I am a postdoc at Regensburg University. In the past I've been a postdoc at Haifa University, a guest researcher at Universität Hamburg, sponsored by the SPP 1786, and a guest at the Max Planck Institute for Mathematics in Bonn, Germany. I completed my PhD in 2014 under Craig Westerland at Melbourne University. I am broadly interested in stable homotopy theory with a particular interest in chromatic homotopy theory.

Publications

Derived completion for comodules - with Tobias Barthel, and Gabriel Valenzuela. Manuscripta Mathematica.(arXiv link).

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.

Local duality in algebra and topology - with Tobias Barthel and Gabriel Valenzuela. Advances in Mathematics. (arXiv link).

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.

Algebraic chromatic homotopy theory for $$BP_*BP$$-comodules. - with Tobias Barthel. Proceedings of the London Mathematical Society (arXiv link).

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.

The algebraic chromatic splitting conjecture for Noetherian commutative ring spectra - with Tobias Barthel and Gabriel Valenzula. Math. Z. (arXiv link).

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.

Vanishing lines for modules over the motivic Steenrod algebra - with Achim Krause. New York Journal of Mathematics (arXiv link).

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 $$\mathcal{A}/\tau$$.

Local duality for structured ring spectra - with Tobias Barthel and Gabriel Valenzula. Journal of Pure and Applied Algebra (arXiv link).

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.

Picard groups of higher real $$K$$-theory spectra at height $$n=p-1$$ - with Akhil Mathew and Vesna Stojanoska. Compositio Mathematica (arXiv link).

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.

The $$E_2$$-term of the $$K(n)$$-local $$E_n$$ Adams spectral sequence - with Tobias Barthel. Topology and its Applications (arXiv link).

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.

$$K$$-theory reality and duality - with Vesna Stojanoska. Journal of $$K$$-theory (arXiv link).

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.

Preprints

Picard groups and duality for real Morava $$E$$-theory - with Guchuan Li, and Danny Shi.

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 $$n=1$$.

On equivariant and motivic slices.

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 $$1 \le n \le \infty$$.

Stratification and duality for homotopical groups - with Tobias Barthel, Natalia Castellana, and Gabriel Valenzuela.

In this paper, we show that the category of module spectra over $$C^*(B\mathcal{G},\mathbb{F}_p)$$ is stratified for any p-local compact group $$\mathcal{G}$$, thereby giving a support-theoretic classification of all localizing subcategories of this category. To this end, we generalize Quillen's $$F$$-isomorphism theorem, Quillen's stratification theorem, Chouinard's theorem, and the finite generation of cohomology rings from finite groups to homotopical groups. Moreover, we show that $$p$$-compact groups admit a homotopical form of Gorenstein duality.

The homotopy limit problem and the Picard group of Hermitian $$K$$-theory.

We use descent theoretic methods to solve the homotopy limit problem for Hermitian $$K$$-theory over very general Noetherian 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.

The Tate cohomology of the higher real $$K$$-theories at height $$n = p-1$$.

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.

Contact

Drew Heard
Zi. M 305
Fakultät für Mathematik
Universität Regensburg
Regensburg, Germany 93040
E-mail: