We investigate to what extent we can descend the classification of localizing, smashing and thick ideals in a presentably symmetric monoidal stable \(\infty\)-category \( \mathcal{C} \) along a descendable commutative algebra \(A\). We establish equalizer diagrams relating the lattices of localizing and smashing ideals of \( \mathcal{C} \) to those of \( \newcommand{\Mod}{\mathop{\rm Mod}\nolimits} \Mod_{A}(\mathcal{C})\) and \(\Mod_{A \otimes A}(\mathcal{C})\). If \(A\) is compact, we obtain a similar equalizer for the lattices of thick ideals which, via Stone duality, yields a coequalizer diagram of Balmer spectra in the category of spectral spaces. We then give conditions under which the telescope conjecture and stratification descend from \(\Mod_{A}(\mathcal{C})\) to \(\mathcal{C}\). The utility of these results is demonstrated in the case of faithful Galois extensions in tensor triangular geometry.

We study the Picard group of Franke's category of quasi-periodic \(E_0E\)-comodules for \(E\) a 2-periodic Landweber exact cohomology theory of height \(n\) such as Morava \(E\)-theory, showing that for \(2p−2>n^2+n\), this group is infinite cyclic, generated by the suspension of the unit. This is analogous to, but independent of, the corresponding calculations by Hovey and Sadofsky in the \(E\)-local stable homotopy category. We also give a computation of the Picard group of \(I_n\)-complete quasi-periodic \(E_0E\)-comodules when \(E\) is Morava \(E\)-theory, as studied by Barthel--Schlank--Stapleton for \(2p−2 \ge n^2\) and \(p−1 \nmid n\), and compare this to the Picard group of the \(K(n)\)-local stable homotopy category, showing that they agree up to extension.

We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. Our theory of stratification is based on the approach of Stevenson which uses the Balmer--Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-to-global principle and establish that the Balmer--Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson--Iyengar--Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite \'{e}tale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria--Sanders--Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height \(0\) and height \(\infty\) chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of \(E(n)\)-local spectral Mackey functors noting that it bijects onto the height \(\le n\) chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

We compare the homological support and tensor triangular support for `big' objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.

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.

We prove the height two case of a conjecture of Hovey and Strickland that provides a \(K(n)\)-local analogue of the Hopkins–Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross &ndash Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava \(E\)-theory is coherent, and that every finitely generated Morava module can be realized by a \(K(n)\)-local spectrum as long as \(2p-2>n^2+n\). Finally, we deduce consequences of our results for descent of Balmer spectra.

Let \(X\) be a topological space with Noetherian mod p cohomology and let \(C^*(X;\mathbb{F}_p)\) be the commutative ring spectrum of \(\mathbb{F}_p\)-valued cochains on \(X\). The goal of this paper is to exhibit conditions under which the category of module spectra on \(C^*(X;\mathbb{F}_p)\) is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever \(X\) admits an \(H\)-space structure. More generally, using Lannes' theory we prove that stratification for \(X\) is equivalent to a condition that generalizes Chouinard's theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.

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

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree \(G\)-spectra. More generally, we show that if \(K\) is a closed subgroup of a compact Lie group \(G\) such that the Weyl group \(W_GK\) is connected, then a certain category of rational \(G\)-spectra `at \(K\)' has an algebraic model. For example, when \(K\) is the trivial group, this is just the category of rational cofree \(G\)-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

We show, at the prime 2, that the Picard group of invertible modules over \(E_n^{hC_2}\) is cyclic. Here, \(E_n\) is the height \(n\) Lubin--Tate spectrum and its \(C_2\)-action is induced from the formal inverse of its associated formal group law. We further show that \(E_n^{hC_2}\) is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when \(n=1\).

We use descent theoretic methods to solve the homotopy limit problem for Hermitian \(K\)-theory over quasi-compact and quasi-separated base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian \(K\)-theory over \( \mathop{Spec}(\mathbb{C})\), showing that the only invertible cellular spectra are suspensions of the tensor unit.

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

We generalize Quillen's \(F\)-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over \(C^*(B\mathcal{G},\mathbb{F}_p)\) is stratified and costratified for a large class of \(p\)-local compact groups \(\mathcal{G}\) including compact Lie groups, connected \(p\)-compact groups, and \(p\)-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that \(p\)-compact groups admit a homotopical form of Gorenstein duality.

Let \(k\) be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over \(Spec(k)\) with the \(C_2\)-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of \(MGL\) and \(MR\), and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of \(MR\) are even in the sense of Hill--Meier, and give a computation of the slice spectral sequence converging to \(\pi_{*,*}BP\langle n \rangle/2\) for \(1 \le n \le \infty\).

The objective of this paper is to study completions and the 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.

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable \(\infty\)-category \(\mathcal{C}\) together with a collection of compact objects \( \mathcal{K} \subset \mathcal{C} \) we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring \(A\) and a suitable ideal in \(A\), we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme \(X\) implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.

As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.

We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum \(R\), and prove it whenever \( \pi_*R \) is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.

We study criteria for freeness and for the existence of a vanishing line for modules over certain sub-Hopf algebras of the motivic Steenrod algebra over \(\mathop{Spec}(\mathbb{C})\) at the prime 2. These turn out to be determined by the vanishing of certain Margolis homology groups in the quotient Hopf algebra \( \mathcal{A}/\tau \).

We use the abstract framework constructed in our earlier paper to study local duality for Noetherian \( \mathbb{E}_{\infty} \)-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real \( K \)-theory spectra of Hopkins and Miller at height \( n=p-1 \). More generally, we determine the Picard groups of the homotopy fixed points spectra \( E_n^{hG}\), where \(E_n\) is Lubin-Tate \(E\)-theory at the prime \( p \) and height \(n=p-1\), and \(G\) is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.

In previous work Devinatz and Hopkins introduced the \(K(n)\)-local \(E_n\)-based Adams spectral sequence and showed that, under certain conditions, the \(E_2\)-term of this spectral sequence can be identified with (continuous) group cohomology. We work with the category of \(L\)-complete \(E^\vee_* E\)-comodules, and show that the \(E_2\)-term of the above spectral sequence can be computed by a relative \(\ \text{Ext}\) functor in this category. We give suitable conditions for when we can identify this \(\ \text{Ext}\) group with continuous group cohomology.

We present a new proof of Anderson's result that the real \(K\)-theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complex \(K\)-theory spectrum \(KU\) is \(C_2\)-equivariantly equivalent to \(\Sigma^4 KU\), where \(C_2\) acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of the \(K(1)\)-local Picard group.

We prove a thick subcategory theorem for the category of \(d\)-excisive functors from finite spectra to spectra. This generalizes the Hopkins-Smith thick subcategory theorem (the \(d=1\) case) and the \(C_2\)-equivariant thick subcategory theorem (the \(d=2\) case). We obtain our classification theorem by completely computing the Balmer spectrum of compact \(d\)-excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum à la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to \(H\mathbb{ℤ}\)-modules and we establish a functor calculus analogue of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone-Ching and Glasman.

We prove that a jointly conservative family of geometric functors between rigidly-compactly generated tensor triangulated categories induces a surjective map on spectra. From this we deduce a fiberwise criterion for Balmer's comparison map to be a homeomorphism. This gives short alternative proofs of the Hopkins--Neeman theorem and Lau's theorem for the trivial action.

We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and discover surprising relations between the theory of costratification and the theory of stratification. We prove that many categories in algebra, topology and geometry are costratified by developing and applying descent techniques. An overarching theme is that cosupport is relevant for diverse questions in tensor triangular geometry and that a full understanding of a category requires knowledge of both its support and its cosupport.

We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group \(G\), generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin--Tate \(E\)-theory \(\underline{E}_n\), for any finite height \(n\) and any finite group \(G\), where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing \(\otimes\)-ideals of the category of equivariant modules over \(\underline{E}_n\), thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.

Let \(E_n\) be Morava \(E\)-theory and let \(G \subset \mathbb{G}_n\) be a finite subgroup of \( \mathbb{G}_n\), the extended Morava stabilizer group. Let \( E_{n}^{tG} \) be the Tate spectrum, defined as the cofiber of the norm map \( N:(E_n)_{hG} \to E_n^{hG}\). We use the Tate spectral sequence to calculate \(\pi_*E_{p-1}^{tG}\) for \( G \) a maximal finite \( p\)-subgroup. We show that \(E_{p-1}^{tG} \simeq \ast\) and so the norm map gives equivalence between homotopy fixed points and homotopy orbit spectra. Our methods also give a calculation of \(\pi_*E_{p-1}^{hG}\), which is a folklore calculation of Hopkins and Miller.