Derived completion for comodules

The objective of this paper is to introduce and study completions and local homology of comodules over Hopf algebroids, extending previous work of Greenlees and May in the discrete case. In particular, we relate module-theoretic to comodule-theoretic completion, construct various local homology spectral sequences, and derive a tilting-theoretic interpretation of local duality for modules. Our results translate to quasi-coherent sheaves over global quotient stacks and feed into a novel approach to the chromatic splitting conjecture.


Introduction
Completion of non-finitely generated modules is pervasive throughout stable homotopy theory, as amply demonstrated in [GM97], for example.The goal of this paper is to study the local homology of comodules over Hopf algebroids, which has come into recent focus due to its central role in an algebraic approach to Hopkins' chromatic splitting conjecture [BBP18].Algebraically, this extends the work of Greenlees and May [GM92] on derived functors of completion, and in geometric terms it is akin to the passage from affine schemes to quotient stacks.However, while local cohomology admits a canonical and well-behaved extension from modules over commutative rings to comodules over Hopf algebroids, the corresponding generalization of local homology is considerably more complicated.
This complication is already visible at the non-derived level: Unlike the case of modules, for a Hopf algebroid (A, Ψ) the naive completion C I (−) = lim k (A/I k ⊗ −) at an ideal I ⊆ A does not usually define an endofunctor on the category of comodules Comod Ψ , but rather takes values in a category of completed comodules [Dev95].To remedy this, one has to replace the limit lim k of the underlying A-modules by the inverse limit in comodules, which leads to a comodule completion functor C I Ψ .We thus begin in Section 1 with an analysis of these non-derived completion functors and in particular the relation between C I and C I Ψ .Given an inverse system of Ψ-comodules, the key problem thus becomes to compare the comodule limit with the underlying module limit, and our first result provides conditions under which the former can be computed from the latter.This motivates the introduction of a class of Hopf algebroids which we call true-level (with respect to the ideal I), see Definition 1.10.We then use a theorem of Enochs to deduce concrete conditions that imply the true-level property; a particular example of a true-level Hopf algebroid highly relevant for applications to stable homotopy theory is given by (A, Ψ) = (E * , E * E) for a variant E of Johnson-Wilson theory due to Baker [Bak00].Note that we always write ⊠ for the underived tensor product.
Let N be a complete Ψ-comodule.If (A, Ψ) is a true-level Hopf algebroid with respect to I, then we prove that the A-module ιN defined by the following pullback square acquires a natural structure as a Ψ-comodule.Here the natural map ιN → N is injective and ιN is the largest possible A-submodule of N that carries a natural Ψ-comodule structure.We note that ι roughly speaking plays the role of a (non-existent) right adjoint to C I on the category of comodules.
Theorem A (Theorem 1.13).If (A, Ψ) is true-level, then for any Ψ-comodule M , there is an equivalence of Ψ-comodules C I Ψ (M ) ≃ ιC I (M ).We then move on to a study of derived completion.For a suitable Hopf algebroid (A, Ψ) and ideal I ⊆ A we construct a local homology functor for comodules Λ I .Our construction is dictated by the general local duality framework of [BHV18], and Section 2 studies the properties of the resulting functors.In particular, we work with a suitable enlargement Stable Ψ of the derived category of comodules with some desirable categorical properties; geometrically speaking, this corresponds to the passage from quasi-coherent to ind-coherent sheaves.
One of the first new phenomena we encounter is that the local homology of a comodule can be non-zero both in positive and negative degrees, which may be interpreted as a measure of the stackyness of the Hopf algebroid under consideration.Consequently, the relation between derived functors of completion and local homology turns out to be more subtle.
For an arbitrary Hopf algebroid we construct a spectral sequence of the form E p,q 2 = lim p Ψ,k Tor Ψ q (A/I k , M ) =⇒ H q−p (Λ I (M )) computing the local homology of a comodule M in terms of more familiar functors.In parallel to the equivalence of local homology with I-adic completion when restricted to finitely presented modules, if A is Noetherian and M is a compact comodule, we show that this spectral sequence collapses to yield ≃ / / lim s Ψ,k M ⊠ A/I k for all s ≥ 0.Moreover, we give an example to show that, contrary to the case of A-modules, the functors H * (Λ I (M )) cannot, in general, be computed by the left or right derived functors of C I Ψ (−).
In the case of Morava E-theory E = E n , we construct a spectral sequence computing the Ehomology of the K(n)-localization of a finite spectrum from the local homology of E * X, where m is the maximal ideal of E * : Theorem B (Theorem 2.32).For X ∈ Sp E compact, i.e., the E-localization of a finite spectrum, there exists a strongly convergent spectral sequence of E * E-comodules abutting to the uncompleted E-homology of L K(n) X.
This spectral sequence is the crucial ingredient in relating algebraic results about local homology in Comod E * E to their homotopical applications, e.g., the chromatic splitting conjecture.
In the case of a discrete Hopf algebroid (A, A) there is a natural equivalence Stable A ≃ D(A).We can thus use the relationship between Λ I and the left-derived functors of completion (known as the derived functors of L-completion) to prove some results about the derived functors of C I on Mod A .We also produce a criterion for when an A-module is L-complete, i.e., for when M is in the category Mod A of L-complete A-modules, generalizing Bousfield and Kan's Ext-p completeness criterion.
Theorem C (Ext-I completeness criterion, Theorem 2.28).Let A be a commutative ring and I ⊆ A an ideal generated by a regular sequence x 1 , x 2 , . . ., x n .If M is an A-module, then M is L-complete ⇐⇒ Ext q A (x −1 i A/(x 1 , . . ., x i−1 ), M ) = 0 for all 1 ≤ i ≤ n and all q ≥ 0. In fact, this is a consequence of a more general result that characterizes those M ∈ Stable Ψ for which Λ I M ≃ M , see Corollary 2.27.
In the final section, we turn to torsion and complete objects in derived categories of comodules.A priori, there are at least three different notions of what it could mean for an object M ∈ D(Ψ) to be torsion with respect to an ideal I ⊆ A: (1) M is in the smallest localizing ideal of D(Ψ) generated by A/I, denoted D I−tors (Ψ).
(2) M is in the image of the canonical functor from the derived category of the abelian category of I-torsion Ψ-comodules, denoted D(Comod I−tors
(3) The homology groups H n M are I-torsion Ψ-comodules for all n ∈ Z.
One gets analogous definitions for complete objects by replacing localizing with colocalizing and I-torsion with I-complete where appropriate.
The goal of Section 3 is to compare these notions and use this to prove a tilting-type equivalence between torsion and complete objects in D(Ψ).When working with comodules, the difficulties intrinsic to complete objects persist at the level of the derived category; while we can show that the three notions above coincide in the case of torsion objects, we can only conclude the same for complete objects when working over a discrete Hopf algebroid.
Theorem D (Theorem 3.6).Let (A, Ψ) be an Adams Hopf algebroid and I ⊆ A a finitely generated invariant ideal.Recall that the abelian categories of torsion and L-complete modules are not equivalent in general; for example, the former is Grothendieck while the latter is not.Nonetheless, we deduce from the previous theorem the following tilting-theoretic interpretation of local duality for commutative rings.

Theorem E. For any commutative ring A and I ⊆ A a finitely generated ideal, local homology and local cohomology induce mutual inverse symmetric monoidal equivalences
where D( Mod A ) denotes the right completion of D − ( Mod A ).
Once again, in the case of a general Hopf algebroid we were unable to obtain such a result.Indeed, there seems to be no good candidate for a derived category fitting the right hand side in the equivalence above.
Conventions.We always assume that our Hopf algebroids are flat.Moreover, we will write ⊠ for the underived tensor product of comodules and ⊗ for the derived tensor product.We will denote the internal hom in a category by Hom.For a cocomplete category C, we let C ω denote the full subcategory of compact objects in C.
We work with ∞-categories, in particular the quasi-categories of Lurie and Joyal throughout this document [Lur09,Lur17].Unless otherwise noted, all functors between stable ∞-categories are assumed to be exact and all subcategories of stable ∞-categories are assumed to be stable subcategories.We follow the convention of [Lur09] and say that a functor between presentable stable ∞-categories is continuous if it preserves filtered colimits.
Acknowledgments.We would like to thank Andrew Blumberg, Mark Hovey, Henning Krause, and Hal Sadofsky for helpful discussions related to this work.
A preliminary version of the results in the first two sections of the present paper was previously contained in the author's joint work [BHV18], while the equivalence between D I−cmpl (A) and a suitable derived category of complete comodules was also considered (via different methods) in the third author's PhD thesis [Val15].
The first author was supported by the Danish National Research Foundation Grant DNRF92 and the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 751794, and acknowledges the hospitality of the Newton Institute in Cambridge, UK, where part of this work was carried out.The second author thanks Haifa University for its hospitality.

Completion for comodules
In this section we study the completion functor for comodules.As we shall see, this differs from the A-module completion functor, as in general the forgetful functor to A-modules does not preserve limits.Nonetheless, under suitable conditions we show the comodule completion functor is the composite of the A-module completion functor and a functor ι that is defined by a certain pullback diagram, which, informally speaking, extracts the largest possible subcomodule of the A-module completion functor.
Remark 1.1.Hovey realized that, under suitable conditions, the comodule product can be defined as the largest possible subcomodule of the A-module product; see the remark after Proposition 1.2.2 of [Hov04].An alternative proof (under slightly more general conditions) is given in the thesis of Sitte [Sit14], and our approach follows his closely.Similar ideas are contained in unpublished work of Sadofsky.
1.1.Limits.Since the forgetful functor ǫ * : Comod Ψ → Mod A from comodules to modules does not preserve arbitrary limits (indeed, it does not even preserve products), the existence of limits in the category of comodules is not immediate.Hovey has shown that the category of comodules is complete [Hov04, Prop.1.2.2], by constructing the product of a system of comodules.Following his argument, we explain briefly how to construct the inverse limit of a system of comodules.
The first step is to define inverse limits for extended comodules, where an adjointness argument shows that, for an inverse system (N k ) of A-modules, we must have where we write lim Ψ,k (−) for the limit in Comod Ψ .One can then construct lim Ψ,k (f ), where f is a map of extended comodules.For a general inverse system (M k ) of comodules, there are exact sequences of comodules where T k is the cokernel of the coaction map of M k .This enables us to construct the inverse limit of (M k ) as lim Ψ,k (M k ) = ker(lim Ψ,k (f k )), see [BHV18,Sec. 4.1] for details.
Lemma 1.3.Let (M k ) be an inverse system of comodules, then the natural morphism of A- is an injection when Ψ is a projective A-module.
Proof.From the discussion above we know that lim Ψ,k (M k ) is given by the kernel where T k is the cokernel of the coaction map of M k .We then have a commutative diagram of A-modules (where we omit writing ǫ * for simplicity) which induces the natural map τ .This is an injection provided the two right hand vertical arrows are, which we claim is true when Ψ is a projective A-module.Indeed, it is not hard to check that the corresponding statement is true for products, and we can then write the inverse limit as a kernel of maps between the product to deduce the claimed result.
The next result shows that, under certain conditions, the inverse limit in comodules can be determined by first taking the inverse limit of the underlying modules and then extracting a subcomodule using a pullback.Proposition 1.4.Suppose (M k ) is an inverse system of comodules such that the canonical maps are monomorphisms, then the inverse limit in comodules, lim Ψ,k (M k ), can be computed by the following pullback of A-modules: Proof.Let P be the pullback of the span part of (1.5).We will omit the standard verification that P naturally admits the structure of an Ψ-comodule.Thus, it remains to show that P satisfies the universal property of the limit.So, suppose we have a comodule N , along with compatible comodule morphisms f k : N → M k for all k.By the universal property of the inverse limit in A-modules, we obtain an A-module morphism f : N → lim k (M k ), and a diagram: One can check that this diagram commutes, and so we obtain a (unique) morphism g : N → P , which can be shown to be a morphism of comodules.The morphism π i : P → M k is the composite of i : P → lim k (M k ) and the A-module projection maps; once again, these can be checked to be comodule morphisms making the required diagrams commute.
Remark 1.6.In the case of products a detailed proof of the corresponding statement can also be found in the PhD thesis of Sitte [Sit14, Lem.3.5.12].1.2.Completion.Let (A, Ψ) be a Hopf algebroid and fix a finitely generated invariant ideal I ⊆ A. Undecorated notation will usually refer to the module-theoretic as opposed to comoduletheoretic constructions.
Definition 1.7.The I-adic completion As an application of Lemma 1.3, we obtain the following coarse comparison between the two notions of completion.
Lemma 1.8.For M ∈ Comod Ψ there is a natural morphism of A-modules This is an injection when Ψ is a projective A-module.
If (A, Ψ) is discrete, then C I Ψ = C I A , but they differ in general.In contrast to C I Ψ , the functor C I does not in general take values in the category of comodules again, because the completed coaction map takes values in a completed tensor product.Following Devinatz [Dev95], we therefore introduce a category of complete comodules over a complete Hopf algebroid.Suppose that A is a k-algebra, and let A be the I-adic completion of A. We write ⊠ for the completed tensor product.Let Ψ be a complete commutative A-biunital ring object with left and right units η L , η R : A → Ψ, along with the following k-algebra homomorphisms: satisfying the usual identities for a Hopf algebroid.The triple Ψ = ( A, Ψ, I) together with the above structure maps is called a complete Hopf algebroid if Ψ is flat as a left A-module and the ideal Definition 1.9.A (left) complete Ψ-comodule M is a complete A-module M together with a left A-linear map ψ = ψ M : M → Ψ ⊠ A M which is counitary and coassociative.A morphism of complete Ψ-comodules is, as usual, a morphism of complete modules that commutes with the structure maps.We will write Comod c Ψ for the category of complete Ψ-comodules.Suppose now that (A, Ψ) is a Hopf algebroid, then for any finitely generated invariant ideal I, the completion ( A, Ψ, I) = (C I (A), C I (Ψ), I • C I (A)) is a complete Hopf algebroid.Inspired by Proposition 1.4, we consider the functor ι : Comod c Ψ → Mod A defined on N ∈ Comod c Ψ by the pullback diagram Informally speaking, ιN extracts the largest subcomodule of N ; however, it not clear that the map i : ιN → N is injective nor that ιN admits a natural Ψ-comodule structure.We therefore introduce a type of Hopf algebroid for which these problems do not arise.In the next subsection, we exhibit a sufficient criterion for verifying these conditions and provide an example.
By abuse of notation, for a true-level Hopf algebroid (A, Ψ), we will denote the functor Comod c Ψ → Comod Ψ given in Lemma 1.11 by ι as well.Proof.Consider the following diagram of A-modules and note that since Ψ is flat, the square part is a pullback.Hence, to obtain a candidate ρ for the coaction map of ιN it is enough to show that the outer part of (1.12) commutes; we may do this after composing with the monomorphism j ′ : Ψ ⊠ Ψ ⊠N → Ψ ⊠ Ψ ⊠N .A routine diagram chase then yields the desired commutativity.A further careful diagram chase, using the fact that N → Ψ ⊠N is counital and coassociative, shows that ρ is indeed a coaction for ιN , so that ιN is a Ψ-comodule.Finally, since j is assumed to be injective, it follows that i : ιN → N is injective.
We are now ready to prove the main result of this section.

Theorem 1.13. If (A, Ψ) is true-level, then there is an isomorphism C
while the right square is a pullback diagram by definition of ι.Using the natural maps between these diagrams, the natural isomorphism of A-modules then furnishes a natural isomorphism C I Ψ (M ) ∼ = ιC I (M ) between pullbacks of Amodules.Finally, the argument of Lemma 1.11 shows that the canonical comparison isomorphism is compatible with the Ψ-coactions on both sides.
Remark 1.14.The ad hoc construction of ι in this section can be understood more conceptually in the setting of Ψ-comodules with L-complete underlying A-modules, which form a category Comod Ψ .Since L-complete A-modules are an abelian subcategory of Mod A , the associated completion functor L 0 : Comod Ψ → Comod Ψ preserves colimits and hence admits a right adjoint ι ′ .Note that this fails for C I because it is not right exact in general, so that Ψ does not preserve colimits.One can then prove that, for true-level Hopf algebroids, ι ′ can be computed by a pullback analogous to our definition of ι.

Examples of true-level Hopf algebroids. In order to provide examples of true-level
Hopf algebroids, we will make use of the following result.

Proposition 1.15. Let A be a regular local Noetherian ring, I = m the maximal ideal of A, and N an A-module satisfying one of the following two conditions:
(1) N is a projective A-module.
(2) A is complete, N is flat, and N → N is injective.
If M is a complete A-module, then the natural completion map Proof.First assume Condition (1), i.e., that N is a projective A-module.We claim that the map N ⊠ M → N ⊠M is injective for all complete modules M .It suffices to consider free modules, so let F = A be free and consider the canonical map This an injection, which can be checked by forgetting down to abelian groups.Since M is complete by assumption and A/I k is finitely presented for all k ≥ 1, the possibly infinite product M is complete as well.Hence, the map above factors through a monomorphism F ⊠ M → F ⊠M .Now assume condition (2) of Proposition 1.15 holds.We first claim that N is pure in A = J A for some indexing set J. Indeed, let E be the cotorsion envelope [BBE01] of N , which is flat because N is.As a special case of the main result of [Eno84], E is thus of the form p∈Spec(A) T p , where T p is the p-completion of a free A p -module.It follows from [HS99, Thm.A.2(b)] that madic completion and L 0 -completion with respect to m coincide on flat modules.Consequently, by [BF15, Prop.A.15], we know that if N is flat, then N is pro-free, i.e., N = C m ( A).Furthermore, C m ( A) → A is a split monomorphism by [HS99, Prop.A.13], thus it suffices to show that N ⊆ N is pure.Since A is complete, it is cotorsion, and so is A, because Ext commutes with direct products in the second variable if A is Noetherian.It follows that the monomorphism N → A factors through E, i.e., Note that m p =m T p = p =m T p , so the only possible map from p =m T p into A is the zero map, as A is complete.It follows that T p must be trivial for all p = m, i.e., T m is the cotorsion envelope of N .Note that N is pure in T m , and since we have Using [Lam99,Prop. 4.44] and a colimit argument, we can verify that for all A-modules M and any indexing set I, the canonical map But M is complete, so the map above factors through N ⊠ M → N ⊠M , which thus must be injective.
Applying the proposition to N = Ψ and the complete comodules M or Ψ ⊠M yields: Corollary 1.16.Let (A, Ψ) be a Hopf algebroid with A a regular local Noetherian ring.If (A, Ψ) satisfies one of the following two conditions: (1) Ψ is a projective A-module, or (2) A is a complete ring, Ψ is flat, and the completion map In particular, E(n) * is local regular Noetherian and he proves that the associated cooperations E(n) * E(n) form a free module over E(n) * .Moreover, using [HS05a, Thm.C], it follows that Comod E(n) * E(n) is equivalent to the category of E(n) * E(n)-comodules.We thus have a true-level model of the category of quasi-coherent sheaves on the height n Lubin-Tate stack.

Derived completion
In the previous section we studied I-adic completion on the abelian category of comodules.In this section, we work in the derived setting and consider torsion and completion functors on suitable derived categories of comodules.In the case of a discrete Hopf algebroid (i.e., in the case of A-modules) the derived functor of completion we construct has a well-known relationship with completion on the abelian level, where it computes the left derived functors of completion.As we shall see, this is not true for an arbitrary Hopf algebroid, and the situation is more complicated in this case.
By applying our methods to the case of A-modules, we obtain alternative proofs of some structural results of Hovey and Strickland [HS99] about derived functors of completion for complete regular local Noetherian rings, and deduce a new criterion for L-completeness.

The stable category of comodules and derived torsion and completion.
In this section we briefly recall the stable category of comodules as well as the basic features of derived torsion and completion that we need.We refer the reader to [BHV18, Sec. 4 and Sec.5] for more details and for proofs.
The category of comodules Comod Ψ over a flat Hopf algebroid (A, Ψ) is a symmetric monoidal Grothendieck abelian category.As such, its derived ∞-category D(Ψ) = D(Comod Ψ ) exists; however, it has been noted [Hov04, Kra05,BHV18] that it has some undesirable properties -for example, the tensor unit A need not be compact.Based on Hovey's work [Hov04], the authors constructed a symmetric monoidal stable ∞-category Stable Ψ which is compactly generated by the set G Ψ of (isomorphism classes of) dualizable Ψ-comodules.
Following [Hov04], we will need the following addtional condition to guarantee that the category of Ψ-comodules is generated by G Ψ .Definition 2.1.A Hopf algebroid (A, Ψ) is said to be an Adams Hopf algebroid if Ψ = colim i Ψ i for some filtered system {Ψ i } of comodules, which are finitely generated and projective over A.
The ring of cooperations of the ring spectra M U , M Sp, K, KO Let I ⊆ A be an ideal which we assume to generated by an invariant regular sequence {x 1 , . . ., x n }.These conditions ensure that A/I is a Ψ-comodule and that its image in Stable Ψ is compact and dualizable.Remark 2.4.In the special case of a discrete Hopf algebroid (A, A) the conditions on the ideal I can be weakened.Here, instead of A/I we can use a suitable Koszul complex, see Remark 3.10 and Theorem 3.11 of [BHV18].
At times we will need a stronger notion of an invariant ideal, which we call strongly invariant.Definition 2.5.We call the ideal I strongly invariant if, for 1 ≤ k ≤ n and every comodule M which is I k−1 -torsion as an A-module, there is a comodule structure on x −1 k M such that the natural homomorphism M → x −1 k M is a comodule morphism.Note that in the case of a discrete Hopf algebroid, this condition is automatic.The full subcategory Stable I−tors Ψ is compactly generated, and hence ι tors admits a right adjoint Γ I which is smashing, i.e., Γ I (M ) ≃ Γ I (A) ⊗ M for any M ∈ Stable Ψ .Therefore, we can apply the results of [BHV18, Sec.2] to obtain localization and completion adjunctions with respect to the ideal I.When considered as endofunctors of Stable Ψ , the functors (Γ I , Λ I ) define an adjoint pair, so that we have a natural equivalence for all X, Y ∈ Stable Ψ .Moreover, Γ I and Λ I induce mutually inverse equivalences If I is a finitely generated ideal, generated by a regular sequence, then we have that There are strongly convergent spectral sequences of A-modules: (1) Remark 2.9.The theory of torsion and complete objects in a suitable category C has previously been studied by Hovey-Palmieri-Strickland [HPS97, Thm.3.3.5]and Dwyer-Greenlees [DG02] among others.
Remark 2.10.Let X be the stack associated to (A, Ψ), then there is a symmetric monoidal equivalence of abelian categories between Comod Ψ and QCoh(X), the category of quasi-coherent sheaves on X, see [BHV18, Prop.5.37] and the references therein.This gives rise to an equivalence of symmetric monoidal stable ∞-categories between Stable Ψ and IndCoh X , the stable ∞-category of quasi-coherent sheaves on X [BHV18, Prop.5.40].Moreover, the construction of local homology and cohomology functors on IndCoh X is given analogously to that on Stable Ψ .It follows that our results on local homology apply equally well to the category IndCoh X .
2.2.Derived completion for modules.In the previous section we introduced a derived version of completion for the stable category associated to a suitable Hopf algebroid.In this section we focus on discrete Hopf algebroids, i.e., the derived category of A-modules for a commutative ring A. We will show how the derived completion functor Λ I is related to the derived functors of ordinary I-adic completion.
To that end, let A be a commutative ring and I an ideal in A. Recall that the I-adic completion, defined by C I (M ) = lim s M/I s M , is neither right nor left exact in general for nonfinitely generated A-modules, see [HS99, App.A] for example.We are then led to consider either the left or right derived functors of completion.It turns out that if A is an integral domain, then the higher right derived functors of completion vanish [GM92, Sec.5], and we hence focus on the left derived functors of completion.
Definition 2.11.Let A and I be as above and M ∈ Mod A .For s ≥ 0, let L I s (M ) denote the s-th left derived functor of I-adic completion on M .
If the ideal I is clear from context, then we will usually just write L s (M ).Note that L 0 M is in general not equivalent to C I (M ).Rather, there is a surjection L 0 M ։ C I (M ), whose kernel can be identified with lim 1 k Tor A 1 (A/I k , M ), see Corollary 2.23.In fact, L I s (M ) can be computed in terms of the local homology groups H s I (M ) whenever the ideal I is generated by a weakly proregular sequence.For the definition of a weakly proregular sequence, see [Sch03], or [BHV18, Def.3.14].We note that if A is Noetherian, then every ideal in A is weakly proregular [PSY14, Thm.Theorem 2.12.Let A be a commutative ring.Let I = (x 1 , . . ., x n ) be an ideal generated by a weakly proregular finite sequence and assume that A is x i -bounded torsion for all i.Then, for all A-modules M , and all s ≥ 0, there is a natural isomorphism , where L I s is the s-th left derived functor of I-adic completion.There is a natural homomorphism η : M → L 0 M , and M is said to be L-complete when η is an isomorphism.Let Mod A ⊆ Mod A be the full subcategory of L-complete A-modules.The following is proved under the assumption that A is a complete local ring in [HS99].
Proposition 2.13.The category Mod A of L-complete modules is an abelian subcategory of Mod A .There are enough projectives in Mod A , and each projective object is a retract of a profree module, i.e., the completion of a free A-module.Moreover, for all k ≥ 0, the modules L k M lie in Mod A .
Proof.The fact that Mod A is an abelian subcategory of Mod A is proved in the same way as in [HS99, Thm.A.6(e)] -the proof only uses that L 0 is right exact and that for any short exact sequence M ′ → M → M ′′ there is a long exact sequence Both of these follow from the definition of the functors L k as the left derived functors of I-adic completion.
Let F be a free A-module.Then L 0 F ∼ = F ∧ I is projective in Mod A since for any M ∈ Mod A there is an adjunction Hom Mod A (L 0 F, M ) ∼ = Hom A (F, M ) and because epimorphisms in Mod A are epimorphisms in Mod A .To see that Mod A has enough projectives, let M ∈ Mod A , then there exists a free A-module F and an epimorphism F → M .Since L 0 is right exact, this gives rise to an epimorphism L 0 F → M .If M is itself projective in Mod A , then this must have a retract, and we see that any projective in Mod A is a retract of a pro-free one.
However, Mod A is not a Grothendieck category in general, because filtered colimits are not necessarily exact.Despite the fact that Mod A is not a Grothendieck category, in Section 3 we will define a version of its derived category and show that it is equivalent to a certain full subcategory of the usual derived category D(A).
In the previous subsection we introduced abstract local homology functors Λ I for the stable category of Ψ-comodules.In the case of A-modules, the abstractly constructed completion functor Λ I is related to the total left derived functor of completion.For the proof of the following, see [BHV18, Prop.3.16] Proposition 2.14.Let A be a commutative ring and I = (x 1 , x 2 , . . ., x n ) an ideal in A. Suppose that A is bounded x i -torsion for i = 1, . . ., n.Then the following are equivalent: (1) The sequence {x 1 , x 2 , . . ., x n } is weakly proregular.
As the following result illustrates, we can use Theorem 2.12 and Proposition 2.14 to translate facts about Λ I in the derived category to concrete results about L-complete modules and derived functors of completion.
Proposition 2.16.Assume that I = (x 1 , . . ., x n ) is an ideal of A generated by a weakly proregular sequence and that A is x i -bounded torsion for all i.Let M be an A-module and N an Lcomplete A-module.Then, Ext q A (M, N ) is L-complete.In particular, the category of L-complete modules is closed symmetric monoidal.
Proof.Since N is L-complete, the local homology spectral sequence Proposition 2.8 collapses to show that N ≃ Λ I N .By local duality (2.6), we thus get an equivalence Hom(Γ I M, N ) ≃ Hom(M, N ), and since Γ I is smashing we have where the last equivalence is a consequence of (2.6).Combining these it follows that H q (Λ I Hom(M, N )) ∼ = H q (Hom(M, N )) ∼ = Ext −q A (M, N ).Moreover, using Theorem 2.12 we see that the spectral sequence of Proposition 2.8 takes the form . By Proposition 2.13 each entry on the E 2 -page is L-complete.Moreover, the derived functors L s vanish whenever s > n, and so the spectral sequence has a horizontal vanishing line on the E 2page.Since Mod A is abelian, again by Proposition 2.13, the abutment of the spectral sequence must also be L-complete.
In [HS99, Cor.A.7], Hovey and Strickland show that Mod A is symmetric monoidal with tensor product given by L 0 (M ⊠N ).Note that if M, N , and K are L-complete modules, we have natural isomorphisms Since Hom A (N, K) is L-complete by the discussion above, we have shown that Hom A (−, −) restricted to Mod A × Mod A factors through Mod A , and it follows that Hom A (−, −) is the internal hom-object corresponding to the symmetric monoidal product of Mod A .
As another application, in Theorem 2.28 we will use the derived approach to completion to give a criterion for when a module is L-complete.A, Ψ) is a flat Hopf algebroid and let I ⊆ A be a finitely generated ideal generated by a regular invariant sequence.In this section, we will construct a Grothendieck type spectral sequence calculating local homology H I s (M ) = H s (Λ I M ) for any Ψ-comodule M .However, the abutment of this spectral sequence will not be related to the left or right derived functors of comodule completion C I Ψ as studied in Section 1.To see this we first need the following lemma, which shows that the right derived functors of the completion functor on comodules vanish under mild conditions on the ring A: Lemma 2.17.If A is an integral domain and J ∈ Comod Ψ an injective comodule, then lim Ψ,k A/I k ⊗ J = 0.In particular, all right derived functors of lim Ψ,k A/I k ⊠ − : Comod Ψ → Comod Ψ are zero.

Derived completion for comodules. Suppose now that (
Proof.Since any injective comodule is a retract of an extended comodule on an injective Amodule J ′ [HS05b, Lem.2.1(c)], we can assume without loss of generality that J = Ψ ⊠ J ′ .Hence, by (1.2) there is an isomorphism and we conclude by [GM92, Lem.5.1] which gives that lim k A/I k ⊠ J ′ = 0.
Remark 2.18.Recall that Proposition 2.14 says that, for A-modules, the homology groups of Λ I compute the left-derived functors of completion.For an arbitrary Hopf algebroid, Example 2.31 below shows that, unlike the case of A-modules, there exist negative local homology groups.This, along with Lemma 2.17, implies that for comodules the homology groups of Λ I cannot, in general, be given by the partial left or right derived functors of C I Ψ .To construct the Grothendieck spectral sequence we need the following result [BHV18, Lem.4.26].
Lemma 2.19.Let (A, Ψ) be an Adams Hopf algebroid.If (D i ) i∈I is an inverse system in D(Ψ), then there is a natural isomorphism of p-th derived functors of inverse limit In light of this lemma, we will write lim p Ψ,i D i for both sides of the equivalence above.Remark 2.20.In [BHV18,Lem. 4.26] this was proved under the assumption that A was a Landweber Hopf algebroid with A Noetherian, using [BHV18, Prop.4.18].These conditions guaranteed that ι * is fully faithful; in light of Proposition 2.2 these assumptions can be weakened.
For the following, we let H s (Λ I M ) = Λ I s (M ).Proposition 2.21.Suppose (A, Ψ) is an Adams Hopf algebroid, I is a finitely generated ideal of A generated by a regular invariant sequence, and M ∈ D Ψ .There is a conditionally and strongly convergent spectral sequence of comodules (2.22) E p,q 2 ∼ = lim p Ψ,k Tor Ψ q (A/I k , M ) =⇒ Λ I q−p (M ).Proof.Let A 1 and A 2 be abelian categories and assume that A 1 has enough injective objects.Recall that there is a conditionally convergent spectral sequence for any left exact functor F : A 1 → A 2 and X ∈ Ch(A 1 ), see [Wei94, 5.7.9, Cor.10.5.7].We apply this spectral sequence to the functor lim : Comod N Ψ → Comod Ψ and X = (A/I k ⊗M ) k∈N ∈ D(Ψ) N , the category of towers of chain complexes of Ψ-comodules up to quasi-isomorphism.Switching the grading so that the spectral sequence converges to H q−p RF (X), the E 2 -page is readily identified as ).This spectral sequence then converges conditionally to which is Λ I q−p (M ) by (2.7) and Lemma 2.19.Finally, since I is generated by a finite regular sequence, this spectral sequence has a horizontal vanishing line at E 2 and hence converges strongly.
Since lim p = 0 for p > 1 for any discrete Hopf algebroid, this spectral sequence degenerates to a short exact sequence.Hence we recover as a special case: Corollary 2.23 (Greenlees-May).If M ∈ Mod A , then there is a short exact sequence for any s ≥ 0. In particular, there exists a natural epimorphism L 0 M → C I (M ).
By the Artin-Rees lemma, local homology is concentrated in degree 0 for finitely generated modules over Noetherian rings, where it simply agrees with I-adic completion, see [HS99, Prop.A.4].Although this is no longer the case for comodules, the next proposition provides an appropriate comodule analogue where one needs to use the derived functors of limit to capture local homology on the nonzero degrees.
Proof.It is enough to show that lim s Ψ Tor Ψ t (M, A/I k ) = 0 for all s ≥ 0 and all t ≥ 1, so that the strongly convergent spectral sequence (2.22) collapses.In fact, we will prove the stronger claim that the tower (Tor Ψ t (M, A/I k )) k is pro-trivial for all t ≥ 1; see [Jan88, Lem.1.11] for a proof that this implies that all derived functors of lim vanish.On the one hand, using Lurie's result [Lur11, Not.5.2.18], it follows that the left derived functors L t C of the right exact functor ) k are pro-trivial on compact Ψ-modules M whenever t ≥ 1.On the other hand, it is easy to see that there is an isomorphism of towers k for all M ∈ Comod Ψ , giving the claim.

2.4.
A criterion for L-completeness.We now establish a new criterion for L-completeness, generalizing the Ext-p-completeness criterion due to Bousfield and Kan.We start with a standard lemma.
Lemma 2.25.Suppose f !: C → D is a continuous functor between presentable categories with right adjoint f * and that C is generated by a set of objects G.If f * is conservative, then D is generated by f !G.If we assume additionally that every element of G is compact and that f * admits a further right adjoint f * , then D is compactly generated by f !G.
Proof.Suppose α : X → Y is a morphism in D such that Hom D (f !G, α) is an equivalence for all G ∈ G.By adjunction, Hom C (G, f * α) is an equivalence as well, so f * α is an equivalence because G generates.Since f * is conservative, α must be an equivalence, and it follows that f !G generates D. To see the last claim, note that since f * has a right adjoint, it preserves colimits, hence f !preserves compact objects and thus f !G ⊆ D ω .

Proposition 2.26. Let (A, Ψ) be a flat Hopf algebroid and G Ψ be a set of representative of isomorphism classes of dualizable Ψ-comodules. Suppose I ⊆ A is a strongly invariant ideal in
A generated by a regular sequence (x 1 , . . ., x n ), then Proof.By Lemma 2.25 and the fact that L I is smashing, Stable I−loc Ψ is the localizing subcategory in Stable Ψ generated by L I G Ψ .We therefore have to show that To simplify notation, let G ∈ G Ψ and write and it remains to show the other inclusion.To this end, consider the following fiber sequences of comodules Applying L I to these sequences and starting from the one for i = n, we see by downward induction on i that The next result is an immediate consequence of the characterization of Λ I -acyclics.
Corollary 2.27.Let M ∈ Stable Ψ .Then, In [BK72], Bousfield and Kan define an abelian group A to be Ext-p complete if the map The following result is thus the natural generalization of Ext-p completeness for modules over any commutative ring.

Theorem 2.28 (Ext-I completeness criterion). Let A be a commutative ring and I ⊆ A an ideal generated by a regular sequence x
Proof.First note that we may take G A = {A}, because dualizable A-modules are finitely generated and projective.By the spectral sequence of Proposition 2.8, a module M is L-complete if and only if M is Λ I -local as a complex concentrated in degree zero.But by Corollary 2.27 in the case of a discrete Hopf algebroid, M is Λ I -local if and only if Hom(x −1 i A/(x 1 , . . ., x i−1 ), M ) ≃ 0 for all i = 1, . . ., n.The result thus follows after applying homology to the latter equivalence.
Remark 2.29.Since the projective dimension of x −1 i A/(x 1 , . . ., x i−1 ) as an A-module is at most i, it suffices to check the vanishing of Ext q A (x −1 i A/(x 1 , . . ., x i−1 ), M ) for 0 ≤ q ≤ i.

2.5.
for any spectrum M ∈ Mod E .This can be considered as the local homology spectral sequence associated to the functor π * : Mod E → Mod E * .
While the local homology groups for a discrete Hopf algebroid vanish in negative degrees, this is not true for an arbitrary flat Hopf algebroid (A, Ψ) as the following example shows.Informally speaking, the negative local homology groups measure the "stackiness" of the Hopf algebroid (A, Ψ).
Example 2.31.For p > 2, let K be p-complete K-theory and consider local homology Λ (p)  with respect to the ideal (p) ⊂ K 0 = Z p .Let g be a topological generator of Z × p ; using the fiber sequence one can then calculate the local homology groups of K * : where V Q is an uncountable rational vector space.For the details of this computation, see [BBP18].From this, one can formally deduce that This example implies that even if the inverse limit of comodules can be computed in terms of the inverse limit of the underlying modules and a functor ι as in Section 1, in general there cannot exist a convergent Grothendieck spectral sequence associated to the composite lim Ψ ∼ = ι • lim A .Indeed, using a graded version of the Ext-p-completeness criterion of [BK72] or Theorem 2.28, we see that lim s K * (. . .
Therefore, the E 2 -page of the composite functor spectral sequence would have to be zero, while the abutment is nontrivial.
We now consider the analogue of (2.30) for the functor E * : Sp E → Comod E * E .The convergence of this spectral sequence in the case of comodules is more subtle, and relies on the proof of the smash product theorem due to Hopkins and Ravenel [Rav92, Sec.8].
Theorem 2.32.For X ∈ Sp E(n) compact, i.e., the E(n)-localization of a finite spectrum, there exists a strongly convergent spectral sequence Proof.By [BBP18], there is a strongly convergent spectral sequence , where (M I(k) ) k is a cofinal tower of generalized type n Moore spectra as in [BHV18, Thm.6.3], which is based on [HS99, Prop.4.22].Note that {E * (M I(k) )} k is proisomorphic to the tower {E * /m k } k .It therefore remains to identify the E 2 -term of this spectral sequence in the case X is compact.
To that end, we will show that there is a pro-isomorphism To construct the maps between the two systems in (2.33), we note that by [Ada69, p. 71] the natural pairing of E * -modules is in fact a comodule map, for arbitrary spectra and Y .The next step follows the proof of [Lur11, Lem.5.2.19].For the convenience of the reader, we sketch the argument.The collection C of spectra X for which (2.33) is a pro-isomorphism of comodules clearly contains L n S 0 and is closed under retracts and shifts.It therefore suffices to show that it is also closed under cofiber sequences.To this end, let be a cofiber sequence of compact L n -local spectra with X ′ , X ′′ ∈ C. We claim that (2.33) is a pro-isomorphism of comodules for X, which can be checked degreewise using that The indicated horizontal maps are pro-isomorphisms by assumption.Both vertical sequences are exact: This is obvious for the one on the right, while [Lur11, Not.5.2.18] checks the claim for the left one using the Artin-Rees lemma.We can thus conclude X ∈ C by the five lemma.The theorem thus follows from Proposition 2.24.
Remark 2.34.This spectral sequence is a special case of a strongly convergent spectral sequence computing the E-homology of an inverse limit of E-local spectra from the higher derived functors of inverse limits of comodules, see [BBP18], which is originally due to Hopkins and has also been considered in unpublished work of Sadofsky.It offers an algebraic approach to the chromatic splitting conjecture, reducing it to a claim in Comod E * E .
3. Tilting and t-structures 3.1.Derived categories and tilting.Let (A, Ψ) be an Adams Hopf algebroid and D(Ψ) = D(Comod Ψ ) its derived ∞-category.We recall from Proposition 2.2 that the ∞-category Stable Ψ is related to D(Ψ) via an adjunction Here the symmetric monoidal, continuous functor ω is given by inverting the homology isomorphisms, and the right adjoint ι * is a fully faithful embedding.Given a discrete comodule M ∈ Comod Ψ we can think of it as a complex in D(Ψ) concentrated in degree 0. Via ι * we can also consider M as an object of Stable Ψ .
In contrast to Stable Ψ , the dualizable discrete comodules are not necessarily compact in D(Ψ).Nonetheless, we claim that the smallest localizing subcategory containing them is all of D(Ψ), so that they still form a suitable collection of generators.
Proof.Since D(Ψ) is presentable, the statement of the lemma is equivalent to the claim that Z ∈ D(Ψ) ≃ 0 if and only if Hom D(Ψ) (G, Z) is contractible for each each G ∈ G Ψ , or equivalently that a morphism φ : X → Y is an equivalence if and only if Hom D(Ψ) (G, φ) is an equivalence of mapping spectra for all G ∈ G Ψ .To see this, apply Lemma 2.25 with f != ω and f * = ι * -note that ι * is fully faithful, and so in particular conservative, so that the conditions of the lemma are satisfied.Proof.We will prove the more general statement that for any collection C of objects in Stable Ψ we have

Recall that given
where G Ψ denotes, as usual, the set of dualizable (discrete) comodules.Since ω preserves colimits and is symmetric monoidal, a standard argument shows that In There is a version of the local cohomology spectral sequence for Γ Ψ I .We recall that we let T Ψ I denote the I-torsion functor on Comod Ψ and write T A I for the corresponding functor on Mod A .We stress that we the following construction only gives a spectral sequence of A-modules -we do not know if it can be the given the structure of a spectral sequence of comodules.Lemma 3.5.For any X ∈ Stable Ψ there is a strongly convergent spectral sequence of A-modules For ǫ * X there is a strongly convergent spectral sequence of A-modules ), see Proposition 2.8.We will identify this with the claimed spectral sequence.
The local cohomology groups H s I are equivalent to R s T A I , the derived functor of torsion in A-modules.We then have isomorphisms , where the last equivalence is [BHV18,Lem. 5.12].This identifies the E 2 -page of the spectral sequence of the lemma.
For the abutment, we have , where the first isomorphism follows from [BHV18,Lem. 5.20].
After collecting this preliminary material, we can now state the main theorem of this section.Remark 3.8.Due to the phenomena discussed in Section 2 (see, for example Remark 2.18), we do not have an analogue of Theorem 3.6(1) for non-discrete Hopf algebroids, and thus are currently unable to prove a version of Corollary 3.7 more generally.
As we will see in the next subsections, the proof of Theorem 3.6(2) is actually a consequence of an analogous statement for Stable Ψ in place of the usual derived category.

t-structures and complete modules.
We start this subsection by recalling some material about t-structures.This concept was introduced by Beilinson-Bernstein-Deligne [BBD82].We follow more closely the treatment given by [Lur17] -namely we work with homological indexing, so that X[n] denotes the n-fold suspension Σ n X.Given a stable category C equipped with a t-structure, the left completion of C is defined to be the limit of the tower We say that the t-structure on C is left-complete if C is equivalent to its left completion.We can similarly define the right completion, and a right-complete t-structure.
The following lemma, proved in [BBD82, 1.3.19],will be useful for constructing t-structures on full subcategories of C. Lemma 3.10.Let (D ≥0 , D ≤0 ) be a t-structure on C with heart C ♥ .Let S ⊆ C be a full stable subcategory of C. If τ ≥0 M and τ ≤0 M are in S whenever M is, then (S ∩ D ≥0 , S ∩ D ≤0 ) defines a t-structure on S with heart S ∩ C ♥ .Moreover, the truncation functors for the induced t-structure on S are the same as those for the t-structure on D.
We now begin the proof of Theorem 3.6(1).The category Mod A of L-complete A-modules is abelian; however it is not Grothendieck abelian in general because direct sums and filtered colimits are not exact.There exist enough projectives in Mod A by Proposition 2.13 and so by [Lur17, Sec.1.3.2]we can associate to it the right-bounded derived category D − ( Mod A ).This category comes equipped with a natural left-complete t-structure whose heart is equivalent to Mod A , see [ The following results show that the right completion of this bounded derived category is naturally equivalent to the derived category D I−cmpl (A) of I-complete A-modules constructed abstractly in Section 2.1.We assume that A and I satisfy the conditions of Theorem 2.12 so that local homology computes the derived functors of completion.We claim this induced t-structure is both left and right complete.Indeed, recall that the truncation functors on D I−cmpl (A) are the restriction of the truncation functors of D(A).Since limits in D I−cmpl (A) are the same as those in D(A), we easily see that left completeness of the induced t-structure follows from left completeness of the t-structure on D(A).On the other hand, the colimit in D I−cmpl (A) is not the same as that in D(A) -it is given by first taking the colimit in D(A), and then applying Λ I .However, by right completeness of the t-structure on D(A) we already have that M ≃ colim k τ ≥k M , for any M ∈ D I−cmpl (A), where the colimit is taken over Since the induced t-structure on D I−cmpl (A) has heart Mod A , applying [Lur17, Prop.1.3.3.7]we deduce the existence of a t-exact functor θ : D − ( Mod A ) → D I−cmpl (A).The same proposition shows that θ is fully faithful if and only if for each pair X, Y ∈ Mod A with X projective, the groups Ext i D I−cmpl (A) (X, Y ) = 0 for i > 0. By the characterization of projectives in Proposition 2.13 we have that X is a retract of L 0 F for some free A-module F , and so we can assume X has this form.For a free module F , we have that L i F ≃ 0 for i > 0, as L i can be computed by taking a projective resolution.Thus, the local homology spectral sequence shows that Λ I F ≃ L 0 F , concentrated in degree 0. It follows that Proof of Theorem 3.6(1).Since D I−cmpl (A) is right complete, the previous proposition and [Lur17, Rem.1.2.1.18]show that there is a canonical equivalence between the right completion of D − ( Mod A ) and D I−cmpl (A).
For the second part suppose that M ∈ D I−cmpl (A), i.e., that Λ I M ≃ M .The spectral sequence of Proposition 2.8 converging to H * (Λ I M ) has E 2 page in Mod A .Since the spectral sequence has a horizontal vanishing line and Mod A is abelian, the abutment is in the latter category as well.For the converse, let M ∈ D(A) be a complex whose homology is L-complete.The aforementioned spectral sequence collapses to give an isomorphism H * (Λ I M ) ∼ = H * (M ).This implies that the natural map M → Λ I M is a quasi-isomorphism, from where it follows that Λ I M ≃ M .Proof.Recalling again that Comod I−tors Ψ , this is the same argument as in the second part of the proof of Theorem 3.6(1) given above.
The case of an arbitrary Adams Hopf algebroid is more difficult, and involves passing to the larger category Stable Ψ .The reason for this is that we do not know how to construct the local cohomology spectral sequence of Lemma 3.5 in D(Ψ).Let J be an injective object in the category of I-torsion Ψ-modules.Dualizing [Lur17, Prop.1.3.3.7]we deduce that if π n Hom D I−tors (Ψ) (X, J) ∼ = π n Hom D(Ψ) (X, J) = 0 for all n < 0 and each I-torsion Ψ-comodule X, then θ is fully faithful, with essential image the full subcategory of left-bounded objects of D I−tors (Ψ).It is not hard to check that any such J is a retract of T Ψ I (L), for some injective Ψ-comodule L.Moreover, any such L is a retract of Ψ ⊗ Q for some injective A-module Q [HS05b, Lem.2.1(c)], and so we can assume that J has the form T Ψ I (Ψ ⊗ Q).But an adjointness argument shows that there is an equivalence Under some additional hypothesis we can say more.We require the following lemma.We now show that the standard t-structure on D(Comod I−tors

Ψ
) is left-complete.We first observe that via an adjunction argument the product in the abelian category Comod I−tors Ψ is given by the composite T Ψ I Ψ (−), where Ψ (−) denotes the product in Ψ-comodules.We wish to prove that this composite has only finitely many derived functors.Note that if the Grothendieck spectral sequence for the composite T Ψ I Ψ exists, it will have the form .
By assumption R q Ψ vanishes whenever q is large enough, and the same is true for R p T Ψ I by the corresponding statement for modules and [BHV18, Lem.5.12].It follows that there are only finitely many derived functors of product in Comod I−tors Ψ , so that the t-structure on D I−tors (Ψ) is left complete.
We now complete the proof by showing that the Grothendieck spectral sequence does exist.Let J be a set regarded as a discrete category, then the functor category Comod J A has enough injectives, and these are pointwise injective.It follows that the Grothendieck spectral sequence exists if R s T Ψ I ( j∈J I j ) = 0 for s > 0 where each I j is an injective Ψ-module.Since each I j is injective, so is the product j∈J I j .But R s T Ψ i vanishes for s > 0 on injective Ψ-comodules, so we are done.This gives a conditional proof of Theorem 3.6(2).Theorem 3.16.Suppose that Comod Ψ has only finitely many derived functors of product.Then the conclusion of Theorem 3.6(2) holds, i.e., there is a canonical equivalence D(Comod I−tors Ψ ) ≃ D I−tors (Ψ).
Remark 3.17.For a discrete Hopf algebroid (i.e., for the category of A-modules) products are exact, and so the above result always holds.If (A, Ψ) = (E * , E * E) where E denotes Morava E-theory at height n and at a prime p ≫ n, then Comod E * E has finite injective dimension, see [Mor85] or [Fra96,Thm. 3.4.9],and hence there can only be finitely many derived functors, so the previous result also holds.

( 1 )
If (A, Ψ) = (A, A) is discrete, then there is a canonical equivalence between the right completion of D − ( Mod A ) and D I−cmpl (A).Moreover, a module M ∈ D(A) is I-complete if and only if the homology groups H * M are L-complete.(2) If I is generated by a regular sequence, then there is a canonical equivalence D(Comod I−tors Ψ ) ≃ D I−tors (Ψ).Moreover, a comodule M ∈ D(Ψ) is I-torsion if and only if the homology group H * M are I-torsion.There are a number of results in the literature closely related to Theorem D. For example, in [PSY14, Cor.3.32] it is proven that M ∈ D(A) is cohomologically I-torsion (that is, the canonical morphism from RΓ I M → M is an equivalence, where RΓ I M denotes the total derived functor of I-torsion of M ) if and only if the homology groups H * M are I-torsion.Moreover, in unpublished work Rezk has constructed a version of the derived category of L-complete modules and has proven a version of the second part of Theorem D(1), see [Rez13, Thm.9.2].

Definition 2. 3 .
The subcategory Stable I−tors Ψ ⊆ Stable Ψ is defined as the localizing tensor ideal of Stable Ψ generated by the compact object A/I.The inclusion of the category Stable I−tors Ψ of I-torsion Ψ-comodules into Stable Ψ will be denoted ι tors .

Theorem 3. 6 .
Let (A, Ψ) be an Adams Hopf algebroid and I ⊆ A a finitely generated invariant ideal.(1) If (A, Ψ) = (A, A) is discrete, then there is a canonical equivalence between the right completion of D − ( Mod A ) and D I−cmpl (A).Moreover, a module M ∈ D(A) is I-complete if and only if the homology groups H * M are L-complete.(2) If I is generated by a regular sequence, then there is a canonical equivalence D(Comod I−tors Ψ ) ≃ D I−tors (Ψ).Moreover, a comodule M ∈ D(Ψ) is I-torsion if and only if the homology group H * M are I-torsion.Combining this result with Section 2 yields the following tilting-theoretic interpretation of local duality.Corollary 3.7.For any commutative ring A and I ⊆ A a finitely generated ideal, the functors L I 0 : Mod I−tors A / / Mod A : T A I o o induce mutual inverse symmetric monoidal equivalences Λ I : D(Mod I−tors A ) ∼ / / D( Mod A ) : Γ I , ∼ o o where D( Mod A ) denotes the right completion of D − ( Mod A ). Transferring the standard t-structure on D( Mod A ) via the equivalence of Corollary 3.7 induces a nonstandard t-structure on D(Mod I−tors A ) whose heart is the abelian category of L-complete A-modules.Since the latter category is usually not equivalent to Mod I−tors A , Corollary 3.7 is a non-trivial instance of a tilting equivalence.

Proposition 3. 11 .
There is a t-structure on D − ( Mod A ) along with a fully faithful t-exact inclusion θ : D − ( Mod A ) ֒→ D I−cmpl (A), whose image consists of the right bounded objects of D I−cmpl (A), i.e., (D I−cmpl (A)) ≥n .Proof.We begin by observing that the standard t-structure on D(A) is left and right complete.Indeed, D(A) is right complete by [Lur17, Prop.1.3.5.21], while left completeness follows, for example, from [Lur17, Prop.7.1.1.13]and the equivalence Mod HA ≃ D(A) [Lur17, Prop.7.1.1.15and Rem.7.1.1.16].The local homology spectral sequence of Proposition 2.8 shows that if M ∈ D(A) is I-complete, then so are the truncations τ ≥n M and τ ≤n M .It follows from Lemma 3.10 that there is an induced t-structure on D I−cmpl (A).

3. 3 .
Torsion comodules.A similar argument as given in the previous section for complete modules also works for torsion comodules using the left-bounded derived category D + (Comod I−tors Ψ ) [Lur17, Var.1.3.2.8].By the dual of [Lur17, Prop.1.3.5.24] the left-bounded derived category of Comod I−tors Ψ can be identified as the full subcategory of D(Comod I−tors Ψ ) spanned by the left bounded objects (where we equip D(Comod I−tors Ψ ) with the standard t-structure [Lur17, Def.1.3.5.16 and Prop.1.3.5.18].)In order to prove Theorem 3.6(2) we first introduce another category, namely the category D I−tors (Ψ) of complexes of comodules with cohomology in Comod I−tors Ψ .In the case of a discrete Hopf algebroid (A, A) it is easy to identify D I−tors (A) with D I−tors (A).Proposition 3.12.Let A be a commutative ring and I a finitely generated ideal.There is an equivalence of categories D I−tors (A) ≃ D I−tors (A).

Lemma 3. 15 .
Suppose that there are only finitely many derived functors of product in Comod Ψ .Then the t-structures on D(Ψ), D I−tors (Ψ), and D(Comod I−tors Ψ ) are left complete.Proof.A general result of Hogadi and Xu shows that if A is any Grothendieck abelian category for which there are only finitely many derived functors of product, then D(A) is left complete [HX09, Thm.1.3].This gives the claim for D(Ψ).To see that D I−tors (Ψ) is left complete, observe that limits in D I−tors (Ψ) are given by taking limits in D(Ψ) and then applying Γ I .Since the inclusion is t-exact, and D(Ψ) is left complete, we see that for any M ∈ D I−tors (Ψ), we have Γ I lim τ ≤n M ≃ Γ I M ≃ M , showing that the induced t-structure on D I−tors (Ψ) is left-complete.
[BHV18,(Ψ), where ω is continuous and ι * is fully faithful.In the case that (A, A) is a discrete Hopf algebroid, Stable A is equivalent to D(A), the usual derived category of A-modules.Proof.Everything except that ι * is fully faithful is proven in[BHV18, Sec.4].The fully faithfulness is proved there under some further assumptions on the Hopf algebroid.This conditions can be weakened by using the recent work of Pstrągowski.Indeed by [Pst18, Thm.3.7 and Cor.3.8] we can identify Stable Ψ with the ∞-category of spherical sheaves of spectra on dualizable comodules, and D(Ψ) with the ∞-category of hypercomplete spherical sheaves of spectra.The functor ι * can then be identified with the inclusion of hypercomplete sheaves into all spherical sheaves.
[GM92],s proved in[BHV18, Cor.5.26] under the additional assumption that A was Noetherian, however this assumption is unnecessary in light of the work of Pstrągowski mentioned previously.In particular, by using [Pst18, Cor.3.8]we can remove the Noetherian hypothesis from [BHV18, Prop.5.24], of which [BHV18, Cor.5.26] is a direct corollary.In the discrete case, there are spectral sequences computing the (co)homology of the torsion and completion functors.These are given in terms of local cohomology and local homology of A-modules with respect to an ideal I ⊆ A, denoted H s I and H I s respectively, for which we refer the reader to[GM92]or [BHV18, Sec.3.2].The following is then [BHV18, Prop.3.20].Let A be a commutative ring and I a finitely generated ideal.Let X ∈ D A .
The topological local homology spectral sequence.Let E = E n be Morava E-theory at height n and m the maximal ideal of E 0 ∼ = WF p n u 1 , . . ., u n−1 .As discussed in [BHV18, Sec.3] the construction of torsion and completion functors described in Section 2.1 works equally well for ring spectra.In particular, associated to m, there is a completion functor Λ m : Mod E → Lur17, Prop.1.3.2.19] and [Lur17, Prop.1.3.3.16].