Local Gorenstein duality for cochains on spaces

We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the absolute notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of $k$-algebras. Our main examples are of the form $R = C^*(X;k)$, the ring spectrum of cochains on a space $X$ for a field $k$. In particular, we establish local Gorenstein duality in characteristic $p$ for $p$-compact groups and $p$-local finite groups as well as for $k = \mathbb{Q}$ and $X$ a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.


Introduction
Given a Noetherian commutative local ring (A, m, k), there are numerous equivalent conditions for when A is Gorenstein. In particular, if A has Krull dimension n, then A is Gorenstein if and only if Ext i A (k, A) ∼ = k i = n 0 otherwise.
In the derived category D(A), this can be restated in terms of the derived hom as an equivalence RHom A (k, A) ≃ Σ n k. Inspired by this, Dwyer, Greenlees, and Iyengar [DGI06] introduced the notion of a Gorenstein ring spectrum. More specially, if k is a field and R a commutative ring spectrum, a morphism R → k of ring spectra 1 (always assumed to be commutative) is said to be Gorenstein of shift r if there is an equivalence Hom R (k, R) ≃ Σ r k for some integer r. For example, if R = C * (BG; F p ) is the ring of F p -valued cochains on the classifying space of a finite group G, then C * (BG; F p ) → F p is always Gorenstein of shift 0, even though the cohomology ring π − * C * (BG; F p ) ∼ = H * (BG; F p ) is not Gorenstein in general [DGI06, Section 10.3].
One is particularly interested in the duality that the Gorenstein condition implies. For example, let X be a suitable space and R = C * (X; k), the ring spectrum of mod p cochains. Then, if R → k is Gorenstein then R automatically satisfies the additional property that Cell k (R) ≃ Σ r Hom k (R, k), where Cell k is the k-cellular approximation to R, see Section 2.2. As we shall see, this is the analog of the classical characterization of Gorenstein rings as those Date: January 9, 2020. 1 If k is a field, we will also denote by k the Eilenberg-MacLane spectrum Hk. where I m ∼ = Hom k (A, k) denotes the injective hull of k. Whenever the equivalence Cell k (R) ≃ Σ r Hom k (R, k) is satisfied for a morphism R → k of ring spectra, we say that R satisfies Gorenstein duality. Note however that, in contrast to the algebraic situation, R → k being Gorenstein does not imply that R satisfies Gorenstein duality, see Remark 2.11. The structural implications for π * R when R satisfies Gorenstein duality have been investigated previously by Greenlees and Lyubeznik [GL00]. For example, when R = C * (BG; F p ), one deduces that H * (BG; F p ) is Cohen-Macaulay if and only if H * (BG; F p ) is Gorenstein, a result originally shown by Benson and Carlson [BC94]. If A is a commutative local Gorenstein ring, then the localization A p at any prime ideal p ∈ Spec(A) is still Gorenstein. One way to see this is to use yet another interpretation of Gorenstein rings as those commutative local rings with finite injective dimension as an A-module. Alternatively, we observe that if p has dimension d, then the ring A p is local Noetherian of dimension n − d, and Greenlees-Lyubeznik's dual localization [GL00, Section 2] can be used to show that where I p is the injective hull of A/p, 2 and hence A p is still Gorenstein. Now suppose that R → k is a morphism of ring spectra. As we will explain below, we can form spectral versions of localization at p, local cohomology, and injective hulls. We thus say that R → k satisfies local Gorenstein duality if the spectral version of ( * ) holds. For example, the main result of [BG08] is that C * (BG; F p ) → F p satisfies local Gorenstein duality when G is a finite group, or more generally for certain compact Lie groups. We note that a fundamental difference between the algebraic and topological situations is that in topology we do not know in general that Gorenstein duality implies local Gorenstein duality. The main objective for this work is to identify conditions where local Gorenstein duality holds.
The main techniques to determine whether a ring spectrum R → k is Gorenstein are the Gorenstein ascent and descent theorems of Dwyer-Greenlees-Iyengar, see [Gre18,Section 19] for a summary. In [BHV18b] we proved that, given a finite morphism f : S → R over k, if S satisfies local Gorenstein duality and the relative dualizing complex ω f = Hom S (R, S) is equivalent to a suspension of R (i.e., that f is relatively Gorenstein), then R → k satisfies local Gorenstein duality as well. Inspired by the Gorenstein ascent of Dwyer-Greenlees-Iyengar, we prove the following, which allows us to identify the relative dualizing complex, and hence prove ascent for local Gorenstein duality.
Theorem (Proposition 3.4). Let S f − → R be a finite morphism of augmented k-algebras and Q = R ⊗ S k. Assume that the following conditions are satisfied: (1) R → k and S → k are orientable Gorenstein (Definition 2.4) of shift r and s respectively.
(2) Q → k is cosmall, i.e., Q is in the thick subcategory in Mod Q generated by k.
(3) R and S are dc-complete (Section 2.2). Then, if S satisfies local Gorenstein duality of shift s, then R satisfies local Gorenstein duality of shift r.
Here the technical conditions of orientability and dc-completeness are satisfied automatically for ring spectra such as C * (X; F p ) when X is a suitably nice space.
Local Gorenstein duality. We now briefly explain how to generalize the local Gorenstein condition ( * ) to a ring spectrum R. More details are given in Section 2.3. To begin, we recall that given a commutative ring spectrum R and a homogeneous prime ideal p ∈ Spec h (π * R), one can form a version of the localization at p to produce a new spectrum R p with π * (R p ) ∼ = (π * R) p . Moreover, there is a functor Γ V(p) such that the homotopy π * (Γ V(p) R p ) is the target of a spectral sequence whose E 2 -term is H s,t p ((π * R) p ). We hence take Γ p R = Γ V(p) (R p ) as our spectral analog of the left hand side of ( * ). For the right hand side of ( * ) we recall that given any injective π * Rmodule I, one can form an R-module spectrum T R (I), such that π * T R (I) ∼ = I, see Section 2.3. We are then led to the following definition.
The consequences of local Gorenstein duality for the ring π * R are reviewed in Theorem 2.15 below. For example, it implies that the ring π * R is generically Gorenstein, i.e., the localization of π * R at any minimal prime ideal is Gorenstein.

Examples.
Our main examples come from ring spectra of the form C * (X; k). As already noted, for reasonable spaces X the technical conditions of orientability and dc-completeness are automatically satisfied. In particular, these conditions are satisfied for spaces of Eilenberg-Moore type (EM-type), see Definition 3.5. The previous proposition specializes to the following.
Theorem (Theorem 3.12). Let g : Y → X be a morphism of spaces of EM-type (p-complete if the characteristic of k is p) with fiber F . Suppose that H * (F ; k) is finite-dimensional, and that C * (X; k) is Gorenstein of shift s.
(2) If, in addition, C * (X; k) satisfies local Gorenstein duality of shift s, then C * (Y ; k) satisfies local Gorenstein duality of shift r.
Let G be a compact Lie group such that (if p > 2) the adjoint representation of G is orientable, and consider a unitary embedding f : G → U (n). Taking g = Bf ∧ p in the above theorem, one recovers the result of Benson and Greenlees that C * (BG; F p ) → F p satisfies local Gorenstein duality of shift dim(G) when G is a compact Lie group (orientable if p > 2). The same result (with no orientability hypothesis) holds if Y is the classifying space of a p-compact group G, that is, C * (BG, F p ) → F p satisfies local Gorenstein duality of shift dim p (G), where dim p (X) denotes the F p -cohomological dimension of a space X.
It is not true in general that if G is a compact Lie group, then C * (BG; F p ) satisfies local Gorenstein duality of shift dim(G). A simple example is given by G = O(2) at an odd prime, which satisfies local Gorenstein duality of shift 3, while dim(G) = 1. However, the triple More generally, we say that a compact Lie group is of p-compact type if the triple (Ω(BG ∧ p ), BG ∧ p , id) is a p-compact group (this is true if and only if Ω(BG ∧ p ) is F p -finite). In this case, C * (BG; F p ) → F p will satisfy Gorenstein duality of shift dim p (Ω(BG ∧ p )) by the previous result on p-compact groups. Taking G = O(2) at an odd prime as above, we have dim p (Ω(BO(2) ∧ p )) = 3, as expected. A common generalization of both p-compact groups and compact Lie groups is the p-local compact groups of Broto, Levi, and Oliver [BLO07]. Given such a p-local compact group G, there is an associated classifying space BG, and one can ask if C * (BG; F p ) → F p satisfies local Gorenstein duality, or even if it is Gorenstein. We do not know the answer to this question in full generality, however we identify conditions for this to occur in Section 5. In the case of p-local finite groups [BLO03] we deduce from work of Cantarero, Castellana, and Morales [CCM19], that C * (BG; F p ) → F p satisfies local Gorenstein duality of shift 0.
In summary, we obtain the following results.
Theorem (Theorems 4.12 and 4.16 and Corollary 5.12). Let G be a p-local compact group of one of the following types: (1) associated to a Lie group of p-compact type, (1) and (2)) or 0 (Case (3)), respectively.
In the rational case, a stronger result holds. Using the fact that algebraic Noether normalization can be lifted to the spectrum level, we show that for any simple connected rational space with Noetherian cohomology, Gorenstein implies local Gorenstein duality of the same shift.
Theorem (Theorem 4.18). Let X be a simply connected rational space with Noetherian cohomology. If C * (X; Q) → Q is Gorenstein of shift r, then C * (X; Q) satisfies local Gorenstein duality of shift r.
Convention. Throughout this document, all rings and structured ring spectra will be assumed to be commutative.

Gorenstein ring spectra
In this section we first review the notions of Gorenstein ring spectrum, relative Gorenstein morphism, and Gorenstein duality, as studied by Dwyer, Greenlees, and Iyengar [DGI06]. Then, we prove a result for recognizing when certain morphisms of ring spectra are relative Gorenstein, and use this in the next section to prove an ascent theorem for local Gorenstein duality.
2.1. The Gorenstein condition. Suppose A is a (discrete) Noetherian commutative local ring with residue field k, then it is a theorem of Serre that A is regular if and only if k has a resolution of finite length by free A-modules. For the associated map of Eilenberg-MacLane spectra HA → k, this implies that k is in the thick subcategory of Mod HA generated by HA itself. This leads more generally to the definition of a regular morphism of ring spectra, where we say that a morphism of ring spectra R → k is regular if k is a compact R-module, i.e., k is in the thick subcategory of Mod R generated by R. However, a weaker notion of regularity is often useful.
Definition 2.1. A morphism of ring spectra R → k is called proxy-regular if there exists another R-module K called a Koszul complex, such that K is a compact R-module, K is in the thick subcategory of Mod R generated by k, and k is in the localizing subcategory generated by K in Mod R . If K = R itself, then we say that R → k is cosmall. If K = k then we say that R → k is small.
Returning to the commutative algebra example, if A is a commutative local Noetherian ring as above, then HA → k is always proxy-regular, where we can take K to correspond to the usual Koszul complex [DGI06, Section 5.1] associated to a sequence of generators of the maximal ideal of A.
We recall that a commutative Noetherian local ring A with residue field k is Gorenstein if and only if Ext * A (k, A) is a one-dimensional k-vector space. This leads to the following definition, which is actually a special case of a more general definition due to Dwyer-Greenlees-Iyengar, see [DGI06,Proposition 8.4].
Definition 2.2. We say that a morphism R → k of ring spectra is Gorenstein of shift a if it is proxy-regular and there is an equivalence Hom R (k, R) ≃ Σ a k of k-modules. More generally, we say that a morphism of ring spectra S → R is relatively Gorenstein of shift a if Hom S (R, S) ≃ Σ a R.
2.2. Gorenstein duality. One important observation of Dwyer-Greenlees-Iyengar is that since Hom R (k, R) has an action by E = Hom R (k, k), if R → k is Gorenstein, then k admits the structure of a right E-module. Note that if R admits the structure of a k-algebra, and R → k is Gorenstein, we have where the last equivalence follows by adjunction.
Restricting to the case of augmented k-algebras, this leads to the following definition. 3 Definition 2.4. Let R be an augmented k-algebra. If R is Gorenstein, then it is orientable if the equivalence of (2.3) is as right E-modules, i.e., is an equivalence of right E-modules.
One interesting consequence of the Gorenstein condition is the duality that it often implies. To explain this, we introduce some further terminology. If M is an R-module, we let Cell R k (M ) denote the k-cellular approximation of M ; that is, Cell R k (M ) is in the localizing subcategory in Mod R generated by k, and there is a morphism Cell R k (M ) → M that induces an equivalence on Hom R (k, −). If the ring spectrum R is clear, we will usually just write Cell k (M ). For example, if A is a local Noetherian ring with residue field k, then taking R = HA and M a discrete A-module, we have that π * Cell k (HM ) is the local cohomology H * m (M ) of M . If R → k is proxy-regular, then k-cellularization has a particularly simple formula, namely where, as previously, E = Hom R (k, k). Moreover, k-cellular approximation is smashing, that is, Cell k (M ) ≃ (Cell k (R)) ⊗ R M for any M ∈ Mod R . Proofs of these claims can be found in [Gre18,Lemma 6.3] and [Gre18, Lemma 6.6]. We then have the following, see [Gre18, Section 18.B].
Proposition 2.6. Suppose that R is an augmented k-algebra such that R → k is orientable Gorenstein of shift a. Then, there is an equivalence of R-modules Proof. Using the Gorenstein orientable condition there are equivalences of right E-modules By (2.5), applying − ⊗ E k to the latter equivalence gives rise to an equivalence of R-modules Definition 2.7. Let R be an augmented k-algebra. We say that R has Gorenstein duality of shift a if there is an equivalence of R-modules We can then deduce from Proposition 2.6 that Cell k (R) ≃ Σ a Hom k (R, k).
Finally, we point out a useful way of recognizing when R → k is orientable Gorenstein. We need the following definition [DGI06, Section 8.11].
Lemma 2.10. Let R be an augmented k-algebra which is proxy-regular. If R satisfies Poincaré duality of dimension a, then it is orientable Gorenstein of shift a. If R is additionally cosmall and coconnective, then the reverse implication is true.
Proof. By [DGI06, Proposition 8.12], R is Gorenstein of shift a and we get orientability by applying Hom R (k, −) to the equivalence R ≃ Σ a Hom k (R, k).
On the other hand, if R is also cosmall and coconnective, then by Proposition 2.6 and Lemma 2.8, we have that so that R satisfies Poincaré duality of dimension a.
Remark 2.11. Unlike in algebra, if R → k is Gorenstein, then R does not necessarily satisfy Gorenstein duality. For example, let X be a non-oriented manifold, and take R = C * (X; Z/4) the ring spectrum of mod 4 cochains on X. Then R is Gorenstein, see [Gre07, Example 11.2(ii)], but cannot satisfy Gorenstein duality. Indeed, the assumptions of [DGI06, Proposition 9.3] are satisfied and give rise to a spectral sequence is the homogeneous ideal of elements in positive degree. If R satisfied local Gorenstein duality, then the spectral sequence would collapse to show that H * (X; Z/4) = H 0 m (H * (X; Z/4)) ∼ = H * (X; Z/4), i.e., π * R = H − * (X; Z/4) would satisfy algebraic Poincaré duality.

Local Gorenstein duality. Classically, if
A is a discrete commutative Gorenstein ring, then so is the localization A p for any prime ideal p ∈ Spec(A). The proof involves the characterization of Gorenstein rings as those rings with finite injective dimension, and so there is no obvious generalization to the case of ring spectra. Rather, we will identify conditions where the duality condition of Definition 2.7 localizes. For this we need to explain what we mean by localizing a ring spectrum R at a prime ideal p ∈ Spec h (π * R). Namely, following [BIK08] or [BHV18b] we explain how, given any p ∈ Spec h (π * R), we can define a functor Γ p : Mod R → Mod R , which is a spectral version of classical local cohomology.
We briefly describe one way to construct Γ p . For any such p there exists a ring spectrum R p with homotopy (π * R) p , and a natural morphism R → R p , see [EKMM97, Ch. V.1] for example. By extension of scalars there is a functor Mod R → Mod Rp sending M to M p = M ⊗ R R p . We can then construct a Koszul object R p / /p inside Mod Rp , see [BHV18b, Section 3.1]. The localizing subcategory generated by R p / /p inside of Mod Rp is denoted Mod p−tors

Rp
, the category of p-local and p-torsion objects. The inclusion Mod p−tors Under some conditions we can identify Cell k with a local cohomology functor.
Lemma 2.12. Let k be a field, and R a coconnective commutative augmented k-algebra. Assume that π * R is a Noetherian local ring and that the augmentation map induces an isomorphism π 0 R ∼ = k ∼ = (π * R)/m. Then, the functors Cell k and Γ m are equivalent.
Proof. This is a consequence of the proof of [DGI06, Proposition 9.3]. We sketch the details for the benefit of the reader. First note that Γ m = Γ V(m) . It then suffices to show that By the proof of [DGI06, Proposition 9.3] the assumptions of the lemma give rise to an equivalence K ∞ ⊗ R M ≃ Cell k (M ) and we are done.
Thus, for our local version of Gorenstein duality we will replace Cell k (R) with Γ p R. The next question is what the analog of Hom k (R, k) should be in general. From now on we write I R = Hom k (R, k). Suppose that we are still under the conditions of Lemma 2.12, then π * I R ∼ = π * Hom k (R, k) ∼ = Hom k (π * R, k) ∼ = I m , the injective hull of (π * R)/m ∼ = k, see [Lam99, Example 3.41]. By Brown representability, there is an R-module T R (I m ) such that π * T R (I m ) ∼ = I m . Then I R ≃ T R (I m ). More generally, for any injective π * R-module I, as in [BHV18b, Section 4] we can construct an R-module T R (I) such that π * T R (I) ∼ = I. These spectra are characterized by the property that for any M ∈ Mod R there is an isomorphism of graded π * R-modules π * Hom R (M, T R (I)) ∼ = Hom π * R (π * M, I).
We let I p denote the injective hull of (π * R)/p. The spectra T R (I p ) are our local substitutes for I R . Together, we get the following definition.
Definition 2.13. Let R be a ring spectrum. We say that R satisfies local Gorenstein duality with shift a if, for each p ∈ Spec(π * R) of dimension d, there is an equivalence Γ p R ≃ Σ a+d T R (I p ).
Remark 2.14. As we have discussed, under the conditions of Lemma 2.12, this reduces to the Gorenstein duality condition Cell k (R) ≃ Σ a Hom k (R, k) in the case that p is the maximal ideal m.
Finally, we point out some properties of rings satisfying local Gorenstein duality. Here we denote the internal shift functor in graded modules by Σ as well.
Theorem 2.15. Let R be a ring spectrum satisfying local Gorenstein duality of shift a. Then the following hold.
(1) There is an isomorphism of R-modules π * Γ p R ∼ = Σ a+d I p .
(2) There is a spectral sequence (3) π * R is generically Gorenstein, i.e., the localization at any minimal prime ideal is Gorenstein. (4) There are no nontrivial R-module phantom maps into Γ p R.

Proof.
(1) is an immediate consequence from the definition of local Gorenstein duality and the fact that π * T R (I p ) ∼ = I p .
(3) follows from the spectral sequence in (2). Indeed, if p is minimal, then the localized ring (π * R) p is of dimension 0, and hence H −s,t p (π * R) p = 0 whenever s = 0, and the spectral sequence collapses. For (4), it is shown in [BHV18a, Lemma 3.2] that there are no nontrivial phantom maps into T R (I p ), and hence no phantom maps into Γ p R.
Finally we introduce a useful way of identifying spectra satisfying local Gorenstein duality. The following notion was introduced in [BHV18b, Definition 4.5].
Definition 2.16. A ring spectrum R with π * (R) local Noetherian of dimension n is algebraically Gorenstein of shift a if π * (R) is a graded Gorenstein ring of the same shift.
Proposition 2.17. [BHV18b, Proposition 4.7] Let R be a ring spectrum. If R is algebraically Gorenstein of shift a, then R satisfies local Gorenstein duality of shift a.
2.4. The relative Gorenstein condition. Let R → k and S → k be morphisms of ring spectra and S f − → R be a morphism of ring spectra over k. We have restriction of scalars f * : Mod R → Mod S with left adjoint f * and right adjoint f ! . Note that if f is relatively Gorenstein, then f ! (S) ≃ Σ a R. The purpose of this section is to identify conditions guaranteeing that f : S → R is relatively Gorenstein. First we observe the following: Lemma 2.18. Let R → k and S → k be proxy-regular morphisms of ring spectra, and f : S → R be a relative Gorenstein ring morphism over k. Then S is Gorenstein if and only if R is so.
Proof. This follows from the following identities where the third one holds because f is relative Gorenstein: At this point, we need to recall the notion of dc-completeness. There is a canonical morphism R → End E (k) = R induced from the R-module action on k, and we say that R is dc-complete if this map is an equivalence. Recall that we write I R = Hom k (R, k). We want to describe hypothesis on R and S which allow us to identify when a morphism is relative Gorenstein.
Proposition 2.19. Suppose that R is an augmented k-algebra, and that R → k is orientable Gorenstein of shift a, then Proof. By (2.5) and the orientable Gorenstein condition, there is an equivalence By the definition of I R , Substituting for Cell k (R), we see that as required.
We require two more lemmas. The first follows by a simple adjunction argument.
Lemma 2.20. Let R → k and S → k be ring homomorphisms and S f − → R be a morphism of ring spectra over k. There is an equivalence of S-modules Hom S (R, I S ) ≃ I R .
For the second lemma, we observe that k is both naturally an S-module and an R-module.
The following compares the k-cellularization functor in the two categories.
Lemma 2.21. Let S f − → R be a morphism of ring spectra over k and Q = R ⊗ S k. Assume that S → k is proxy-regular with Koszul object K(S). Consider the following conditions: Assuming (2) then, we have the equivalences According to [Sha09, Lemma 3.1(2)] this follows if R ⊗ S K(S) is K(S)-cellular as an S-module. Since the category of S-modules is generated by S, we deduce that R ⊗ S K(S) ∈ Loc ModS (K(S)), i.e., R ⊗ S K(S) is K(S)-cellular as an S-module, as needed.
We now obtain our result for deducing that a morphism f : S → R is relatively Gorenstein.

and (3)]
This is exactly the claim that Hom S (R, S) is relatively Gorenstein of shift s − r.

Gorenstein ascent
In this section we describe ascent techniques which allow to construct new examples of ring spectra satisfying (local) Gorenstein duality from known examples.
3.1. Ascent for Gorenstein rings. Let R → k and S → k be morphisms of ring spectra. Suppose we are given a morphism f : S → R over k with f relatively Gorenstein, then S is Gorenstein if and only if R is Gorenstein, see Lemma 2.18. Now let Q = R ⊗ S k. We consider the situation where two out of R, S and Q satisfy Gorenstein duality. Part (1) of the following was already shown in [DGI06, Section 8.6] or [Gre18,Lemma 19.3].
Theorem 3.1. Let S f − → R be a morphism over k, and let Q = R ⊗ S k. Suppose that the natural morphism ν : Hom S (k, S) ⊗ S R → Hom S (k, R) is an equivalence of S-modules, and that one of the following conditions is satisfied: (i) S → k is proxy-regular and Q → k is cosmall.
(ii) S → k is small and Q → k is proxy-regular. Then the following hold.
(1) If Q → k and S → k are Gorenstein of shift q and s respectively, then R → k is Gorenstein of shift s + q.
(2) If S → k and R → k are Gorenstein of shift s and r respectively, then Q → k is Gorenstein of shift r − s.
The result follows.
Remark 3.2. The natural map ν is an equivalence if (but not only if) either R or k are small as S-modules.
3.2. Ascent for local Gorenstein duality. In Section 3.1 we recalled the Gorenstein ascent theorem of Dwyer-Greenlees-Iyengar, namely that for a finite morphism S f − → R over k if S and Q = R ⊗ S k are Gorenstein, then so is R. Using the results of [BHV18b], we can give a criterion for descent of local Gorenstein duality. Note that in the following we do not need to assume that R and S are k-algebras.
Proposition 3.3. Let R and S be ring spectra. Suppose that S satisfies local Gorenstein duality of shift s and that f : S → R is a finite morphism and is relatively Gorenstein of shift r − s, then R satisfies local Gorenstein duality of shift r.
Proof. This is essentially a restatement of [BHV18b,Theorem 4.27]. There we assumed that f : S → R had the following properties: (1) S is algebraically Gorenstein of shift ν in the sense of Definition 2.16.
(2) R is a compact S-module, i.e., the morphism f is finite.
(3) The dualizing module ω f = Hom S (R, S) is an invertible R-module. However, Condition (1) is only used to ensure that S satisfies local Gorenstein duality, which follows from [BHV18b, Proposition 4.7]. By assumption, Conditions (2) and (3) are satisfied, and so by [BHV18c,Theorem 4.27] we deduce that R satisfies local Gorenstein duality of shift s + r − s = r, claimed.
Combined with Proposition 2.22 we deduce the following. (1) R → k and S → k are orientable Gorenstein of shift r and s respectively.
(3) R and S are dc-complete. Then, if S satisfies local Gorenstein duality of shift s, then R satisfies local Gorenstein duality of shift r.
3.3. Cochain algebras. In this section we specialize the results of the previous subsection to ring spectra obtained as cochains on spaces. We write C * (X; k) for the ring spectrum of k-valued cochains on X defined as the function spectrum Hom Sp (Σ ∞ + X, Hk). In particular, there is an isomorphism π * C * (X; k) ∼ = H − * (X; k).
It is important to have in mind that the object we are interested in is R = C * (X; k), not X itself, which means that we can have different spaces giving rise to the same ring spectrum R. For example, R = C * (BZ/p; F q ) ≃ C * ( * ; F q ) if p and q are coprime.
If k is a field and X is a space, then we denote the Bousfield k-completion of X by X ∧ k . If X is k-good, then C * (X; k) ≃ C * (X ∧ k ; k) and in this case we can assume that X is k-complete. For example, if π 1 X is finite, then X is k-good for k = Q and k = F p for any prime p, and therefore X ∧ k is k-complete (see [BK72, I.5.2, VII.3.2, VII.5.1]). Given a space X, the ring spectrum of cochains will be well behaved when it satisfies certain hypothesis which we will assume mostly through the rest of the paper: Definition 3.5. [DGI06, Section 4.22] A space X is said to be of Eilenberg-Moore type (EMtype) if X is connected, H * (X; k) is of finite type, and (1) X is simply connected when k = Q, or (2) k is of characteristic p and π 1 X is a finite p-group.
Remark 3.6. A space of EM-type is k-good and so we can assume always that X is k-complete when considering its ring spectrum of cochains with coefficients in k.
We are interested in these properties for the following reason. Suppose we are given a homotopy pullback square of spaces Y × X Z Z Y X. If X is of EM-type then the Eilenberg-Moore spectral sequence shows that C * (Y × X Z; k) ≃ C * (Y ; k) ⊗ C * (X;k) C * (Z; k).
In particular, we obtain: Lemma 3.7. Let F → Y → X be a fiber sequence of spaces where X is of EM-type. Then Under some hypothesis on X, a morphism C * (X; k) → k that is Gorenstein is also automatically orientable.
Lemma 3.8. Let X be a connected space such that H * (X; k) is of finite type. Suppose that: (1) X is simply connected with k = Q, or (2) k is field of characteristic p, and π 1 X is a finite p-group.
(3) k is a finite field of characteristic p and π 1 X is a pro-p group.
Then, if C * (X; k) → k is Gorenstein, it is orientable. In particular, the conditions of the lemma hold if X is a space of EM-type.
Proof. We first check that E = Hom R (k, k) ≃ C * (ΩX; k). If X is simply connected with k = Q then it follows from the strong convergence of the Eilenberg-Moore spectral sequence. Otherwise, the action of π 1 X on H * (X; k) is nilpotent: if k is a finite field of characteristic p, the action factors then through a finite quotient which is a p-group since H * (X; k) is of finite type. Then again the strong convergence of the Eilenberg-Moore spectral sequence shows that E = Hom R (k, k) ≃ C * (ΩX; k). We show that E has a unique action on k. This action factors through π 0 E ∼ = π 1 X since k is an Eilenberg-MacLane spectrum. The case where k is of characteristic p and π 1 X is a finite p-group is [Gre18, Lemma 18.2]. If, π 1 X is a pro-p group and k is finite, then the action map factors through a finite quotient of π 1 X, which is a finite p-group, and hence the result also follows in this case. In the rational case, since X is simply connected, the same argument as [Gre18, Lemma 18.2] shows that k has a unique E-module structure, and hence is orientably Gorenstein.
In the previous subsection we identified conditions to descend local Gorenstein duality along a finite morphism. In light of Remark 3.2, we are interested in conditions which ensure that the induced morphism on cochains for a map of spaces f : Y → X is finite. To this end, we have the following, due to Greenlees-Hess-Shamir [GHS13] in the rational case, and Benson-Greenlees-Shamir [BGS13] in the characteristic p case. We now present a cochain version of Theorem 3.1.
Theorem 3.10. Suppose that g : Y → X is a morphism of spaces of EM-type (p-complete if the characteristic of k is p) with fiber F , such that H * (F ; k) is finite-dimensional, and that C * (X; k) → k is proxy-regular. Then the following hold: (1) If C * (F ; k) → k and C * (X; k) → k are Gorenstein of shift q and s respectively, then C * (Y ; k) → k is Gorenstein of shift s + q.
(2) If C * (Y ; k) → k and C * (X; k) → k are Gorenstein of shift r and s respectively, then C * (F ; k) → k is Gorenstein of shift r − s.
Remark 3.11. It worth pointing out that C * (Y ; k) is Gorenstein if, for example, C * (F ; k) is a Poincaré duality algebra by Theorem 3.10 and Lemma 2.10.
We now specialize the results on relative Gorenstein duality and local Gorenstein duality to cochain algebras.
Theorem 3.12. Let g : Y → X be a morphism of spaces of EM-type (p-complete if the characteristic of k is p) with fiber F . Suppose that H * (F ; k) is finite-dimensional, and that C * (X; k) is Gorenstein of shift s.
(2) If, in addition, C * (X; k) satisfies local Gorenstein duality of shift s, then C * (Y ; k) satisfies local Gorenstein duality of shift r.
Proof. The Eilenberg-Moore condition shows that C * (F ; k) ≃ C * (Y ; k) ⊗ C * (X;k) k. We therefore must verify the three conditions given in Proposition 2.22 and Proposition 3.4, applied to morphisms C * (X; k) f − → C * (Y ; k) and C * (F ; k) → k. We note that by assumption on H * (F ; k), the morphism f is finite by Lemma 3.9.
(1) By assumption C * (X; k) is Gorenstein of shift s. Further, C * (Y ; k) is Gorenstein of shift r, either by assumption, or by Theorem 3.10 in the case that C * (F ; k) is a Poincaré duality algebra. Since X and Y are assumed to be of EM-type, they are automatically orientable Gorenstein by Lemma 3.8. (2) The assumption that H * (F ; k) is finite-dimensional implies that C * (F ; k) → k is cosmall [DGI06, Section 5.5(2)].
(3) Since X and Y are assumed to be of EM-type, they are automatically dc-complete [Gre18, Section 7.B]. Now we apply Proposition 2.22 to see that f is relatively Gorenstein of shift s − r, as claimed. If, in addition, C * (X; k) satisfies local Gorenstein duality of shift s, then Proposition 3.4 implies that C * (Y ; k) satisfies local Gorenstein duality of shift r.

Examples
This section is devoted to relevant examples of type C * (X; k) coming from H-spaces, finite loop spaces, Lie groups, and Noetherian rational spaces.

Spaces with Gorenstein cohomology ring. A first source of examples is given by al-
gebraically Gorenstein ring spectra R. When R = C * (X; k), this means that H * (X; k) is a Gorenstein ring, see Definition 2.16. As recalled in Proposition 2.17, algebraically Gorenstein implies local Gorenstein duality, and so, by virtue of Lemma 2.12, Gorenstein duality holds. Secondly, if R is proxy-regular then we also deduce that C * (X; k) → k is Gorenstein by applying Hom R (k, −) to the Gorenstein duality, see the proof of Lemma 2.10.
Examples of Gorenstein rings are given by finite duality algebras (which are Gorenstein of Krull dimension zero) and polynomial algebras. Other examples come from the theory of H-spaces.
Example 4.2. If X is an H-space with finite cohomology H * (X; k), then by the classification [Hop41,MM65] of finite-dimensional Hopf algebras over a perfect field k, H * (X; k) is a Poincaré duality algebra. In particular this includes mod p finite loop spaces when k = F p , that is, loop spaces with finite mod p cohomology.
Another source of examples is given by spaces with polynomial algebra. If X is k-good with H * (X; k) polynomial then it is also algebraically Gorenstein.
Example 4.3. The group cohomology of U (n) is given by H * (BU (n); k) ∼ = k[c 1 , . . . , c n ], where c i is the i-th Chern class, with degree 2i, and k = F p , Q. In particular, the ring is regular, hence Gorenstein, and therefore satisfies local Gorenstein duality of shift dim(U (n)), see Proposition 2.17.
A less obvious situation is when X is a connected H-space with Noetherian mod p cohomology, which combines the two previous basic examples.
Proposition 4.4. Let X be a connected H-space with Noetherian cohomology, then H * (X; F p ) is Gorenstein. In particular, C * (X; F p ) is algebraically Gorenstein, and therefore local Gorenstein duality holds.

4.2.
Classifying spaces of p-compact groups. We consider the p-compact groups of Dwyer-Wilkerson [DW94]. As such, we fix k = F p , and for a space X we write H * (X) for H * (X; F p ) and similarly for C * (BX). We recall that a p-compact group is a loop space X ≃ ΩBX such that X is F p -finite and BX is a pointed connected F p -complete space. A homomorphism i : Y → X of p-compact groups is a pointed map Bi : BY → BX. Finally, if the homotopy fiber X/Y of Bi is F p -finite, then we say that i is a monomorphism and that Y ≤ i X is a subgroup of X.
Example 4.5. If G is a compact Lie group with π 0 G a finite p-group, then G ∧ p is a p-compact group, using [DW94, Lemma 2.1].
Lemma 4.6. Let X be a p-compact group. Then BX is of EM-type and π 1 X is an abelian pro-p group.
Proof. By definition BX is a connected space, and by [DW94, Lemma 2.1] π 1 BX ∼ = π 0 X is a finite p-group. That H * (BX) is of finite type follows from the main result of [DW94].
For the second part, we can assume that X is connected since X is a loop space and hence all its connected components have the same homotopy type. Let T be a maximal torus of X, then π 1 X is always a quotient of π 1 T ∼ = Z r p , see for example [DW09, Remark 1.3]. Since X is F p -finite, π 1 X is p-complete [BK72, Proposition VI.5.4], and the claim follows.
For the following, we recall that the F p -cohomological dimension of a space X, denoted dim p (X), is the largest integer i for which H i (X) = 0 (with the convention that dim p (X) = ∞ if there is no such integer and H * (X) = 0, and −∞ if H * (X) vanishes).
Proposition 4.7 (Dwyer-Greenlees-Iyengar). Let X be a p-compact group, then C * (BX) → F p is orientable Gorenstein of shift dim p (X).
Proof. This is shown in [DGI06, Section 10.2], except for identifying the shift. The proof relies on the fact that the graded ring H * (X) satisfies algebraic Poincaré duality of dimension a, which is exactly the shift. This implies that a = dim p (X). Finally, it is automatically orientable, because Lemma 4.6 shows that the conditions of Lemma 3.8 are satisfied.
This has the following consequences. First, we consider the relative Gorenstein property for p-compact groups. We show that monomorphisms induce relative Gorenstein morphisms. Then we prove that the homotopy fiber of a monomorphism is a mod p Poincaré duality space.
Proof. We need only verify the conditions of Theorem 3.12(1) for the morphism Bi : BY → BX. We have already seen in Lemma 4.6 that classifying spaces of p-compact groups are always of EMtype. By the definition of a monomorphism the homotopy fiber F of Bi has finite-dimensional cohomology. Since p-compact groups always satisfy Gorenstein duality (Proposition 4.7), Theorem 3.12(1) applies with q = dim p (X) − dim p (Y ) as required.
Corollary 4.10. Let i : Y → X be a monomorphism of p-compact groups.
Proof. We will apply Theorem 3.10(2). We have already seen that the classifying space of a p-compact group is of EM-type and satisfies Gorenstein duality. Thus Theorem 3.10(2) applies to show that C * (X/Y ) → F p is Gorenstein of shift dim p (Y ) − dim p (X).
Recall from Example 4.5 that if G is a compact Lie group with π 0 (G) a p-group, then G ∧ p is a p-compact group. In particular, the p-completion U (n) ∧ p is a p-compact group. In order to prove local Gorenstein duality, we will use ascent along unitary embeddings. We will need the following [CC17, Theorem 6.2], which relies on the classification of p-compact groups.
Theorem 4.11. Any p-compact group X admits a monomorphism X → U (n) ∧ p for some n > 0. Then we deduce that any p-compact group satisfies local Gorenstein.
Theorem 4.12. Let X be a p-compact group, then C * (BX) satisfies local Gorenstein duality of shift dim p (G).
Proof. Choose a monomorphism i : BX → BU (n) ∧ p . The same argument as in Corollary 4.8 along with the fact that C * (BU (n) ∧ p ) satisfies local Gorenstein duality (Example 4.3) shows that the conditions of Theorem 3.12(2) are satisfied, so C * (BX) satisfies local Gorenstein duality of shift dim p (X) as claimed.
4.3. Compact Lie groups. Let G be a compact Lie group and continue to work with F pcoefficients. Following Benson and Greenlees [BG97], given a d-dimensional real representation V of G, we say that it is orientable (with respect to F p ) if the action of G on H d (S V ; F p ) is trivial. The adjoint representation is orientable if for example G is finite or connected. In this case, we have the following, see [DGI06,Section 10.3].
Theorem 4.13 (Dwyer-Greenlees-Iyengar). If G is a compact Lie group whose adjoint representation is orientable, then C * (BG) → F p is Gorenstein of shift dim(G).
More generally, C * (BG) → F p is a 'generalized' Gorenstein morphism, where we allow twists by an invertible element that may not be a suspension of F p , see [Gre20]. In fact, even in the case where the shift is a suspension of F p , it may not be by dim(G), as the following example from [Gre20] demonstrates.
One can explain this example in the following way: Ω(BO(2)) ∧ p is a p-compact group with dim p (Ω(BO(2)) ∧ p ) = 3, and then we can apply Proposition 4.7. In general the classifying space BG of any compact Lie group is p-good since the fundamental group is finite, so that C * (BG ∧ p ) ≃ C * (BG), and moreover π 1 (BG ∧ p ) is a finite p-group, see [BG08,Theorem 7.3]. Thus, if Ω(BG ∧ p ) is F p -finite, then BG ∧ p is the classifying space of the p-compact group Ω(BG ∧ p ). Note that Ω(BG ∧ p ) is not necessarily equivalent to G ∧ p , and in fact, that happens only if π 0 G a finite p-group. We record this in a definition.
Definition 4.15 (Ishiguro). A compact Lie group G is said to be of p-compact type if Ω(BG ∧ p ) is F p -finite.
In particular, if G is of p-compact type, then Ω(BG ∧ p ) is a p-compact group. We can then apply Proposition 4.7 and Theorem 4.12 to Ω(BG ∧ p ) to deduce the following. Theorem 4.16. Let G be a compact Lie group of p-compact type, then C * (BG) → F p is Gorenstein of shift dim p (Ω(BG ∧ p )) and C * (BG) → F p satisfies local Gorenstein duality of the same shift.
It is an interesting open problem to find conditions on a compact Lie group G so that G is of p-compact type. Finite groups provide examples showing that it does not hold in general, for example, consider the case of Σ 3 at the prime 3. It is necessary, but not sufficient, that G satisfies the following conditions (see [Ish01b,Proposition 3.1]): (1) π 0 G is p-nilpotent (2) π 1 ((BG) ∧ p ) is isomorphic to a p-Sylow subgroup of π 0 G. Under some more hypotheses, we have stronger results, for example if the connected component has rank 1, then G is of p-compact type if π 0 G is p-nilpotent, see [Ish01a, Theorem 2]. 4.4. Local Gorenstein duality for rational spaces. We now switch to the rational case, i.e., we take k = Q. In this case, the conditions of Theorem 3.12 become particularly easy to check given the fact that the algebraic Noether normalization can be lifted to spectra.
Theorem 4.18. Let X be a simply connected rational space with Noetherian cohomology. Then, if C * (X; Q) → Q is Gorenstein of shift r, then C * (X; Q) satisfies local Gorenstein duality of shift r.

Proof.
The key observation is due to Greenlees-Hess-Shamir [GHS13, Proposition 3.2]. By assumption on X there exists a Noether normalization R * ∼ = Q[x 1 , . . . , x n ] of H * (X; Q), where the polynomial algebra is concentrated in even degrees. We can realize this polynomial subring via a map X → i K(Q, 2k i ), giving a map of ring spectra R = C * ( i K(Q, 2k i ), Q) → C * (X; Q), which is finite by [Gre18,Lemma 10.2].
We now observe that R * is a regular local ring, and in particular is Gorenstein. The universal coefficient spectral sequence Ext * , * π * R (k, π * R) =⇒ π * Hom R (k, R) shows that R → k is Gorenstein of some shift r. We claim that R satisfies local Gorenstein duality of the same shift. Indeed, since R * is a regular local ring, it is algebraically Gorenstein in the sense of [BHV18b,Definition 4.5], and hence satisfies local Gorenstein duality by Proposition 2.17. That the shifts are the same can be deduced from the collapsing of the local cohomology spectral sequence of Greenlees-Hess-Shamir [GHS13,Proposition A.2].
By [Gre18,Section 7.B] both R and C * (X; Q) are dc-complete. Thus, if C * (X; Q) → Q is Gorenstein of shift r, then Theorem 3.12 applies to show it is satisfies local Gorenstein duality of shift r.
Example 4.19. The following example is taken from [GHS13, Example A.6]. Identify, CP ∞ with BS 1 , and then consider the inclusion BS 1 → BS 3 . This gives a map CP ∞ ×CP ∞ → BS 3 ×BS 3 . We let X denote the fiber, so that there is a fibration The rational cohomology ring H * (X; Q) is isomorphic to Q[u, v, p]/(u 2 , uv, up, p 2 ) where u, v, p have degrees 2, 2, and 5. This ring is not Gorenstein, however, the map C * (X; Q) → Q is Gorenstein of shift −4, and hence satisfies local Gorenstein duality of the same shift by Theorem 4.18.

Local Gorenstein duality for p-local compact groups
In this section we continue to write C * (X) and H * (X) where the coefficients are understood, unless any confusion is likely to arise. Recall that a compact Lie group gives rise to a p-compact group whenever π 0 G is a finite p-group. In order to capture the homotopy theory of compact Lie groups in general, Broto, Levi, and Oliver introduced the concept of a p-local compact group [BLO03,BLO07]. To motivate the definition, let G be a finite p-group, S a Sylow p-subgroup, and consider the category F S (G) with objects subgroups of S and morphisms Hom FS(G) (P, Q) = Hom G (P, Q), those morphisms induced by subconjugation inside G. This is the fusion category of G over S, and many results and concepts in group theory can be stated in terms of this category. The idea of p-local finite groups, and more generally p-local compact groups, is to generalize this where we are given only a finite p-group S (or a discrete p-toral group), and we define a category from this, with similar properties to the category F S (G).
In more detail, we fix a discrete p-toral group S, that is, a group that fits in an extension where π is a finite p-group. We then define a fusion system F on S to be a category whose objects are the subgroups of S, and whose morphisms satisfy the following properties for P, Q ≤ S: (1) Hom S (P, Q) ⊆ Hom F (P, Q) ⊆ Inj(P, Q).
(2) Every morphism in F factors as an isomorphism followed by an inclusion. In general, one is interested in fusion systems which are saturated, a technical condition defined in [BLO07, Definition 1.2]. Given a fusion system on a discrete p-toral group S, Broto, Levi, and Oliver constructed another category, the centric linking system L. One then defines a classifying space |L| ∧ p , as the p-completion of the nerve of the category L. In the case where S is a finite p-group, it was shown by Chermak [Che13] that the data (S, F ) already uniquely determines the category L, while the general case was shown by Levi and Libman [LL15].
Definition 5.1. A p-local compact group G = (S, F ) consists of a discrete p-toral group S and F a fusion system on S. The classifying space BG is defined as the p-completion of the nerve of the centric linking systems L associated to (S, F ). If S is a finite p-group, then G is called a p-local finite group.
We point out that this classifying space comes with a canonical morphism θ S : BS → BG. The notion of monomorphism in this context involves a more general condition on the homotopy fiber of a map between classifying spaces than the one for p-compact groups introduced by Dwyer and Wilkerson, but it just specializes to that one when the spaces involved are p-compact groups. We claim now that if f is a homotopy monomorphism, then ρ is a monomorphism. We first observe that using [BLO07, Theorem 6.3] one can deduce that θ S and θ P are homotopy monomorphisms (see also [CC17,Proposition 3.3]). We also have that the composite of homotopy monomorphisms is a homotopy monomorphism, and so the diagram above implies that the composite θ P •Bρ is a homotopy monomorphism. Now suppose that ρ was not a monomorphism. Let F denote the fiber of θ P • Bρ. Let σ ∈ Hom(Z/p, S) be an injection into ker(ρ). This implies that Bσ : BZ/p → BS is null-homotopic in BH, and hence Bσ lifts to a map BZ/p → F which is not null-homotopic. This is a contradiction to θ P • Bρ being a homotopy monomorphism, and so ρ is a monomorphism as claimed.
Taking cohomology, we obtain a commutative diagram By [BCHV19, Cororally 4.20 and Proposition 5.5] the cohomology rings in this diagram are Noetherian, and the horizontal morphisms are injections which exhibit the target as a finitely generated module over the source. We claim that in order to show that H * (BG) is a finitely generated H * (BH)-module, it suffices to show that H * (BS) is finitely generated as H * (BP )module via Bρ * . Indeed, suppose that Bρ * is finite, then H * (BS) is a finitely generated H * (BH)module. Because H * (BH) is Noetherian, it follows that H * (BG) is a finitely generated H * (BH)module, so φ * is finite. To see that Bρ * is finite, consider the p-compact toral group S = Ω((BS) ∧ p ) (a p-compact group which is an extension of a p-compact torus and a finite p-group) and the natural F p -equivalence f : BS → B S. By [MN94, Proposition 3.4], the morphism ρ : S → P is a monomorphism of p-compact groups since ρ : S → P is an algebraic monomorphism. This implies that H * (B S) is a finitely generated H * (B P )-module via B ρ * by [DW94, Proposition 9.11]. Therefore, H * (BS) is a finitely generated H * (BP )-module via Bρ * .
We would like to identify conditions that ensure that, given a homotopy monomorphism φ : BG → BH as above, the induced morphism φ * : C * (BH) → C * (BG) is finite. The next results follows from combining [BGS13, Lemma 3.4 and Lemma 3.5].
Lemma 5.6. Let f : Y → X be a map between p-complete spaces with fundamental groups finite p-groups and let F denote the fiber of f . If H * (ΩX) is finite dimensional and H * (Y ) is finitely generated over H * (X) via the induced map, then f * : C * (Y ) → C * (X) is finite and H * (F ) is finite-dimensional.
Proposition 5.7. Let G = (S, F ) and H = (P, E) be p-local compact groups and φ : BG → BH a homotopy monomorphism with homotopy fiber F , where BH is the classifying space of a pcompact group of dimension h. Then: (1) C * (BG) is Gorenstein if and only if H * (F ) is a Poincaré duality algebra.

Proof.
We have already seen that classifying spaces of p-local compact groups are always of EM-type, and they are p-complete by definition. The first point follows directly from Theorem 3.10, by using that H * (F ) is finite (Lemma 5.6) and Lemma 2.10. For the second point, as usual, we wish to apply Theorem 3.12. By Proposition 5.5 and Lemma 5.6 we have that f * : C * (BH) → C * (BG) is finite and H * (F ) is finite-dimensional, where F is the fiber of φ. By Theorem 4.12, C * (BH) satisfies local Gorenstein duality of shift h. Thus, if C * (BG) satisfies Gorenstein duality of shift g, then Theorem 3.12 applies to give the result.
Definition 5.8. Let X be a connected p-complete space, then a unitary embedding of X is a homotopy monomorphism X → BU (n) ∧ p for some n > 0. A p-local compact group G is said to admit a unitary embedding if its classifying space BG does.
Remark 5.9. Since classifying spaces of p-compact groups always admit unitary embeddings by Theorem 4.11, any homotopy monomorphism as in Proposition 5.7 gives rise to a unitary embedding of BG.
Corollary 5.10. Let G be a p-local compact group with a unitary embedding φ : BG → BU (n) ∧ p . The homotopy fiber U (n)/G is a Poincaré duality space of dimension q if and only if C * (BG) is Gorenstein of shift n 2 + q and satisfies local Gorenstein duality of the same shift.
Proof. The condition on the homotopy fiber ensures that C * (BG) is Gorenstein of shift n 2 + q by Theorem 3.10.
Remark 5.11. Suppose that there exists a unitary embedding φ : BG → BU (n) ∧ p with Poincaré duality fiber. Then, any homotopy monomorphism f : BG → BH into the classifying space of a p-compact group will have the property that the homotopy fiber is a Poincaré duality space. Indeed, C * (BG) will be Gorenstein by Corollary 5.10, and hence by Proposition 5.7(1) the homotopy fiber of f will be a mod p Poincaré duality space. For example, this holds for any other unitary embedding of G. In this situation, the Gorenstein shift can be interpreted as a notion of dimension for the p-local compact group.
We do not know in general that p-local compact groups admit unitary embeddings, and even when they do we do not know that the homotopy fiber is a Poincaré duality space. However, in the case of a p-local finite group (i.e., when S is a finite p-group), we do in fact know this is the case.
Corollary 5.12. Any p-local finite group G satisfies local Gorenstein duality of shift 0.