The $\mathop{Sp}_{k,n}$-local stable homotopy category

Following a suggestion of Hovey and Strickland, we study the category of $K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra. When $k = 0$, this is equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is the category of $K(n)$-local spectra, both of which have been studied in detail by Hovey and Strickland. Based on their ideas, we classify the localizing and colocalizing subcategories, and give characterizations of compact and dualizable objects. We construct an Adams type spectral sequence and show that when $p \gg n$ it collapses with a horizontal vanishing line above filtration degree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study the Picard group of $K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra, showing that this group is algebraic, in a suitable sense, when $p \gg n$. We also consider a version of Gross--Hopkins duality in this category. A key concept throughout is the use of descent.


Introduction
In their memoir [HS99b] Hovey and Strickland studied the categories of K(n)-local and E nlocal spectra in great detail.Here K(n) is the n-th Morava K-theory; the spectrum whose homotopy groups are the graded field n ], |v n | = 2(p n − 1) and E n is the n-th Lubin-Tate spectrum, or Morava E-theory, with As explained in the introduction of [HS99b], the Morava K-theories are the prime field objects in the stable homotopy category -for a way to make that precise, see [HS98] or more specifically [Bal10, Corollary 9.5] -and are one of the fundamental objects in the chromatic approach to stable homotopy theory.
A deep result of Hopkins and Ravenel [Rav92] is that Bousfield localization with respect to E n is smashing, which simplifies the study of the category of E n -local spectra considerably.On the other hand, localization with respect to K(n) is not smashing [HS99b,Lemma 8.1], and the monoidal unit L K(n) S 0 ∈ Sp K(n) is dualizable, but not compact.In the language of tensor-triangulated geometry, Sp K(n) is a non-rigidly compactly generated category.Because of this, much of the work in [HS99b] is therefore dedicated to understanding the more complicated category of K(n)-local spectra.
By a Bousfield class argument, the category of E n -local spectra is equivalent to the category of K(0) ∨ . . .∨ K(n)-local spectra.In this paper we study the categories of K(k) ∨ . . .∨ K(n)local spectra for 0 ≤ k ≤ n, which were suggested as 'interesting to investigate' by Hovey and Strickland, see the remark after Corollary B.9 [HS99b].We write L k,n for the associated Bousfield localization functor.As we shall see, when k = 0, the category Sp k,n of K(k) ∨ . . .∨ K(n)-local spectra behaves much like the category Sp K(n) = Sp n,n of K(n)-local spectra.For example, it is an example of a non-rigidly compactly generated category; as soon as k = 0, the monoidal unit L k,n S 0 ∈ Sp k,n is dualizable, but not compact.However, the categories Sp k,n for k = n are in some sense more complicated than the case k = n; for example, Sp K(n) has no non-trivial (co)localizing subcategories, while this is not true for Sp k,n as long as k = n.
1A. Contents of the paper.We now describe the contents of the paper in more detail.We begin with a study of Bousfield classes, constructing some other spectra which are Bousfield equivalent to K(k) ∨ • • • ∨ K(n).In particular, we show that there is a Bousfield equivalence between a localized quotient of BP , denoted E(n, J k ) and K(k) ∨ • • • ∨ K(n).For this reason, as well as brevity, we often say that X is E(n, J k )-local, instead of As was already noted by Hovey and Strickland [HS99b, Corollary B.9] Sp k,n is an algebraic stable homotopy theory in the sense of [HPS97] with compact generator L k,n F (k), the localization of a finite spectrum of type k.We investigate some consequences of this; for example, analogous to Hovey and Strickland's formulas for L K(n) X, in Proposition 2.24, we prove some formulas for L k,n X in terms of towers of finite type k Moore spectra.Some of these results had previously been obtained by the author and Barthel and Valenzuela [BHV18].
In Section 3 we investigate the tensor-triangulated geometry of Sp k,n .We begin by characterizing the compact objects in Sp k,n , culminating in Theorem 3.8 which is a natural extension of Hovey and Strickland's results in the cases k = 0, n.A classification of the thick ideals of Sp ω k,n is an almost immediate consequence of this classification, see Theorem 3.16 for the precise result.Of course, here we rely on the deep thick subcategory theorem in stable homotopy [HS98] and its consequences.Finally, we classify the localizing and colocalizing subcategories of Sp k,n in Theorem 3.33.We obtain the following.
1.1.Theorem.There is an order preserving bijection between (co)localizing subcategories of Sp k,n and subsets of {k, . . ., n}.Moreover, the map that sends a localizing subcategory C of Sp k,n to its left orthogonal C ⊥ induces a bijection between the set of localizing and colocalizing subcategories of Sp k,n .The inverse map sends a colocalizing subcategory U to its right orthogonal ⊥ U.
We also compute the Bousfield lattice of Sp k,n (Proposition 3.39) and show that a form of the telescope conjecture holds (Theorem 3.46).
In Section 4 we show that, as a consequence of the Hopkins-Ravenel smash product theorem, the commutative algebra object E n ∈ Sp k,n is descendable, in the sense of [Mat16].This has a number of immediate consequences.For example, it implies the existence of a strongly convergent Adams type spectral sequence, which we call the E(n, J k )-local E n -Adams spectral sequence, computing π * (L k,n X) for any spectrum X.Moreover, descendability implies this collapses with a horizontal vanishing line at a finite stage (independent of X).In the case of X = S 0 it is known that, in the cases k = 0 and k = n, this vanishing line already occurs on the E 2 -page so long as p ≫ n.In order to generalize this result, we first show that when X = S 0 , the E 2 -term of the E(n, J k )-local E n -Adams spectral sequence spectral sequence can be given as the inverse limit of certain Ext groups computed in the category of (E n ) * E n -comodules, see Proposition 4.16 for the precise result.We are then able to utilize a chromatic spectral sequence and Morava's change of rings theorem to show the following (Theorem 4.24): 1.2.Theorem.Suppose p − 1 does not divide k + s for 0 ≤ s ≤ n − k (for example, if p > n + 1), then in the E 2 -term of the E(n, J k )-local E n -Adams spectral sequence converging to L k,n S 0 we have E s,t 2 = 0 for s > n 2 + n − k.
In the case k = 0, this recovers a result of Hovey and Sadofsky [HS99a, Theorem 5.1].
As noted previously, so long as k = 0, the categories of dualizable and compact spectra do not coincide in Sp k,n ; every compact spectrum is dualizable, but the converse does not hold, with the unit L k,n S 0 being an example.In Section 5 we study the category of dualizable objects in Sp k,n .As a consequence of descendability, we show that X ∈ Sp k,n is dualizable if and only if L k,n (E n ∧ X) is dualizable in the category of E(n, J k )-local E n -modules.In turn, we show that this holds if and only if L k,n (E n ∧ X) is dualizable (equivalently, compact) in the category of E n -modules.We deduce that X ∈ Sp k,n is dualizable if and only if its Morava module (E k,n ) ∨ * (X) := π * L k,n (E n ∧ X) is finitely-generated as an (E n ) * -module, see Theorem 5.11.This generalizes a result of Hovey and Strickland, but even in this case our proof differs from theirs.
It is an observation of Hopkins that the Picard group of invertible K(n)-local spectra is an interesting object to study [HMS94].Likewise, Hovey and Sadofsky have studied the Picard group of E(n)-local spectra [HS99a].In Section 6 we study the Picard group Pic k,n of E(n, J k )local spectra.Our first result, which is a consequence of descent, is that X ∈ Sp k,n is invertible if and only if its Morava module (E k,n ) ∨ * (X) is free of rank 1.We then study the Picard spectrum [MS16] of the category Sp k,n .Using descent again, we construct a spectral sequence whose abutment for π 0 is exactly Pic k,n .The existence of this spectral sequence in the case k = n is folklore.We say that this spectral sequence is algebraic if the only non-zero terms in the spectral sequence occur in filtration degree 0 and 1.Using Theorem 1.2 we deduce the following result (Theorem 6.8).In the case k = n, this is a theorem of Pstrągowski [Pst18].
There is an interesting element in the K(n)-local Picard group, namely the Brown-Comenetz dual of the monochromatic sphere [HS99b, Theorem 10.2].In Section 7 we extend Brown-Comentz duality to the E(n, J k )-local category.We do not know when the Brown-Comenetz dual of the monochromatic sphere defines an element of Pic k,n ; this is not true when k = 0, and we provide a series of equivalent conditions for the general case in Proposition 7.10.
The case n = 2, k = 1.The first example that has essentially not been studied in the literature is when n = 2 and k = 1, i.e., the category of K(1) ∨ K(2)-local spectra.In Section 5B we give a computation of the Balmer spectrum of K(1) ∨ K(2)-locall dualizable spectra.For this, we recall that Hovey and Strickland have conjectured a description of the Balmer spectrum Spc(Sp dual K(n) ) of dualizable objects in K(n)-local spectra [HS99b,p. 61].This was investigated by the author, along with Barthel and Naumann,in [BHN22].This admits a natural generalization to k,n and some finite spectrum Z of type at least i.We also set D n+1 = (0).The conjecture is that these exhaust all the thick tensor-ideals of Sp dual k,n .We show in Theorem 5.21 that if this holds K(n)-locally (i.e., in Sp dual n,n ), then it holds for all Sp dual k,n .In particular, since it is known to hold K(2)-locally by [BHN22, Theorem 4.15] we obtain the following, see Corollary 5.22.
Conventions and notation.We let X denote the Bousfield class of a spectrum X.The smallest thick tensor-ideal containing an object A will be denoted by thick ⊗ A (it will always be clear in which category this thick subcategory should be taken in).Likewise, the smallest thick (respectively localizing) subcategory containing an object A will be written as Thick(A) (respectively Loc(A)).
Acknowledgments.It goes without saying that this paper owes a tremendous intellectual debt to Hovey and Strickland, in particular for the wonderful manuscript [HS99b].We also thank Neil Strickland for a helpful conversation, as well as his comments on a draft version of this document.
2. The category of Sp k,n -local spectra 2A.Chromatic spectra.We begin by introducing some of the main spectra that we will be interested in.
2.1.Definition.Let BP denote the Brown-Peterson homotopy ring spectrum with coefficient ring 2.2.Remark.The classes v i are not intrinsically defined, and so the definition of BP depends on a choice of sequence of generators; for example, they could be the Hazewinkel generators or the Araki generators.However, the ideals By taking quotients and localizations of BP (for example, using the theory of structured ring spectra [EKMM97, Chapter V]), we can form new homotopy ring spectra.In particular, let J k denote a fixed invariant regular sequence p i0 , v i1 1 , . . ., v Then we can form the homotopy associative ring spectrum BP J k with These were first studied by Johnson and Yosimura [JY80].A detailed study on the product structure one obtains via this method can be found in [Str99].

Definition.
We let E(n, J k ) for n ≥ k denote the Landweber exact spectrum with Here Landweber exact means over BP J k (as studied by Yosimura [Yos83]), that is, there is an isomorphism 2.6.Lemma.
2.8.Notation.We will follow standard conventions and write Sp 0,n as Sp n and Sp n,n as Sp K(n) .
Similarly, the corresponding Bousfield localization functors will be denoted by L 0,n = L n and L n,n = L K(n) , respectively.
2.9.Remark.By [Rav84, Theorem 2.1] we have E(n) = K(0) ∨ . . .∨ K(n) .In fact, let E be a BP -module spectrum that is Landweber exact over BP , and is v n -periodic, in the sense that v n ∈ BP * maps to a unit in E * /(p, v 1 , . . ., v n−1 ).Then Hovey has shown that In particular, this applies to the Lubin-Tate E-theory spectrum E n (see [Rez98]) with or the completed version of E-theory used in [HS99b] with In .2B. Bousfield decomposition.In the previous section we introduced the spectra E(n, J k ) for n ≥ k and an invariant regular sequence p i0 , . . ., v i k−1 k−1 of length k.We now give Bousfield decompositions for E(n, J k )-local spectra.

Proposition. There are equivalences of Bousfield classes
(1 (2) can be deduced from [Yos85] as we now explain.First, by [Yos85, Corollary 1.3 and Proposition 1.4] along with (1), we have By [Yos85, Corollary 1.8] we have v −1 i P (i) = K(i) , and hence (2) follows.For (3), we first note that by the thick subcategory theorem E n /I k = E n ∧ F (k) for some finite type k spectrum.Since E n = n i=0 K(i), (3) then follows from the definition of a type k spectrum.2.11.Remark.In other words, the category of E(n, J k )-local spectra is equivalent to the category of K(k)∨• • • K(n)-local spectra.Note that this implies this category only depends on the length of the sequence, and not the integers i 0 , . . ., i n−1 .We will therefore sometimes say that a spectrum 2C. Algebraic stable homotopy categories.We now begin by recalling the basics on algebraic stable homotopy theories, see [HPS97] in the triangulated setting.
2.12.Definition.A stable homotopy theory is a presentable, symmetric monoidal stable ∞category (C, ⊗, 1) where the tensor product commutes with all colimits.It is algebraic if there is a set G of compact objects such that the smallest localizing subcategory of C containing all G ∈ G is C itself.2.13.Remark.The assumptions on C imply that it the functor − ∧ Y has a right adjoint F (Y, −), i.e., the symmetric monoidal structure on C is closed.2.14.Remark.The associated homotopy category Ho(C) is then an algebraic stable homotopy theory in the sense of [HPS97].We note that compactness can be checked at the level of the homotopy category, see [Lur17,Remark 1.4.4.3].
2.15.Proposition (Hovey-Strickland, Hovey-Palmieri-Strickland).Sp k,n is an algebraic stable homotopy category with compact generator L k,n F (k).The symmetric monoidal structure in Sp k,n is given by Colimits are computed by taking the colimit in spectra and then applying L k,n , while function objects and limimts are computed in the category of spectra.
2.16.Remark.The most difficult part of the above proposition is that L k,n F (k) is a compact generator of Sp k,n .Indeed, one must show that the conditions of [HS99b,Proposition B.7] are satisfied and to do this, one at some point needs to invoke the thick subcategory theorem [HS98], or one its consequences (such as the Hopkins-Ravenel smash product theorem [Rav92]).
2.17.Remark.The localization L n = L 0,n is smashing (that is L n X ≃ L n S 0 ∧X) by the Hopkins-Ravenel smash product theorem [Rav92] and in this case X∧Y ≃ X ∧ Y .However, if k = 0, then localization L k,n is not smashing as the following lemma shows, and so X∧Y ≃ X ∧ Y in general.
2.18.Lemma.If k = 0, then L k,n is not smashing, and L k,n S 0 is not compact in Sp k,n .
Proof.We first claim that L k,n S 0 = E(n) .To see this, note that we have ring maps , so that these inequalities are actually equalities, and all three are Bousfield equivalent to E(n).
Suppose now that L k,n were smashing, so that L k,n S 0 = n i=k K(i) [Rav84, Proposition 1.27].Then, since E(n) n i=k K(i) as soon as k = 0, we have obtained a contradiction.The second part is then a consequence of [HPS97, Theorem 3.5.2].
We call such a tower {M k (j)} j a tower of generalized Moore spectrum of type k.
2.20.Remark.The tower as above is constructed in the homotopy category of spectra.However, as explained in [HS98, Page 9] (see equation ( 15)), such sequential diagrams can always be lifted to a sequence of cofibrations between cofibrant objects, and in particular to a diagram in the ∞-category of spectra (the point is that such diagrams have no non-trivial homotopy coherence data).Then, the (co)limit in the ∞-categorical sense, agrees with the homotopy (co)limit used in Definition 2.2.3 and Definition 2.2.10 of [HPS97].
2.21.Notation.We write M k,n X for the fiber of the localization map L n X → L k−1 X.By definition, we set M 0,n = L n .
2.22.Lemma.We have an equality of Bousfield classes Proof.Recall that, by definition, there is a cofiber sequence Applying L n to this and using L i L j ≃ L min(i,j) we see that where the last equivalence follows as both functors are smashing.It follows from [HS99b, Proposition 5.3] that M k,n S 0 = n i=k K(i) as claimed.2.23.Remark.In [HS99b, Proposition 7.10(e)] Hovey and Strickland give a formula for L K(n) X in terms of towers of generalized Moore spectra.We show now that their proof extends to L k,n X.

2.24.
Proposition.There are equivalences where the limit is taken over a tower {M k (j)} of generalized Moore spectra of type k.
Proof.We first note that Moreover, by [HS99b, Proposition 7.10(a)] this is equivalent to L F (k) L n X.
To finish the proof, we will show that L k,n X ≃ L F (k) L n X.First, note that X → L n X is an L n S 0 -equivalence, and has also been obtained in [BHV18, Proposition 6.21] using the theory and complete and torsion objects in a stable ∞-category.The next result is also contained in [BHV18, Corollary 6.17].
2.26.Proposition.For any spectrum X there is a pullback square Proof.This is a standard consequence of the Bousfield decomposition 2.28.Remark.These types of iterated chromatic localizations have been investigated by Bellumat and Strickland [BS19].Results such as the chromatic fracture square can be recovered from their work, however we do not investigate this in detail.
2.29.Corollary.Suppose M j is a generalized Moore spectrum of type at least k, then Proof.By definition, L k−1 M j ≃ * , and so by the pullback square of Proposition 2.26 we must show that It follows that there is an equivalence of categories M k,n ≃ Sp k,n given by L k,n , with inverse given by M k,n .
Proof.The proof of Hovey and Strickland in the case k = n generalizes essentially without change.
By definition M k,n X fits into a cofiber sequence so applying L k,n gives a cofiber sequence Using Proposition 2.24 we have L k,n X ≃ F (M k,n S 0 , L n X), and so applying F (−, L n X) to the defining cofiber sequence for M k,n S 0 we obtain a cofiber sequence 2.32.Remark.Once again, this result was obtained (by different methods) in [BHV18, Proposition 6.21].

Thick subcategories and (co)localizing subcategories
In this section we compute the thick subcategories of compact objects in Sp k,n and (co)localizing subcategories of Sp k,n .When k = 0 or k = n both results have been obtained by Hovey and Strickland.Along the way we give a classification of the compact objects in Sp k,n .
3A. Compact objects in Sp k,n .In this section we characterize the compact objects in Sp k,n .We will use this in the next section to compute the thick subcategories of Sp ω k,n .We begin by recalling the notions of thick and (co)localizing subcategories.
(1) D is called thick if it is closed under extensions and retracts.
(2) D is called localizing if it is thick and closed under arbitrary colimits.
(3) D is called colocalizing if it is thick and closed under arbitrary limits. ( We will also speak of localizing (or thick) tensor-ideals and colocalizing coideals.
3.2.Remark.In Sp n the dualizable and compact objects coincide, and are precisely those that lie in the thick subcategory generated by the tensor unit L n S 0 .In categories whose tensor unit is not compact, such as Sp k,n for k = 0, the dualizable and compact objects do not coincidefor example, the tensor unit is always dualizable, but is not compact (Lemma 2.18).In [HS99b] Hovey and Strickland gave numerous characterizations of compact objects in Sp K(n) .In this section we extend some of these characterizations to Sp k,n .We first recall the concept of a nilpotent object in a symmetric monoidal category.
from which the result easily follows.
3.5.Remark.In other words, we can talk unambiguously about the category of E n /I k -nilpotent spectra.
We will also need the following generalization of [HS99b, Lemma 6.15].
Proof.The argument is only a slightly adaptation of that given by Hovey and Strickland.By a thick subcategory argument, we can assume that X = M k is a generalized Moore spectrum of type k.By [Rav92] we have L n S 0 ∈ thick ⊗ E n , and it follows that The compact objects in Sp k,n can be characterized in the following ways, partially generalizing [HS99b, Theorem 8.5].We note that every compact object in Sp k,n is automatically dualizable by [HPS97, Theorem 2.1.3];we investigate the dualizable objects in Sp k,n in more detail in Section 5.
(3) implies (2) because every finite spectrum of type at least k lies in the thick subcategory generated by F (k).The implication (1) implies (3) is the same as given by Hovey and Strickland [HS99b, Theorem 8.5].Namely, suppose that The first and last equivalence follow by adjunction, the second because while the third equivalence follows because X ∈ Sp ω k,n by assumption and because L k,n lim − − →i is the colmit in Sp k,n .We have K(i) * Y = 0 for i < k and so [HS99b, Corollary 6.11] implies that Y , and hence X, is a retract of L n Z ≃ L k,n Z for a finite spectrum Z of type at least k.This shows that (1),(2) and (3) are equivalent.
Assume now that (4 . By [HPS97, Theorem 2.1.3]the smash product of a dualizable and compact object is compact, and so X is a retract of a compact E(n, J k )-local spectrum, and so is also compact, i.e., (1) holds.
To see that (3) implies (5), we use a thick subcategory argument to reduce to the case that is a localized generalized Moore spectrum of type k.Such an X is clearly dualizable and is additionally E n /I k -nilpotent by Lemma 3.6.Now suppose that X satisfies (5).Following Hovey and Strickland [HS99b, Proof of Corollary 12.16] let J be the collection of those spectra Z ∈ Sp k,n such that Z is a module over a generalized Moore spectrum of type i (for a fixed i, k ≤ i ≤ n).By [HS99b, Proposition 4.17] J forms an ideal.Because K(i) ∧ Z is non-zero and a wedge of suspensions of K(i), J contains the ideal of K(i)-nilpotent spectra.Moreover, it follows from the Bousfield decomposition In particular, X ∈ J , so that X is retract of a spectrum of the form Y ∧ X where Y is a generalized Moore spectrum of type i, and so (4) holds.
Finally, we prove the subsidiary claims.It is immediate from (2) that Sp ω k,n ⊆ Sp dual k,n is thick, and it is an ideal by [HPS97, Theorem 2.1.3(a)].Because generalized Moore spectra are self-dual (see [HS99b,Proposition 4.18]), (c) implies that Sp ω k,n is closed under Spanier-Whitehead duality.Therefore, 3.9.Remark.When k = 0, then X is compact if and only if X is dualizable [HS99b, Theorem 6.2].To reconcile this with (5) of the previous theorem, we note that every spectrum 3B.The thick subcategory theorem.We now give a thick subcategory theorem for Sp ω k,n .As we shall see, given Theorem 3.8 this is an immediate consequence of the classification of thick subcategories of Sp ω n , which ultimately relies on the Devinatz-Hopkins-Smith nilpotence theorem.
3.10.Definition.For 0 ≤ j ≤ n + 1 let C j denote the thick subcategory of Sp n consisting of all compact spectra X such that K(i) * X = 0 for all i < j, i.e., 0 for all i < j}.3.11.Remark.By [HS99b, Proposition 6.8] we equivalently have and moreover We now present the result of Hovey and Strickland [HS99b, Theorem 6.9].
3.13.Theorem (Hovey-Strickland).If C is a thick subcategory of Sp ω n , then C = C j for some j such that 0 ≤ j ≤ n + 1.

3.14.
Remark.This result can be restated in terms of the Balmer spectrum of Sp ω n [Bal05].In particular, we have Spc(Sp ω n ) ∼ = {C 1 , . . ., C n+1 } with topology determined by the closure operator {C j } = {C i |i ≥ j}.This is in fact equivalent to Theorem 3.13, essentially by the same argument as in [BHN22, Proposition 3.5].
Given the classification of compact E(n, J k )-local spectra in Theorem 3.8, we deduce the following.with inverse given by sending a specialization closed subset

Lemma. The category of compact
This follows by combining Theorem 3.13 and Lemma 3.15.
3.17.Remark.Note that Sp ω k,n is not a tensor-triangulated category when k = 0, as it does not have a tensor unit.Therefore, we cannot speak of the Balmer spectrum of Sp ω k,n .We also have a nilpotence theorem.
3C. Localizing and colocalizing subcategories.In this section we calculate the (co)localizing (co)ideals of Sp k,n .We first observe that every (co)localizing subcategory is automatically a (co)ideal, so it suffice in fact to concentrated on (co)localizing subcategories.
Proof.We prove the case of localizing subcategories -the case of colocalizing subcategories is similar.1 To that end, let C ⊆ Sp k,n be a localizing subcategory, and consider the collection D = {X ∈ Sp | X∧C ⊆ C}.This is a localizing subcategory of Sp containing S, and hence D = Sp itself.It follows that C is a localizing ideal.
3.20.Remark.We remind the reader that 1 is not compact in Sp k,n unless k = 0 (see Lemma 2.18).Therefore, in all other cases, 1 is a non-compact generator of Sp k,n .
We begin by defining a notion of support and cosupport in Sp k,n , extending the notion of support defined previously for Sp ω n .3.22.Definition.For a spectrum X ∈ Sp k,n we define the support and cosupport of X by , and so cosupp(K(i)) = i as well.
3.24.Remark.The notion of support is slightly ambiguous, as objects can live in multiple categories.For example L K(n) S 0 ∈ Sp i,n for all 0 ≤ i ≤ n, and in fact has different support in each category.However, it should also be clear in which category we are considering the support.
Support and cosupport are well behaved with respect to products and function objects in Sp k,n .
3.28.Notation.For an arbitrary collection C of objects we set For a subset T ⊆ Q we also define 3.29.Lemma.For a subset T ⊆ Q, supp −1 (T ) and cosupp −1 (T ) are localizing and colocalizing subcategories of Sp k,n respectively.
Proof.We simply note that We will see that these are bijections.We need the following local-global principle, which is a slight variant of that given by Hovey and Strickland [HS99b, Proposition 6.18].

Proposition (Local-global principle).
For any X ∈ Sp k,n we have Proof.Because X ∈ Sp k,n ⊆ Sp n applying [HS99b, Proposition 6.18] we have The result for colocalizing subcategories is then clear, as we get the same result taking the colocalizing subcategories in Sp k,n .For localizing subcategories we apply [BCHV19, Lemma 2.5] to (3.3) with the colimit preserving functor where we have used that As previously noted (Remark 3.26) supp HS (X) ∩ Q = supp(X), and the result follows.
3.32.Corollary.We have The argument for colocalizing categories is similar.
We now give the promised classification of localizing and colocalizing subcategories. 3.33.Theorem.
Proof.Let C ⊆ Sp k,n be a localizing subcategory and T ⊆ {k, . . ., n} a subset.Then via Corollary 3.32 and basic properties of support Now suppose that X ∈ C, so that supp(X) ⊆ supp(C).It follows from the definitions that X ∈ supp −1 (supp(C)), and so C ⊆ supp −1 (supp(C)).We are therefore reduced to showing that supp −1 (supp(C)) ⊆ C. To that end, let Y ∈ supp −1 (supp(C)), so that supp(Y ) ⊆ supp(C).Using the local-global principle, Proposition 3.30, we then have where the last equality follows from Proposition 3.30 again.The proof for colocalizing subcategories is analogous.
3.34.Notation.For the following, we recall that for C ⊆ Sp k,n the right orthogonal C ⊥ is defined as Similarly, the left orthogonal Moreover, the right orthogonal is a colocalizing subcategory, and the left orthogonal is a localizing subcategory.The inverse map sends a colocalizing subcategory U to ⊥ U.
Proof.We follow [BIK12, Corollary 9.9].Let C be a localizing subcategory, then using Remark 3.31 and Lemma 3.27 Similarly, if U is a colocalizing subcategory, then It follows that under the equivalences of Theorem 3.33, the maps C → C ⊥ and U → ⊥ U correspond to the map Q → Q sending a subset to its complement, and are thus mutually inverse bijections.
3D.The Bousfield lattice.We recall the basics on the Bousfield lattice of an algebraic stable homotopy theory.In order to avoid confusion with the (localized) categories of spectra considered previously we let (C, ∧, 1) denote a tensor triangulated category.
3.36.Definition.The Bousfield class of an object X ∈ C is the full subcategory of objects 3.37.Remark.We always assume that our categories are compactly-generated and hence there is a set of Bousfield classes [IK13, Theorem 3.1].
3.38.Remark.We let BL(C) denote the set of Bousfield classes of C. As is known, this has a lattice structure, which we now describe.We say that Hence, 0 is the minimum Bousfield class, and 1 is the maximum.The join is defined by i∈I X i = i∈I X i , and the meet is the join of all lower bounds.3.39.Proposition.The Bousfield lattice BL(Sp k,n ) is isomorphic to the lattice of subsets of Q via the map sending X to supp(X).
Proof.Define a map that sends T ⊆ Q to i∈T K(i) in BL(Sp k,n ).We claim that this gives the necessary inverse map.By the local-global principle (Proposition 3.30) we have In particular, X∧W ≃ 0 if and only if K(i)∧W ≃ 0 for all i ∈ supp(X), so that (3.5) The result then follows by direct computation.
3E.The telescope conjecture and variants.We begin by considering variants of the telescope conjecture in the localized categories Sp k,n using work of Wolcott [Wol15].
In particular, we have We also consider the following Bousfield localization on Sp k,n .

Definition. For
3.44.Remark.Following Wolcott [Wol15] we consider the following variants of the telescope conjecture on Sp k,n for i ∈ Q.
Every smashing localization is generated by a set of compact objects.SDGSC : Every smashing localization is generated by a set of dualizable objects.
Here LTC stands for the localized telescope conjecture, GSC is the generalized smashing conjecture, and SDGSC is the strongly dualizable generalized smashing conjecture.We emphasize the difference here because compact and dualizable objects do not coincide in Sp k,n when k = 0.
3.45.Proposition.On Sp k,n , we have that LTC1 i , LTC2 i and LTC3 i hold for all i ∈ Q.
Proof.By [Wol15, Theorem 3.12] it suffices to prove that LTC1 i holds, and by Proposition 3.39 this will follow if we show that L k,n Tel(i) and K(i) have the same support in Sp k,n .To see this, we have supp(K(i)) = {i} by Example 3.23, while supp(Tel(i)) = {i} by [Wol15, Lemma 2.10 and Lemma 3.7].
We now classify all smashing localizations on Sp k,n and show that all variants of the telescope conjecture hold.
3.46.Theorem.Let L be a non-trivial smashing localization functor on Sp k,n , then L ≃ l f j ≃ l j for some j ∈ Q.In particular, the GSC and SDGSC both hold in Sp k,n .
Proof.We closely follow [Wol15, Theorem 4.4].Throughout the proof we let 1 denote L k,n S 0 , the monoidal unit in Sp k,n , so that L = L1 .By (3.5) we have Note that supp(L1) is non-empty because we assume L = 0. Hence, we can fix j ∈ supp(L1) so that K(j) ≤ L(1) in BL(Sp k,n ).It follows that L K(j) L ≃ LL K(j) ≃ L K(j) , and By [HS99b, Proposition 5.3] we have L K(j) S 0 = j i=0 K(i) in BL(Sp), and it follows easily that L K(j) S 0 = We deduce that L1 = j i=k K(i) , where j = max{supp(L1)}, and hence by Proposition 3.45 that L ≃ l f j ≃ l j .Finally, because L n F (j + 1) is compact and therefore also dualizable in Sp k,n , both the GSC and SDGSC hold in Sp k,n .

theory and the E(n, J k )-local Adams spectral sequence
In this section we use descent theory to construct an Adams-type spectral sequence in the E(n, J k )-local category.Using descent, we shall see that this has a vanishing line at some finite stage.Moreover, for p ≫ n, we show that the E(n, J k )-local Adams spectral sequence computing π * L k,n S 0 has a horizontal vanishing line on the E 2 -page, and there are no non-trivial differentials.4A.Descendability.We begin with the notion of a descendable object in an algebraic stable homotopy category.4.1.Remark.We recall that in C there is an ∞-category CAlg(C) of commutative algebra objects, see [Lur17, Chapter 3].Moreover, given A ∈ CAlg(C) we can define a stable, presentable, symmetric monoidal ∞-category Mod A (C) of A-modules internal to C, with the relative A-linear tensor product [Lur17, Section 4.5].We will mainly focus on the case

Remark.
Note that E n ∈ CAlg(Sp) by the Goerss-Hopkins-Miller [GH04] theorem, and so E n ∈ CAlg(Sp k,n ) as well.On the other hand, E(n, J k ) will not, in general, be a commutative ring spectrum (for example, K(n) is never a commutative ring spectrum).

Definition. ([Mat16, Definition 3.18]) A commutative algebra object
One reason to be interested in descendable objects is the following [Mat16, Proposition 3.22].4.4.Proposition (Mathew).Let A ∈ CAlg(C) be descendable, then the adjunction C ⇆ Mod C (A) given by tensoring with A and forgetting is comonadic.In particular, the natural functor from C to the totalization / / / / / / is an equivalence.

Proposition (Mathew). If
4B. Morava modules and L-complete comodules.The following theorem, essentially due to Hopkins-Ravenel [Rav92], shows that the results of the previous section can be applied in Sp k,n .We note that E n ∈ Sp is a commutative algebra object; this is the Goerss-Hopkins-Miller theorem [GH04].It follows that E n ∈ CAlg(Sp k,n ).

4.6.
Theorem.E n ∈ CAlg(Sp k,n ) is descendable, and there is an equivalence of symmetricmonoidal stable ∞-categories We therefore define the following.
We recall that L k,n X ≃ lim ← − −j (L n X ∧ M k (j)) (Proposition 2.24).The Milnor sequence then gives the following.4.9.Lemma.There is a short exact sequence 4.11.Remark.As the short exact sequence shows, (E k,n ) ∨ * X is not always complete with respect to the I k -adic topology.However, it is always L I k 0 -complete in the sense of [HS99b, Appendix A] -this the same argument as given in [HS99b, Proposition 8.4(a)].
4C.The E(n, J k )-local E n -Adams spectral sequence.In this section we construct an Adams type spectral sequence in the E(n, J k )-local category.When k = 0, this is the E n -Adams spectral sequence, while when k = n this is the K(n)-local E n -Adams spectral sequence considered in [DH04, Appendix A].
To begin, we recall that the cobar (or Amitsur) complex for E n in Sp k,n is , it is the cohomology of the complex ). 4.13.Proposition.For any spectrum X there is a strongly convergent spectral sequence as used above is a relative Ext group in the category of L I k 0 -complete comodules.We will not use this in what follows, so we leave the details to the interested reader.4.15.Remark.In [HMS94, Section 7] the authors construct the K(n)-local Adams spectral sequence for dualizable K(n)-local X as the inverse limit of the E n -Adams spectral sequences for X ∧ M n (j).The following result recovers the identification of the E 2 -term in the case k = n.4.16.Proposition.Let M k (j) be a tower of generalized Moore spectra of height k, then there is an isomorphism Proof.By definition Ext ) by Proposition 2.24, and there is a corresponding Milnor exact sequence of the form We note that E ∧t n is Landweber exact, as the smash product of Landweber exact spectra (see for suitable integers i 0 , . . ., i k−1 .In particular, the maps in the tower are surjections by the construction of the tower {M k (j)} (see Remark 2.19), and so the lim ← − − 1 j -term vanishes, and Note that the cohomology of the complex We will see below in Corollary 4.22 that Ext q, * (En) * En (( ) is finite, and so the lim ← − − 1 j -term vanishes in the exact sequence, and the result follows.4.17.Remark.It follows that when k = 0, the groups Ext and showed that, as a consequence of descendability, this has a horizontal vanishing line at some finite stage.In the extreme cases of k = 0 and k = n it is known that when p ≫ n and X = S 0 , this vanishing line occurs on the E 2 -page, and occurs at s = n 2 + n and s = n 2 , respectively, see [HS99a, Theorem 5.1] and [Rav86, Theorem 6.2.10].In this section, we show (Theorem 4.24) that the analogous result occurs in general; for p ≫ n there is a vanishing line on the E 2 -page of the spectral sequence of Proposition 4.13 above s = n 2 + n − k in the case X = S 0 .The proof relies on a variant of the chromatic spectral sequence [Rav86, Chapter 5], which we now construct.Along the way we prove Corollary 4.22, which also completes the proof of Proposition 4.16.

Remark (The chromatic spectral sequence). Fix k ≤ n, and for
Applying [Rav86, Theorem A.1.3.2]there is then a chromatic spectral sequence of the form 4.19.Proposition.In the chromatic spectral sequence (4.2) we have If particular, if p − 1 does not divide k + s, we have E s,r, * Proof.This is similar to the proof by Hovey and Sadofsky [HS99a, Theorem 5.1], which is the case where k = 0. We first recall the change of rings theorem of Hovey and Sadofsky [HS99a, Theorem 3.1]; if M is a BP * BP -comodule, on which v j acts isomorphically, and n ≥ j, then there is an isomorphism Applying this change of rings theorem twice to the BP * BP -comodule . ., u ∞ k+s−1 ) with j = k + s and j = n shows that the E 1 -term has the claimed form.
For brevity, let us denote . By Morava's change of rings theorem, we have Along with an argument similar to that given by Hovey and Sadofsky's, using standard exact sequences and taking direct limits we find that Ext r, * (E k+s ) * (E k+s ) ((E k+s ) * , (E k+s ) * /I) = 0 for r > (k + s)2 as well.4.20.Remark.Let M k denote a generalized Moore spectrum of type k, then there is an obvious analog of this spectral sequence, whose abutment is ) for a suitable sequence (i 0 , . . ., i k−1 ) of integers.The result for the sequence (1, . . ., 1) holds by Proposition 4.19 and therefore in general by taking appropriate exact sequences.
Proof.Combine Proposition 4.16 and Corollary 4.23.4.25.Remark.The condition on the prime is always satisfied if p is large enough compared to n (in fact p > n + 1 suffices).This suggests the following, which we do not attempt to make precise: for large enough primes, the cohomological dimension of (E n ) * in a suitable category of (completed)-(E k,n ) ∨ * (E n )-comodules is finite, and equal to n 2 + n − k.We also have the following expected sparseness result.4.26.Proposition.Let q = 2(p − 1), then Ext s,t (En) ∨ * (En) ((E n ) * , (E n ) * ) = 0 for all s and t unless t ≡ 0 mod q.Consequently, in the spectral sequence of Proposition 4.13, d r is nontrivial only if r ≡ 1 mod q and E * , * mq+2 = E * , * mq+q+1 for all m ≥ 0. Proof.Using Proposition 4.16 it suffices to show the first statement for the E 1 -term of the chromatic spectral sequence of Proposition 4.19.Again using the Hovey-Sadofsky change of rings theorem this E 1 -term is isomorphic to

Dualizable objects in Sp k,n
In this section we use descendability to characterize the dualizable objects in Sp k,n .As noted previously, as long as k = 0, these differ from the compact objects studied in Section 3A.5.1.Definition.Let (C, ∧, 1) be a symmetric-monoidal ∞-category, then X ∈ C is dualizable if there exists an object D C X and a pair of morphisms are the identity on X and D C X, respectively.5.2.Remark.The definition makes it clear that X ∈ C is dualizable if and only if it is dualizable in the homotopy category of C.Moreover, a formal argument shows that, if it exists, we must have D C X ≃ F (X, 1).Finally, for the equivalence with other definitions of dualizability the reader may have seen, see [DP80, Theorem 1.3].
5.3.Definition.We let C dual ⊆ C denote the full subcategory consisting of the dualizable objects of C.
We have the following relationship between descent theory and dualizability.
5.5.Proposition.Let A ∈ CAlg(C) be descendable, then the adjunction C ⇆ Mod A (C) gives rise to an equivalence of symmetric monoidal ∞-categories Proof.The first claim follows from Proposition 4.4 because passing to dualizable objects commutes with limits of ∞-categories [Lur17, Proposition 4.6.1.11].The second is then an easy consequence, using that all the maps in the totalization are symmetric monoidal.
5A. Dualizable objects in the E(n, J k )-local category.Using Theorem 4.6 and Proposition 5.5 we deduce the following.5.6.Proposition.The adjunction Sp k,n ⇆ Mod En (Sp k,n ) gives rise to an equivalence of symmetric monoidal ∞-categories This proposition suggests we begin by studying dualizable objects in the category Mod En (Sp k,n ).Fortunately, these have a nice characterization.We begin with the following.
Proof.We first note that for any M ∈ Mod En (in particular, for M = X) the Bousfield localization L K(n) M is the spectrum underlying L En En∧X , where the latter denotes the Bousfield localization with respect to E n ∧ X internal to the category of E n -modules.In particular, the lo- Because X ∈ Mod En (Sp k,n ) dual , using [HPS97, Lemma 3.3.1],we see that there are equivalences 5.8.Remark.For the following, we let , and so K(n) = K n .We use this only because K n is naturally an E n -module.5.9.Proposition.For X ∈ Mod En (Sp k,n ) the following are equivalent: (1) X is dualizable in Mod En (Sp k,n ).
(3) The spectrum underlying X is K(n)-local, and the homotopy groups π * (K n ∧ En X) are finite.
Conversely, assume that (1) holds.As above we have a symmetric-monoidal localization which preserves dualizable objects (as any symmetric-monoidal functor does).Using Lemma 5.7 it follows that L Finally, the equivalence of (2) and ( 3) is well-known, see for example [HL17, Proposition 2.9.4].4 In other words, for dualizable X, there is an isomorphism . We now give our characterization of dualizable spectra in Sp k,n -see [HS99b, Theorem 8.6] for the case k = n.We note that even in this case our proof, which uses descendability, differs from that of Hovey and Strickland.5.11.Theorem.The following are equivalent for X ∈ Sp k,n : (1) X is dualizable.
Finally, we show that there is only a set of isomorphism classes of dualizable objects.5.12.Lemma.There are at most 2 ℵ0 isomorphism classes of objects in Sp dual k,n .Proof.This is the same as the argument given in [HS99b, Propositon 12.17].Namely, there are only countably many finite spectra X ′ of type at least k, and for each one [L n X ′ , L n X ′ ] is finite, so L n X ′ has only finitely many retracts.By Theorem 3.8 it follows that there is a countable set of isomorphism classes of objects in Sp ω k,n .If U and V are finite, then [U, V ] is finite, and so there are at most . Therefore, X is the inverse limit of a tower of spectra in Sp ω k,n , and hence there are at most 2 ℵ0 isomorphism classes of objects in Sp dual k,n .
5B.The spectrum of dualizable objects.In Theorem 3.16 we computed the thick subcategories (equivalently, thick tensor-ideals) of compact objects in Sp k,n .One could also ask for a classification of the thick tensor-ideals of dualizable objects in Sp k,n , or equivalently a computation of the Balmer spectrum Spc(Sp dual k,n ) (which is well-defined by Lemma 5.12).Based on a conjecture of Hovey and Strickland, the author, along with Barthel and Naumann, investigated Spc(Sp dual K(n) ) in detail in [BHN22], showing that the Hovey-Strickland conjecture holds when n = 1 and 2, and that in general it is implied by a hope of Chai in arithmetic geometry.In this section, we make some general comments regarding Spc(Sp dual k,n ).5.13.Remark.The following full subcategories were considered in the case k = n by Hovey-Strickland [HS99b, Definition 12.14].5.14.Definition.For i ≤ n, let D i denote the category of spectra X ∈ Sp dual k,n such that X is a retract of Y ∧ Z for some Y ∈ Sp dual k,n and some finite spectrum Z of type at least i.It is also useful to set D n+1 = (0).5.15.Remark.We note that D k ≃ Sp ω k,n ; this is a consequence of the characterization of compact objects given in Theorem 3.8, and that D 0 = Sp dual k,n .The following is [HS99b, Proposition 4.17] 5.16.Lemma.X is in D k if and only if X is a module over a generalized Moore spectrum of type k.Moreover, D k ⊆ Sp dual k,n is a thick tensor ideal.Hovey and Strickland conjecture that in the case k = n these exhaust the thick-tensor ideals of Sp dual K(n) .This has been investigated in detail in [BHN22].We conjecture this holds more generally in Sp k,n .5.17.Conjecture.If C is a thick tensor-ideal of Sp dual k,n , then C = D i for some 0 ≤ i ≤ n + 1. Equivalently, Spc(Sp dual k,n ) = {D 1 , . . ., D n+1 } with topology determined by the closure operator {D i } = {D j | j ≥ i}.
In this section we show that if Conjecture 5.17 holds K(n)-locally, i.e., for Sp dual n,n , then it holds for all Sp dual k,n .We first recall the following definition.5.18.Definition.Suppose F : K → L is an exact tensor triangulated functor between tensortriangulated categories.We say that F detects tensor-nilpotence of morphisms if every morphism f : X → Y in K such that F (f ) = 0 satisfies f ⊗m = 0 for some m ≥ 1.
We will use the following.5.19.Proposition.Suppose A ∈ CAlg(C) is descendable, then extension of scalars C → Mod A (C) detects tensor-nilpotence of morphisms.
Proof.Let I denote the fiber of 1 η − → A, and let ξ : I → 1 denote the induced map.By [MNN17, Proposition 4.7] if A is descendable, then there exists m ≥ 1 such that I ⊗m → 1 is null-homotopic, i.e., ξ is tensor-nilpotent.We can now argue as in (ii) implies (iii) of [Bal16]: suppose we are given f : X → Y a morphism in C with A ⊗ f : A ⊗ X → A ⊗ Y null-homotopic.Now consider the diagram of fiber sequences: We see that (η ⊗ id Y )f is null-homotopic, so f factors through ξ ⊗ id Y , which is tensor-nilpotent.
The following is our key observation.
5.20.Proposition.Let i > k, then the map induced by localization Proof.By [Bal18, Theorem 1.3] it suffices to show that the functor L i,n : Sp dual k,n → Sp dual i,n detects tensor-nilpotence of morphisms.To that end, let f : X → Y be a morphism in Sp dual k,n with L i,n (f ) = 0, so that we must show f ∧m = 0 for some m ≥ 1.Because E n ∈ Sp k,n is descendable, Proposition 5.19 shows that detects tensor-nilpotence of morphisms, and hence so does its restriction to dualizable objects, i.e., L k,n (E n ∧ f ) = 0 =⇒ f ∧m = 0 for some m ≥ 1.In other words, it suffices to show that L k,n (E n ∧ f ) is trivial.By Lemma 5.7 however, this is a morphism in Mod En (Sp n,n ).In particular,

The Picard group of the Sp k,n -local category
In this section we study invertible objects in the Sp k,n -local category.We show that invertibility of an object can be detected by its Morava module.We construct a spectral sequence computing the homotopy groups of the Picard spectrum of Sp k,n and use this to show that if p is large compared to n, then the Picard group of Sp k,n is entirely algebraic, in a sense we make precise.
6A. Invertible objects and Picard spectra.We recall that if C is a symmetric monoidal category we denote by Pic(C) the group of isomorphism classes of invertible objects; a priori this could be a proper class, but if C is a presentable stable ∞-category (which it will always be in our cases), then it is actually a set [MS16, Remark 2.1.4].
The following standard lemma will be useful for us.Here we write D C (X) for the dual of an object in a category C, i.e., D C (X) = F (X, 1).Note that an invertible object is always dualizable [HPS97, Proposition A.2.8].
6.1.Lemma.Let F : C → D be a symmetric-monoidal conservative functor between stable ∞categories, then X ∈ C is invertible if and only if F (X) ∈ D is invertible.
Proof.We first note that X is invertible if and only if the natural morphism X ⊗ C D C (X) → 1 C is an equivalence, see [HPS97, Proposition A.2.8].Because F is assumed to be symmetric monoidal and conservative, this is an equivalence if and only if it is so after applying F , i.e., if and only if ), as symmetricmonoidal functors preserve dualizable objects, and the result follows.6.2.Remark.To our symmetric monoidal category C we can instead associate the Picard spectrum pic(C) [MS16, Definition 2.2.1]; this is a connective spectrum with the property that The key advantage of using the Picard spectrum is that, as a functor from the ∞-category of symmetric monoidal ∞-categories to the ∞-category of connective spectra, pic commutes with limits [MS16, Proposition 2.2.3].6.3.Example.Let C be a category, A ∈ CAlg(C).Then the Picard spectrum of the category Mod A (C) of A-modules internal to C satisfies This follows because A is the tensor unit in Mod A (C). Indeed, writing F : C → Mod C (A) for the extension of scalars functor (so that A ≃ F (1 C )), we have Proof.We always have Pic(Mod En ) ⊆ Pic(Mod En (Sp k,n )) because any invertible E n -module is compact, and hence E(n, J k )-local.The other inclusion follows if any M ∈ Pic(Mod En (Sp k,n )) is compact as an E n -module.Such an M is automatically dualizable in Mod En (Sp k,n ), and hence compact in Mod En by Proposition 5.9.This gives the first of the above isomorphisms, and the others hold by work of Baker and Richter [BR05].
We now give criteria for when X ∈ Pic k,n is invertible.This (partially) extends work of Hopkins-Mahowald-Sadofsky [HMS94], who considered the case k = n.6.8.Theorem.Let X ∈ Sp k,n , then the following are equivalent.
(1) X ∈ Pic k,n . ( The equivalence of (1) and (2) follows from Corollary 4.7 and Lemma 6.1, while the equivalence of (2) and (3) follows from Lemma 6.7, which also shows that (3) implies (4).Finally, to see that (4) implies (3), we note that if M is any E n -module whose homotopy groups are free of rank one over (E n ) * , then M is equivalent to a suspension of E n (for the elementary proof, see [HS14, Proposition 2.2]).Thus, (4) implies that E n ∧X ≃ E n , up to suspension, and hence (3) holds.6.9.Remark.When n = 1, there are two possibilities, the K(1) and E(1)-local Picard groups, both of which are known.We record the results here: These are due to Hovey-Sadofsky [HS99a] and Hopkins-Mahowald-Sadofsky [HMS94] respectively.
Proof.The morphism in question factors through the morphism Pic 0,2 → Pic 2,2 and so it suffices to show that this is an injection.When p > 3 this is clear, and so we focus on the case p = 3.In this case, the calculations of Goerss-Henn-Mahowald-Rezk [GHMR15] show that this map is an injection.
6.11.Remark.As noted in the proof the interesting case in the above propsition is the case p = 3.
In fact, for all n and p ≫ n we have that Pic 0,n → Pic i,n is an injection for i ≥ 0. However, here Pic 0,n ∼ = Z (by [HS99a]), so this is not particularly helpful.
6C. Descent and Picard groups.In Remark 6.2 we recalled that we can associate a connective Picard spectrum pic(C) to a symmetric-monoidal ∞-category C. Using descent for the E(n, J k )-local category, we now construct a spectral sequence whose π 0 computes Pic k,n .We need to introduce another algebraic gadget to describe the spectral sequence.
6.12.Definition.We let Pic alg k,n denote the first cohomology of the complex / / / / / / • • • induced by taking the units in degree 0 in the cobar complex.6.13.Theorem.There exists a spectral sequence with In particular, when t = s, this computes Pic k,n .The differentials in the spectral sequence run d r : E s,t r → E s+r,t+r−1 r .
Proof.Because pic commutes with limits (Remark 6.2), Theorem 4.6 implies that / / / / / / (6.1) We have (compare Example 6.3) The Bousfield-Kan spectral sequence associated to (6.1) has the form / / / / / / By (6.2) when t ≥ 2, the spectral sequence is just a shift of the E(n, J k )-local Adams spectral for X = S 0 sequence considered in Proposition 4.13.When t = 0 and i = 0, we have Pic(Mod En (Sp k,n )) ∼ = Z/2 by Lemma 6.7.We do not know the higher terms, but this does not matter as only the Z/2 is relevant for the s = t = 0 part of the spectral sequence.
Finally, we consider the t = 1 part of the spectral sequence.Again using (6.2), we have By definition, when s = 1 this is Pic alg k,n .
6.14.Remark.The proof shows that when t = 1, we can compute E s,1 2 as the s-th cohomology of the complex in Definition 6.12.However, unless k = 0 or n we do not have a convenient description of this group (for the case k = n, see Example 6.18 below).6.15.Definition.We will say that Pic k,n is algebraic if the only contributions come from the s = 0 and s = 1 lines of the spectral sequence.6.16.Remark.The E 0,0 2 term of the spectral sequence always survives the spectral sequence, as it is the Picard group of E n -modules.It is however possible that there is a non-trivial differential in the E 1,1 r spot.
By Theorem 4.24 and the assumption that p − 1 does not divide k + s for 0 ≤ s ≤ n − k we have that E s,s 2 = 0 for s > n 2 + n − k.Therefore, if additionally 2p − 2 ≥ n 2 + n − k there can be no non-algebraic contributions to the spectral sequence.

6.18.
Example.Let us spell out the details in the case k = n.We first claim that the spectral sequence of Theorem 6.13 takes the follows form: The identification is much as in Remark 4.14.For the t = 1 term, we note that π ).The existence of such a spectral sequence is folklore, see [DGI11, Remark 6.10] or [Pst18, Remark 2.6].In fact, the latter also proves Theorem 6.17 in the case k = n.

E(n, J k )-local Brown-Comenetz duality
We recall the classical definition of Brown-Comenetz duality.The group Q/Z is an injective abelian group, and so the functor X → Hom(π 0 X, Q/Z) defines a cohomology theory on spectra represented by a spectrum I Q/Z ; this is the Brown-Comentz dual of the sphere.The Brown-Comenetz dual of a spectrum X is then defined as I Q/Z X := F (X, I Q/Z ), and satisfies It is an insight of Hopkins [HG94] that there is a good notion of Brown-Comenetz duality (also known as Gross-Hopkins duality) internal to the K(n)-local category, given by defining I n X = F (M n X, I Q/Z ) for a K(n)-local spectrum X.For details on this, see [Str00].As we will see, this definition can also naturally be made in the E(n, J k )-local category.We begin with the following generalization of a result of Stojanoska [Sto12, Proposition 2.2].We recall that, by definition, M 0,n = L n .In this case, the following lemma just says that F (L n X, Y ) is already L n -local.7.1.Proposition.For any X, Y the natural map F (L n X, Y ) → F (M k,n X, Y ) is E(n, J k )localization.
Proof.We can repeat Stojanoska's argument.First, we show that F (M k,n X, Y ) is E(n, J k )-local.Indeed, let Z be E(n, J k )-acyclic, then we must show that is contractible.Here we have used that M k,n is smashing.But M k,n Z ≃ M k,n L k,n Z by Theorem 2.31 and this is contractible because Z is E(n, J k )-acyclic.
We now show that the fiber F (L k−1 X, Y ) is E(n, J k )-acyclic.By Proposition 2.15 it suffices by a localizing subcategory argument to show this after smashing with a generalized Moore spectrum M (k) of type k.Then (up to suspension), as required.7.2.Definition.The E(n, J k )-local Brown-Comenetz dual of X is I k,n X = I Q/Z (M k,n X) = F (M k,n X, I Q/Z ).We let I k,n denote the E(n, J k )-local Brown-Comenetz dual of L k,n S 0 .7.3.Remark.It does not matter if we ask that X be E(n, J k )-local in the previous definition, as I k,n X only depends on the E(n, J k )-localization of X.Indeed, we have equivalences In particular, I k,n = I k,n (L k,n S 0 ) ≃ I k,n S 0 .
From the definition of I Q/Z , we deduce the following.

Lemma.
There is a natural isomorphism As a consequence of Proposition 7.1 we deduce the following.7.5.Lemma.I k,n X is always E(n, J k )-local.In fact, I k,n X ≃ L k,n I Q/Z (L n X) and moreover, I k,n X ∼ = L k,n I j,n X for any j ≤ k.
It follows that we have natural maps given by localization: 7.6.Example.Let n = 1 and p > 2, then I 0,1 ≃ L 1 (S 2 p ), the localization of the p-completion of S 2 .On the other hand, when p = 2 we have I 0,1 ≃ Σ 2 L 1 (DQ ∧ 2 ) where DQ is the dual question mark complex [Dev90, Remark 1.5].Similarly, I 1,1 ≃ L K(1) S 2 if p > 2, while I 1,1 ≃ Σ 2 L K(1) DQ. 7.7.Example.We always have I k,n (K n ) ≃ K n .Indeed, first note that L k,n K n ≃ K n , so Lemma 7.5 allows us to reduce the case where k = 0 (although the proof is no more difficult in the other cases -just note that M k,n K n ≃ K n ).Using the fact that we argue as in [HS99b, Theorem 10.2(a)] to see that This implies that I 0,n (K n ) ≃ K n , as claimed.7.8.Theorem.Let X ∈ Sp k,n , then the natural map X → I 2 k,n X is an isomorphism when π * (F (k) ∧ X) ∼ = π * (L k,n F (k) ∧ X) is finite in each degree.In particular, this holds for X = L k,n S 0 .
Proof.Let κ X : X → I 2 k,n X denote the natural map.We first note that I 2 k,n (F (k)∧X) ≃ I 2 k,n (F (k) ∧ X) ≃ F (k) ∧ I 2 k,n (X), because F (k) is compact (and hence dualizable) in Sp.As in [HS99b, Proof of Theorem 10.2] this identifies κ F (k)∧X ≃ id F ∧ κ X , and so it is enough to show that κ Y is an equivalence, where Y = F (k) ∧ X.
Because The equivalence of (1) and (3) is just Theorem 5.11.Finally to see that (1) ⇐⇒ (4) we note that it suffices to show that (K n ) * I k,n is finite.In fact, because (K n ) * is a graded field, it suffices to see that (K n ) * I k,n is finite.For this, we compute using Example 7. 7.12.Remark.Condition (4) clearly holds in the case k = n.Of course, Proposition 7.10 is precisely Hovey and Strickland's proof in this case.However, due to the p-completion this does not hold at n = 1 and k = 0 (Example 7.6).In fact, this fails at all heights when k = 0, as we now explain.7.13.Remark.Fix n > 1, k = 0 and take p ≫ n.Then Pic 0,n ∼ = Z, generated by L n S 1 [HS99a, Theorem 5.4].Therefore, if Proposition 7.10 held for k = 0 we must have I 0,n ≃ L n S k for some k ∈ Z.On the other hand, work of Hopkins and Gross [HG94], as written up by Strickland [Str00], and known results about the K(n)-local Picard group [HMS94, Proposition 7.5] show that I n,n ≃ Σ n 2 −n S det , where S det is the determinant sphere spectrum [BBGS22].It cannot then be the case that L K(n) I 0,n ≃ I n,n , a contradiction to Lemma 7.5.We do not know what occurs in the cases k = 0, n.
3.26.Remark.In [HS99b, Definition 6.7] Hovey and Strickland define the support of an object by supp HS

3. 35 .
Corollary.The map that sends a localizing subcategory C of Sp k,n to C ⊥ induces a bijection Localizing subcategories of Sp k,n

3. 40 .
Definition.For i ∈ Q, let l f i : Sp k,n → Sp k,n denote finite localization away from L k,n F (i+ 1).3.41.Remark.Because L k,n F (i + 1) is in Sp ω k,n by Theorem 3.8, this is a smashing localization.3.42.Remark.By [Wol15, Proposition 3.8] we have an equivalence of endofunctors of Sp k,n (recall that Tel(n) the Bousfield class of a telescope of a finite type n spectrum)

F
(k) has type k, L k−1 F (k) ≃ * , and M k,n F (k) ≃ L n F (k), so that M k,n Y = M k,n (F (k) ∧ X) ≃ M k,n F (k) ∧ X ≃ L n F (k) ∧ X ≃ Y .Likewise, M k,n (I k,n Y ) ≃ M k,n (DF (k) ∧ I k,n X) ≃ DF (k) ∧ I k,n X ≃ I k,n Y .This implies that π * I 2 k,n Y ∼ = Hom(Hom(π * Y, Q/Z), Q/Z), which is the same as π * Y because π * Y is finite in each degree.Therefore κ Y is an equivalence, as required.7.9.Remark.The Gross-Hopkins dual I n,n is always an invertible K(n)-local spectrum.We do not know what happens for I k,n in general, however we note the following result.7.10.Proposition.The following are equivalent:(1)I k,n ∈ Sp dual k,n (2) I k,n ∈ Pic k,n (3) (E k,n ) ∨ * (I k,n ) is a finitely generated E * -module.(4) E n ∧I k,n is K(n)-local.Proof.Suppose first that (1) holds.Then, F (I k,n , I k,n ) ≃ DI k,n ∧I k,n , but on the other handF (I k,n , I k,n ) ≃ I 2 k,n (L k,n S 0 ) ≃ L k,n S 0 by Theorem 7.8.It follows that I k,n ∈ Pic k,n , i.e., that (2) holds.The converse, (2) =⇒ (1), always holds, see for example [HPS97, Proposition A.2.8].
7 and Theorem 7.8[I k,n , K n ] * ≃ [I k,n , I k,n (K n )] * ≃ [I k,n , F (K n , I k,n )] * ≃ [K n , F (I k,n , I k,n )] * ≃ [K n , L k,n S 0 ] * .By[HS99b, Lemma 10.4] if M n denotes a generalized Moore spectrum of type n, then[E n , L K(n) M n ] * ≃ [E n , L k,n M n] * is finite (the last equivalence follows, for example, from the fact that L n M n ≃ L K(n) M n for a generalized Moore spectrum of type n).As in [HS99b, Corollary 10.5] it follows that [E n ∧DM n , L k,n S 0 ] is finite, and hence so is [K n , L k,n S 0 ], as K n lies in the thick subcategory generalized by E n ∧DM n (note that DM n is also the localization of a generalized Moore spectrum of type n, cf.[HS99b, Proposition 4.18]).7.11.Question.For which values of k and n do the conditions of Proposition 7.10 hold?
The inclusion Sp k,n ֒→ Sp has a left adjoint adjoint L k,n , and Sp k,n is a presentable, stable ∞-category.Proof.This is a consequence of [Lur09, Proposition 5.5.4.15].2.7.Remark.The category Sp k,n and localization functor L k,n only depend on the Bousfield class , and the result follows.2.30.Definition.Let M k,n denote the essential image of the functor M k,n : Sp → Sp.2.31.Theorem.For any spectrum X we have natural equivalences M which are clearly (co)localizing subcategories of Sp k,n .
are profinite, i.e., either finite or uncountable.Contrast the case k = 0, where Ext s,t (En) * En ((E n ) * , (E n ) * ) is 21. Remark.The following completes the proof of Proposition 4.16.4.22.Corollary.Let M k denote a generalized Moore spectrum of type k, then the group By taking appropriate exact sequences it suffices to show this for (E n ) * /I k (alternatively, one can argue directly using the spectral sequence of Remark 4.20).Given the chromatic spectral sequence, we can reduce to showing that H r (G k+s , (E k+s ) * /I) is finite, with I as in the proof of the previous proposition.For this, see [SW00, Proposition 4.2.2].4.23.Corollary.Let M k denote a generalized Moore spectrum of type k, then if p − 1 does not divide k + s for 0 ≤ s ≤ n − k we have * , (E n ) * (M k )) is finite.Proof.
an injection, and hence a bijection by Proposition 5.20.Using that Spc(L n−1,n ) is continuous and the topologies on each space, we see that it is fact a homeomorphism.It follows that Spc(Sp dual n−1,n ) → Spc(Sp dual 0,n ) is a homeomorphism, and we can now repeat the argument.
Invertible objects in the Sp k,n -local category.Our main group of interest is the Picard group local spectra.6.5.Remark.By [KS07, Lemma 2.2] the localization functors induce natural morphisms Pic 0,n → Pic 1,n → • • • → Pic n,n .6.6.Remark.The morphism X → E n ∧X induces a functor Pic k,n → Pic(Mod En (Sp k,n )).We can fully understand the latter Picard group.6.7.Lemma.For all 0 ≤ k ≤ n we have Pic(Mod En