use this observation to show that the diagram commutes, where µ : E ⊕ V → V is the map sending (e, v) to ev. Indeed, from the observation above it suffices to show that both composites gives h andj after composition with the relevant projection maps. To that end, we have isomorphisms Only the last third and fourth displayed equations require comment; the third uses that κ R,(E,f ) makes T E (R; f ) into a H * E -comodule so that (1 ⊗ ǫ E ) • κ R,(E,f ) ∼ = id, while the fourth follows because h is central in A TE (R;f ) (compare the proof of Proposition 3.4 and Proposition 3.3).
To finish the lemma, we apply [DW92, Proposition 3.3] which implies that where the second last step uses the commutative diagram above. Since µ is surjective we have by [HLS95, Lemma 4.8]. Combining the previous two isomorphisms gives the result.

Examples
We finish with examples from group theory, and homotopical group theory, giving results analogous to Kuhn's in the case of compact Lie groups.
6.1. Group theory. We now focus on unstable algebras of the form R = H * G where G is a group. In this case, Rector's category will take a particularly nice form. We will need the following definition.
Definition 6.1. The Quillen category associated to a group G at the prime p is the category A G with objects elementary abelian p-subgroups E ≤ G and with morphisms E → V those monomorphisms induced by conjugation in G.
While most of the groups we study should be familiar to the reader, we first explain the class of groups considered by Broto and Kitchloo [BK02].
Definition 6.2 (Broto-Kitchloo). Let X be a class of compactly generated Hausdorff topological groups, and let K 1 (X ) be a new class of groups, such that a compactly generated Hausdorff topological group G belongs to K 1 (X ) if and only if there exists a finite G-CW complex X with the following two properties: (1) The isotropy subgroups of X belong to the class X .
(2) For every finite p-subgroup π < G, the fixed point space X π is p-acyclic.
If X is the class of compact Lie groups, then Kac-Moody groups are an example of a group in K 1 (X ), see [BK02,Section 5].
With this we get the following, which is a compendium of results of Quillen [Qui71], Rector [Rec84], Lannes [Lan86,Lan92] Henn [Hen98a] and Broto-Kitchloo [BK02], see [Hea20, Theorem 4.1 and Theorem 4.8] for the precise details. Theorem 6.3. Assume we are in one of the following cases: (1) G is a compact Lie group.
(2) G is a discrete group for which there exists a mod p acyclic G-CW complex with finitely many G-cells and finite isotropy groups. (3) G is a profinite group such that the continuous mod p cohomology H * G is finitely generated as an F p -algebra. (4) G is a group of finite virtual cohomological dimension such that H * G is finite generated as an F p -algebra.
Definition 6.4 (Mislin [Mis92]). An elementary abelian subgroup E < G is said to be cohomo- Under the equivalence of categories A G ≃ A H * C G (E) , these are precisely the central elements as considered throughout this paper. We use the terminology cohomological p-central so as to not conflict with the usual group theoretic notion of central elementary abelian p-subgroup. The two are related in the following way, where we let C p (G) denote the maximal cohomologically p-central subgroup of G (which is only unique up to conjugacy, see Theorem 3.12), and Z(G)[p] the maximal central elementary abelian p-subgroup in the usual sense. Proof. The first claim is clear because in this case C G (E) ∼ = G. The injective homomorphism φ is constructed exactly as by Mislin [Mis92]. We recall his argument now. Let x ∈ Z(G)[p] be represented by a mapφ(x) : Z/p → G, and write f for the induced map f : We then set φ(x) = f . This is clearly injective, because if φ(x) = φ(y), then x and y are conjugate in G, and hence equal, as they are central.
Remark 6.6. If G is a finite p-group, then the main result of [Mis92] implies that Z(G)[p] ∼ = C p (G), however in general φ is not surjective. A counterexample is given by the group Σ 3 at p = 2, as in Example 3.14. This means that the definition of CEss(H * G ) does not necessarily agree with Kuhn's definition of CEss(G). For example, we have CEss(H * Σ3 ) ∼ = H * Σ3 (i.e., CEss(Σ 3 ) has pcentral defect 0), while CEss(Σ 3 ) is trivial, as it is the kernel of the restriction map H * Σ3 → H * C2 . Of course, in any case one gets the same result, namely that d 0 (H * Σ3 ) = 0.
Remark 6.7. In Parts (1)-(3) of Theorem 5.19 we determined d 0 (H * G ) for certain classes of groups in the case that they have p-central defect 0. This is the unstable algebra analog of being pcentral, i.e., that the maximal central elementary abelian p-group is equal to the p-rank of G (equivalently, the Krull dimension of H * G ). To be more specific, the previous lemma implies that if G is p-central, then it has p-central defect 0, and in this case Z(G)[p] ∼ = C(G). On the other hand, the usual counter-example of Σ 3 at p = 2 has 2-central defect 0, but is not 2-central, as it has trivial center, but p-rank 1.
Theorem 6.8. Let G be one of the groups considered in Theorem 6.3, then for any prime p we have where c(C G (E)) is the rank of the maximal central cohomologically p-central subgroup of G. Moreover, if G is a compact Lie group, then Reg(H * CG(E) ) ≤ − dim(C G (E)), with equality if π 0 (C G (E)) is a finite p-group.
Proof. This will be a consequence of Theorems 5.24 and 6.3, but we first explain why we are able to prove this for any prime p, and not just p = 2, using an observation of Nick Kuhn. 6 The point is that for a group we can always assume that the Duflot algebra is polynomial (this has already been observed by Kuhn in the case of compact Lie groups, see [Kuh13,Page 160]). Indeed, since the action of G on F p is trivial H 1 G ∼ = Hom Z (G, Z/p) (these homomorphisms need be continuous in the case G is a profinite group). In particular, elements in the image of res * G,Cp(G) : are exactly homomorphisms from C p (G) → Z/p that factor through G. Recall that the image of . , x c ) Using the observation above it is not hard to see that c − b is the rank of the largest subgroup of C splitting off G as a direct summand (compare the discussion on page 158 of [Kuh13]). Write and similar for e prim and e indec . Thus, we can assume reduce to the case of the group L, which necessarily has polynomial Duflot algebra. Thus, in this case Theorem 5.24 is valid for all primes p.
Finally, the regularity statement is due to Symonds. [Sym10].
to obtain a result that is analogous to that obtained by Kuhn in the case of compact Lie groups.
Example 6.11. Consider the profinite group GL 2 (Z 3 ). This admits a splitting GL 2 ( ), we observe that this has a single elementary abelian subgroup Z/3 whose centralizer in SL 2 (Z 3 ) is isomorphic to Z/2 × Z/3 × Z 3 , so that H * , with |y| = 2 and |x| = |e| = 1, see the discussion after Proposition 5.5 (as well as Theorem 5.2) of [Hen98a]. It follows that H * SL2(Z3) has trivial 3-cohomological center, and hence that CEss(H * SL2(Z3) ) is the kernel of the restriction map By [Hen98a,Proposition 5.6] we deduce that CEss(H * SL2(Z3) ) is trivial. Since we work at p = 3, we have H * . This has depth 1, and c(Z/3 × Z 3 ) = 1. Moreover, it is of 3-central defect 0, so that Putting these observations together, we conclude that Example 6.12. Consider the 2nd Morava stabilizer group S 2 at the prime 3. This admits a decomposition S 2 ∼ = S 1 2 × Z 3 and so The group S 1 2 has two conjugacy classes of elementary abelian 3-subgroups E i for i = 1, 2 with C S 1 2 (E i ) ∼ = Z/3 × Z 3 in both cases. We note that both S 1 2 and C S 1 2 (E i ) are 3-adic Lie groups. We also observe that H * S 1 2 has trivial 3-cohomological center, and hence CEss(H * S 1 2 ) is the kernel of the product of restriction maps By [Hen98a,Proposition 4.3] we deduce that CEss(H * ) and so We deduce that d 0 (H * S2 ) = 2. Remark 6.13. One could also use Theorem 5.17 and Remark 5.18 to compute d 0 (H * S 1 2 ). Indeed, this has 3-central defect 1 and is an almost Cohen-Macaulay ring and we find that 2 ) = 0 + 1 = 1. 6.2. Homotopical groups. We now move onto the case of homotopical groups, namely the p-local finite and compact groups of Broto, Levi, and Oliver [BLO03b,BLO07]. Once we have set up the right language, the results take essentially the same form as for ordinary groups. The canonical references for both p-local finite and compact groups are the aforementioned papers of Broto, Levi, and Oliver, however the reader may also find the survey article [BLO04] valuable.
To begin, we recall the definition of the fusion system F p (G) associated to a finite group G. This is a category whose objects are the p-subgroups of G, and where Hom Fp(G) = Hom G (P, Q) := {α ∈ Hom(P, Q) | α = c x , for some x ∈ G}.
i.e., α is a homomorphism induced by conjugation in G. To this one can associate another category, the centric linking system L c p (G). Then, by [BLO03a, Proposition 1.1] there is a homotopy equivalence |L c p (G)| ∧ p ≃ BG ∧ p . The idea of p-local finite groups is to begin with a finite p-group S, and try and mimic the constructions above. Thus, a fusion system F associated to S is a category whose objects are subgroups of S, and whose morphism sets Hom F (P, Q) satisfy the following conditions: (1) Hom S (P, Q) ⊆ Hom F (P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.
(2) Every morphism in F factors as an isomorphism in F followed by an inclusion. This is not quite enough; Broto, Levi, and Oliver additionally require that the fusion system is saturated, see [BLO03b, Definition 1.2]. A centric linking system L associated to F is another category whose objects are a certain subset of S. The centric linking system contains the additional data to associate a classifying space to the fusion system F .
A p-local finite group is a triple G = (S, F , L) where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F . The classifying space of G is defined as BG = |L| ∧ p , the p-completed nerve of the category L. We write H * G := H * (BG) for the mod p cohomology of G.
If instead of a finite p-group we begin with a discrete p-toral group S -that is a group that contains a normal subgroup T ∼ = (Z/p ∞ ) r , and such that T has finite index in S -then we can define saturated fusion systems F over S, and centric linking systems over F , see [BLO07]. A p-local compact group is a triple G = (S, F , L) where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a centric linking system associated to F . In fact, it was later shown that every saturated fusion system over a discrete p-toral group has an associated centric linking system which is unique up to isomorphism [Che13,LL15]. Thus, we often define our p-local compact groups as simply a pair G = (S, F ).
Example 6.14. Here we list some examples of p-local compact groups.
(1) If G is a compact Lie group, with p-toral subgroup S, then there exists a p-local compact group G = (S, F S (G)) along with an equivalence of classifying spaces BG ∧ p ≃ BG [BLO07, Theorem 9.10].
(2) Suppose that X is a p compact group, that is a triple (X, BX, e) where X is a space with H * (X; F p ) finite, BX is a pointed p-complete space, and e : X → Ω(BX) is an equivalence [DW94]. There is a notion of a Sylow subgroup f : S → X, and moreover, there exists a p-local compact group G = (S, F S,f (X)) with BG ≃ BX [BLO07, Theorem 10.7]. More generally, the p-completion of any finite loop space gives rise to a p-local compact group [BLO14].
Remark 6.15. Because, up to p-completion, every compact Lie group can be modeled by a p-local compact group, this section recovers the result for compact Lie groups in the previous section. This follows because the classifying space of a compact Lie group is always p-good (see [BK72, Proposition VII.5.1]) and so H * . We now identify Rector's category A H * G and present the relevant T -functor calculations. We let F e be the full subcategory of F whose objects are elementary abelian p-subgroups E ≤ S which are fully-centralized in F in the sense of [BLO07, Definition 2.2]. This assumption ensures that the centralizer p-compact group C G (E) = (C S (E), C F (E)) exists [Gon16, Section 1.2], where C F (E) is the fusion system over C S (E) with objects Q ≤ C S (E) and morphisms Moreover, we note that any elementary abelian p-subgroup E ≤ S is isomorphic in F to one that is fully F -centralized.
For the following, we note that there is a canonical map θ : BS → BG.
which is the restriction of the homomorphism φ E . Definition 6.18. Let G = (S, F ) be a p-local compact group, then E ∈ F e is called central if Map(BE, BG) Bι → BG is a homotopy equivalence.
This does not conflict with the notion of centrality used previously in this paper, by the following lemma.
Proof. If E ∈ F e is central then this is clear from the discussion before the definition of centrality. For the converse, suppose that ρ is an equivalence. Because the classifying space of a p-local compact group is p-good (combine [BLO07,Proposition 4.4] and [BK72, Proposition I.5.2]), the map Map(BE, BG) Bι → BG is a homotopy equivalence.
Remark 6.20. By [BLO07,Theorem 7.4] if E ∈ F e is central, then the p-local compact groups G and C G (E) are isomorphic in the sense discussed on [BLO07,. In particular, there are isomorphisms of groups and categories α : S → C S (E) and α F : F → C F (E) which are compatible in a certain sense.
Note that we have a natural definition of p-centrality for a p-local compact group. For the following, we let C(G) denote the maximal central elementary abelian p-subgroup E ∈ F e , which exists by Theorem 3.12 and the previous lemma.
The central essential ideal is defined in the obvious way, namely as the kernel One deduces, as in Theorem 4.3 that the Krull dimension of CEss(G, X) is at most the rank of C. The proof is essentially the same. One just needs to observe that it suffices to take the product over C ≤ E ≤ G in (A.1), which follows from a diagram chase and the observation that C C•E (G) ∼ = C E (G) and X C•E ≃ X E , the latter because C acts trivially on X. Alternatively, one can use the methods of [Kuh07, Section 4.4] to prove an even stronger result (in particular, apply [Kuh07, Lemma 4.5 ] with F (E 1 → E 2 ) = H * E2 and F (E 1 → E 2 ) = H * CG(E2) (X E2 ) to deduce that the analog of [Kuh07, Theorem 4.4] holds).
The H * G -module CEss(G, X) is also naturally a H * C -comodule, and as in Section 4.1 we can define e prim (CEss(G, X)) to be the spectrum of degrees in which the module of primitives P C (CEss(G, X)) is non-zero. Similarly, we can define e indec (CEss(G, X)) to be the largest nonzero degree of the space of indecomposables Q B CEss(G, X), where B is a Duflot algebra.
Proof. The observation on the Krull dimension of CEss(G, X) and Lemma 4.16 show that the analog of Theorem 4.17 holds. Thus, whenever CEss(G, X) = 0, we have d 0 (CEss(G, X)) = e prim (CEss(G, X)) and e prim (CEss(G, X)) ≤ e indec (CEss(G, X)) < ∞. Once we show that Proposition 4.19 holds, i.e., that Q B CEss(G, X) = H 0 m (Q B CEss(G, X)) = H 0 m (Q B H * G (X)). then the same argument as given Theorem 4.20 proves that d 0 (CEss(G, X)) ≤ e(G, X) + Reg(H * G (X)). We note that here m is the maximal ideal of H * G , i.e., the ideal generated by elements of positive degree. The first equality is clear, and the second holds if for each E with C E, we have H 0 m Q B H * CG(E) (X E ) = 0. This is shown by the identical argument as in Proposition 4.19, where one replaces Theorem B.6 with [Hea20, Theorem 4.4].
Along with Theorem A.3 this completes the proof of Theorem A.2.
Appendix B. Depth, dimension, and regularity B.1. Depth and dimension. We begin by considering the depth and dimension of gradedcommutative connected Noetherian k-algebras for k a field. Given such a k-algebra we write R j for the degree j part of R. Hence, R connected means that R 0 ∼ = F p and R i = 0 for i < 0. We let m = R >0 denote the maximal homogeneous ideal of R. With these assumptions, the commutative algebra of R is much like that of a local ring.  where M is an R-module by restriction of scalars. In particular, depth R (R ′ ) = depth(R ′ ).
Finally, we will need the following version of a group theoretic theorem of Carlson [CTVEZ03], which is proved by the author in [Hea20, Theorem 3.5].
Theorem B.6. Let R be a Noetherian unstable algebra, and suppose (E, f ) ∈ A R is central. If x 1 , . . . , x n is a sequence of homogeneous elements in R such that the restrictions of x 1 , . . . , x n form a regular sequence in H * E , then x 1 , . . . , x n is a regular sequence in R.
An easy consequence is the following, see [Hea20, Corollary 3.6], which was originally proved in the case R = H * G for G a finite group by Duflot [Duf81].
Corollary B.7. Let R be a Noetherian unstable algebra with center (C, g), then depth(R) ≥ rank(C).