For a long time I resisted maintaining a webpage for research papers, with the idea that it would be redundant, since you can find these things on the preprint arXiv. What changed my mind is that, on a webpage like this one, it’s possible to organize papers into the broader projects they belong to, and to explain what those broader projects are about. I have put some explanations along those lines on this page; maybe you will find something useful in it.

Unless otherwise noted, all these papers are somewhere in the journal system between “submitted” and “in print,” and you can find them on the preprint arXiv too. Versions on this page may be more current (with corrections of minor typos which don’t warrant a full arXiv update) than the arXiv versions.

**List of projects:**

- Complex multiplication and homotopy groups of spheres.
- THH and algebraic K-theory of K(F_q).
- Other papers.

**“Complex multiplication and homotopy groups of spheres” project:**

This is the main project I work on. The basic theme is to study formal groups with complex multiplication — that is, “formal modules” — and exploit some of their properties to both make deeper (that is, higher height) computations in the stable homotopy groups of spheres, and to establish descriptions of the zeta-functions associated to a number field K in terms of the flat cohomology groups of the moduli stack of formal groups with complex multiplication by the ring of integers of K. The project is far from finished, but already there are advances made in both of these two directions, which you can read in the papers below. I have also been working on a book on this material which is intended as a companion piece to Ravenel’s book “Complex cobordism and stable homotopy groups of spheres.”

- The structure of the classifying ring of formal groups with complex multiplication, 2015.

This paper computes the classifying ring L^A, modulo torsion, of formal groups with complex multiplication by a Dedekind domain A. Earlier work of Lazard and Drinfeld in special cases, and Hazewinkel in general, showed that L^A is a polynomial algebra as long as the class group of A vanishes. This implies qualitative features of the moduli theory of formal A-modules: every formal A-module over a quotient ring R/I lifts to a formal A-module over R, and every formal A-module n-bud extends to a formal A-module (n+1)-bud. The main result in this paper is a proof that, for an arbitrary Dedekind domain A, the classifying ring L^A is a*symmetric*(although not necessarily*polynomial*) algebra generated by a particular graded A-module, modulo torsion. This implies that the same qualitative features (lifting of formal A-modules from quotient rings, and extension of formal module buds) hold, without any restriction on the class group of A, as long as the coefficient rings of the formal A-modules in question are torsion-free.Along the way, we work out some general features of the functor that sends a commutative ring A to the classifying ring L^A of formal A-modules, particularly its behavior under colimits and completion. - Computation of the classifying ring of formal groups with complex multiplication, 2015.

This paper explicitly computes the classifying ring L^A of formal groups with complex multiplication by A, for A the ring of integers of a number field. (This differs from the results of the previous paper in that the computations are “on the nose,” not modulo torsion, but the methods are specific to number rings, not general Dedekind domains.) Lazard, Drinfeld, and Hazewinkel computed the ring L^A in the case of A a field or a Dedekind domain of class number one; in either case, L^A is a polynomial algebra over A. Hazewinkel described a case (specifically, the case of A the ring of integers in Q with a fourth root of -18 adjoined) where L^A could not possibly be a polynomial algebra, but Hazewinkel did not compute that ring L^A; it seems that, before this paper, L^A had never been computed in any situation in which it failed to be polynomial.In this paper we get that L^A is the symmetric algebra on a particular projective A-module. The proof of this fact is a pretty enjoyable one: it involves producing a homology theory which measures the failure of a certain comparison map between L^A and a symmetric algebra to be an isomorphism, and then using a May-type spectral sequence to prove a rigidity theorem for this homology theory.This leads to explicit presentations for L^A. You can look at the paper for really explicit presentations (which I’m not going to try to render in HTML for this webpage!) for L^A in the case of A the ring of integers in a quadratic extension of Q, and for A the ring of integers in Q with a fourth root of -18 adjoined, that is, Hazewinkel’s original example.It also leads to qualitative results about the moduli theory of formal A-modules, namely, any formal A-module over R/I lifts to a formal A-module over R, and any formal A-module n-bud extends to a formal A-module (n+1)-bud.In this paper, there are also some results on L^A for some rings A other than number rings. Most importantly, when A is the group ring Z[C_p], that is, the integral group ring of the cyclic group with p elements, we give an explicit presentation for the group ring L^A with p inverted. As this L^A is the classifying ring of formal groups with an action by C_p, these ought to have some relationship to the “C_p-equivariant FGLs” studied by Abram, Cole, Greenlees, Kriz, and Strickland, and perhaps some bearing on Greenlees’s conjecture about the relationship between the classifying ring of C_p-equivariant FGLs and the coefficient ring of C_p-equivariant complex bordism, but those connections are speculative at this point.

- Ravenel’s algebraic extensions of the sphere spectrum do not exist, 2015.

In a 1983 paper, Ravenel asked whether there exists a spectrum S^A such that the complex bordism of S^A is isomorphic to L^A as a module over the coefficient ring MU_* of complex bordism. Here A is the ring of integers in a finite extension of the p-adic rationals. Such a spectrum S^A would then be an “algebraic extension of the sphere spectrum.” In the base case, when A is the ring of p-adic integers, S^A exists: it is simply the p-complete sphere spectrum.Some later work was done on this problem: A. Pearlman computed the Morava K-theories of the spectra S^A, assuming they exist, and T. Lawson showed that, if S^A exists, then it cannot be a A_p-ring spectrum, that is, a spectrum with a ring structure that is associative up to pth-order homotopies. Lawson also showed that S^A does not exist if A is the ring of integers in the 2-adic rationals with a primitive fourth root of unity adjoined, but in all other cases, Ravenel’s original problem remained open.This paper settles Ravenel’s question: S^A does not exist except in the base case, when A is the ring of p-adic integers. The natural global and p-typical analogues of Ravenel’s question are also stated and then shown to have negative answers. These all ultimately follow from the paper’s main technical result, a topological nonrealizability theorem for modules over the coefficient ring BP_* of Brown-Peterson homology, which implies that the only spectra X such that BP_*(X) is a V^A-module are extensions of rational spectra by dissonant spectra. Here V^A is the classifying ring of A-typical formal A-modules.(While formal A-modules are not topologically realizable in the way that Ravenel asked about, they are still very useful for making computations in topology: see the papers below, where I use them to compute the homotopy groups of the K(4)-local Smith-Toda complex V(3).) - Ravenel’s Global Conjecture is true, 2015.

In a 1983 paper, Ravenel made a conjecture, the “Global Conjecture,” describing the orders of the Ext^1 groups of the classifying Hopf algebroid of formal A-modules, or equivalently, the orders of the flat cohomology groups H^1 of the moduli stack of formal A-modules, in terms of numbers generalizing Adams’s numbers h(f,t) from “On the groups J(X) II.” In this paper we get a proof of the Global Conjecture by a combination of spectral sequence methods and a kind of Hasse principle for ideals satisfying conditions generalizing the characterizing properties of ideals generated by Adams’ numbers.Some interesting things come out of this which go beyond what Ravenel originally conjectured. In this paper we see that certain classes of Hecke L-functions can be recovered (via their Euler products) from the flat cohomology group H^1 of the moduli stack of formal modules. For example, if K/Q and L/Q are finite Galois extensions with rings of integers A and B respectively, and if we suppose that 2 ramifies in both A and B and that [K : Q] and [L : Q] are odd primes, then the flat cohomology group H^1 of the moduli stack of formal A-modules is isomorphic (as a graded abelian group) to the flat cohomology group of the moduli stack of formal B-modules if and only if the Dedekind zeta-function of K is equal to the Dedekind zeta-function of L. (The assumptions on degree and ramification can be lifted somewhat; see the paper for details.)The representations of Galois groups in the cohomology of Lubin-Tate spaces, and more “globally,” Shimura varieties, are often used to study the zeta-functions associated to those representations. The moduli stack of formal A-modules is an alternative “globalization” to a Shimura variety: like a Shimura variety, the formal completion at each point is a deformation space of a Barsotti-Tate module (e.g. a Lubin-Tate space), and in this paper we show that zeta-function data is also visible in the cohomology of the moduli stack of formal A-modules. - Moduli of formal A-modules under change of A, 2016. (The earliest version of this paper was one of the first things I wrote, but in 2016 I went back and rewrote the paper from scratch, with many new results.)

This one sets up the machinery you need in order to explicitly compute where classes in the cohomology of the height n Morava stabilizer group are sent, under the restriction map in cohomology, into the cohomology of the group of automorphisms of a height n formal group which commute with complex multiplication by A, where A is the ring of integers in a field extension of the p-adic rationals of degree dividing n. This is a pretty useful thing to be able to do: in the paper we show that, using some local class field theory, this gets you an action of the Galois group Gal(K^ab/K^nr) on the Morava/Lubin-Tate spectrum E_n, for each degree n field extension K of Q_p. The paper also computes the map from the K(2)-local homotopy groups of the Smith-Toda complex V(1) to the (automatically K(2)-local) homotopy groups of the homotopy fixed-points of that Galois action on E_2, for each of the quadratic extensions K/Q_p, for p>3. Another point of view: this construction is getting you a map from K(n)-local homotopy into the Galois cohomology of K^ab/K^nr for each degree n field extension K/Q_p. Here K^ab is the maximal abelian extension of K and K^nr is the maximal unramified extension of K.This map detects a lot: in the paper we show that, in the n=2 case, for each element in the K(2)-local homotopy of V(1), either that element or its Poincare dual (exactly one or the other, not both) maps nontrivially into the Galois cohomology of K^ab/K^nr for some quadratic K/Q_p. For example, the element zeta_2 from the chromatic splitting conjecture maps nontrivially into the Galois cohomology of K^ab/K^nr for K/Q_p unramified, while all the other elements in the K(2)-local homotopy of V(1) map nontrivially into the Galois cohomology of K^ab/K^nr for K/Q_p totally ramified.(I don’t mention it in this paper, but this phenomenon does appear to happen more generally than in just the p>3 quadratic cases; it is more difficult to describe the phenomenon, however, in the p=3, n=2 and p=5, n=4 cases, where it also seems to be occuring, since the Morava stabilizer group at those heights and primes has infinite cohomological dimension and so its cohomology does not have the “easy” Poincare duality.) - Height four formal groups with quadratic complex multiplication, 2016.

In this paper you can find the more technical (and powerful) machinery for computing cohomology of automorphism groups of formal groups with complex multiplication. In Ravenel’s “green book” he sets up a filtration on the Morava stabilizer algebras so that their associated graded Hopf algebras are (dual to) primitively generated Hopf algebras, so that the methods of Milnor-Moore and Peter May’s thesis can be used to compute the cohomology of strict automorphism groups of formal groups (since the Morava stabilizer algebras are the continuous linear duals of the group rings of those strict automorphism groups). This paper does the same thing for formal groups with complex multiplication, reducing the problem of computing the cohomology of their automorphism groups to the problem of computing the cohomology of solvable Lie algebras using a Chevalley-Eilenberg resolution, and then running certain May spectral sequences.Then in this paper we actually carry out the computations to compute the cohomology of the group G of the strict automorphisms of a height four formal group which commute with complex multiplication by Z_p[sqrt(p)], for primes p>5. This group turns out to have cohomological dimension 8 and its cohomology has rank 80. (That computation also appears in the draft version of “The cohomology of the height four Morava stabilizer group at large primes,” below; when that paper is ready for journal submission, I’ll trim that computation in it, with the idea that “Height four formal groups with quadratic complex multiplication” is the paper with the authoritative version. This also tells you one reason why the cohomology of this particular group G matters: it is a stepping stone on the way to computing the cohomology of the height four Morava stabilizer group.)Finally, we show that the automorphism group of a formal group with complex multiplication is a*closed*subgroup of the Morava stabilizer group, so we can use the work of Devinatz-Hopkins to take the homotopy fixed-points of the action of this closed subgroup on a Lubin-Tate/Morava E-theory spectrum. In this paper we carry out that computation to get the V(3)-homotopy of the homotopy fixed points of the group G, above, acting on E_4, for primes p>5 (since otherwise V(3) does not exist). The result is 2(p^2 – 1)-periodic, and interpolates between the K(4)-local homotopy groups of V(3) (which are quite complicated, and 2(p^4 – 1)-periodic) and the Lubin-Tate fixed points arising from Galois cohomology of totally ramified quartic extensions of Q_p (which are quite simple, and 2(p – 1)-periodic), as in the paper “Moduli of formal A-modules under change of A,” above. - The cohomology of the height four Morava stabilizer group at large primes, 2016.

This is, for the moment, an announcement and not a finished paper; I will update the version of the document available here once I have a complete write-up of the results. Despite this, the main results are stated and the main tools are developed; the only content missing from this document is the computation of certain spectral sequence differentials which I worked out using a computer, and which I am still figuring out how to explain and typeset concisely and intelligibly in a paper.This paper features the computation of the mod p cohomology of the height 4 Morava stabilizer group, that is, the automorphism group of a p-height 4 formal group law over the field with p elements, at primes p > 5. This uses some new “height-shifting” techniques: we construct a sequence of spectral sequences whose input is the cohomology of the height n Morava stabilizer group, and whose output is the cohomology of th e automorphism group of an A-height n formal A-module, where A is the ring of integers in a quadratic extension of the p-adic rationals; and then we construct a sequence of spectral sequences whose input is the cohomology of an A-height n formal A-module, and whose output is the cohomology of the height 2n Morava stabilizer group. This generalizes, by replacing the quadratic extension with an arbitrary extension K/Q_p and replacing 2n with [K : Q_p] times n, but it is the quadratic case that this paper is primarily concerned with, since we actually then run the spectral sequences in the case n = 2 and p > 5.In the end, the rank (i.e., vector space dimension over F_p) of the cohomology of the height 4 Morava stabilizer group at large primes is 3,440. (Compare this to 152 for height 3, 12 for height 2, and 2 for height 1.) These ranks fit into a pattern, given in the paper, which suggests a conjecture describing the rank of the cohomology of the height n Morava stabilizer group at large primes, for all n. As far as I know, this is the first attempt at giving a conjectural description of the ranks of these cohomology groups for all heights.There is no room for differentials in the E(4)-local Adams spectral sequence for the Smith-Toda complex V(3), so the cohomology of the height 4 Morava stabilizer group computed in this paper is also (after a regrading) the homotopy groups of the K(4)-local Smith-Toda V(3). I’m told that Mahowald conjectured that V(3) is the last Smith-Toda complex to exist, i.e., that V(n) fails to exist for n > 3 and for all primes. If that conjecture is true, then the computation in this paper finishes the problem of computing the K(n)-local homotopy groups of V(n-1).

**“THH and algebraic K-theory of K(F_q)” project:**

Gabe Angelini-Knoll and I have been working on the problem of computing the Waldhausen algebraic K-groups of the algebraic K-theory spectra of certain finite fields. This is an example of “iterated K-theory” and Rognes’ redshift conjecture is not known in these cases. Thus far, this project has led to (with the aid of a new “THH-May” spectral sequence for computing topological Hochschild homology) the computation of the homotopy groups of THH(K(F_q)) smashed with the p-primary Smith-Toda complex V(1), for p > 2 and for many (but not all) prime powers q. Parts of this project are joint work, and other parts are Gabe’s work entirely, so some of the resulting papers have been coauthored and some are authored solely by Gabe. Gabe is working on the computations of the homotopy groups of the C_p fixed points of this spectrum (part of which were computed in Gabe’s thesis), with the goal of using trace methods to recover the K-groups of K(F_q).

- Maps of simplicial spectra whose geometric realizations are cofibrations, joint with Gabe Angelini-Knoll, 2016.

Given a map of simplicial topological spaces, mild conditions on degeneracies and the levelwise maps imply that the geometric realization of the simplicial map is a cofibration. These conditions are not formal consequences of model category theory, but depend on properties of spaces, and similar results have not been available for any model for the stable homotopy category. In this paper, Gabe and I prove such results for symmetric spectra. Consequently, we get a set of conditions which ensure that the geometric realization of a map of simplicial symmetric spectra is a cofibration. These conditions are very “user-friendly” in that they are simple, often easily checked, and do not require computation of a latching object or any other knowledge of Reedy theory.Gabe and I use the results in this paper in our next paper in this series, in which we construct the May spectral sequence for topological Hochschild homology. - A May-type spectral sequence for higher topological Hochschild homology, joint with Gabe Angelini-Knoll, 2016.

Given a filtration of a commutative monoid A in a symmetric monoidal stable model category C, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of A, and whose output is the higher order topological Hochschild homology of A. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring R, we get an upper bound on the size of the THH-groups of E_infty-ring spectra A such that pi_*(A) = R.Gabe uses this spectral sequence in his computation of the topological Hochschild homology of the algebraic K-theory spectrum of certain finite fields, which is input for his further computations in the iterated algebraic K-theory of finite fields. You can find Gabe’s computation of the V(1)-homotopy of THH(K(F_q)) using our THH-May spectral sequence, for a certain family of prime powers q, in the paper*On topological Hochschild homology of the K(1)-local sphere*on Gabe’s website, here.

**Other papers.**

These are miscellaneous papers, most of which are addressed at algebraic and categorical problems which arose while dealing with some computational problem in homotopy theory.

- Totalization of simplicial homotopy types, joint with Crichton Ogle, 2013.

It is fairly well-known that, given a simplicial object in the homotopy category of a stable model category (i.e., a “simplicial homotopy type”), there are obstructions to building the geometric realization of that simplicial homotopy type; these obstructions are given by Toda brackets; and these obstructions vanish, of course, if the simplicial object lifts to a simplicial object in the stable model category itself.Now given two simplicial homotopy types which*are*geometrically realizable, and a map of simplicial homotopy types between them, can that map be geometrically realized? In this paper we build a sequence of “secondary Toda brackets” which are the obstructions to geometric realization of a map of simplicial homotopy types. We also give an example from cyclic homology, due to Ogle, where these obstructions do not vanish. - Relative homological algebra, Waldhausen K-theory, and quasi-Frobenius conditions, 2013.

An “allowable class” on an abelian category is the structure required to do relative homological algebra in that category. (If you have ever worked with comodules over Hopf algebroids in order to compute generalized Adams spectral sequences, then this is a familiar story, the generalized Adams E_2-term is a relative Ext.) This paper works out when an allowable class on an abelian category also defines a Waldhausen structure with cylinder functor, so that one can consider and compute algebraic K-groups of that category which split all the short exact sequences in that allowable class.As an application, this paper proves the following: let k be a finite field with n elements, and let C be the category of finitely generated k[x]/x^n-modules, with the Waldhausen structure in which the cofibrations are the monomorphisms and the weak equivalences are the stable equivalences. Let K(C) denote the Waldhausen K-theory spectrum of C. Then K(C) is a complex-oriented ring spectrum. This is a partial answer to a question asked by J. Morava about which K-theory spectra admit complex orientations. - Homotopy colimits in stable representation theory, 2013.

In representation theory of algebras (and, for a stable homotopy theorist, the motivating examples are often the Steenrod algebra or its subalgebras, or the group ring of the Morava stabilizer group or its subalgebras) one encounters the following construction: given an abelian category C, two maps are homotopic if their difference factors through a projective object, and a map is a stable equivalence if it admits an inverse up to homotopy. If C is quasi-Frobenius, then there exists a model structure on C whose weak equivalences are the stable equivalences, and one can make use of the usual theory of homotopy colimits in a model category.If C is not quasi-Frobenius, then it is not known that C admits a model structure in which the weak equivalences are the stable equivalences, and so it is not clear whether C admits homotopy-invariant colimits. This paper proves that the answer is*no*: if C has enough projectives and at least one object of finite, positive projective dimension, then C does not admit well-defined homotopy cofibers. Consequently C does not admit a model structure in which the weak equivalences are the stable equivalences.On the other hand, when C has enough projectives, when every projective object in C is injective, and when every object can be embedded appropriately into a projective object, then in this paper it is proven that C*does*admit homotopy-invariant colimits. This includes cases where it is not known that C admits a model category structure whose weak equivalences are stable equivalences. The results in the paper are also somewhat more general than this, allowing the use of relative projectives (in the sense of relative homological algebra) in places of projectives, throughout. - A recognition principle for the existence of descent data, 2014.

Let R and S be rings. Given a faithfully flat ring map from R to S and an R-module M, well-known descent theory describes (in terms of a cohomology group, for example) the set of isomorphism classes of S/R-descent data on M base-changed to S, i.e., the set of isomorphism classes of R-modules whose base-change to S is isomorphic to that of M. But, given an S-module N, there do not seem to be any tools for determining whether N admits a S/R-descent datum at all. In other words: given an S-module N, how can we tell whether N is isomorphic to some R-module base-changed to S?This paper develops a tool for answering questions of this kind. The main theorem is phrased in a high level of abstraction, in terms of extensions of comonads, but the concrete special case I like best is the following: given an augmented commutative algebra A over a field k and a ring map from A to a commutative k-algebra B, we get reasonable sufficient conditions on the map for the following condition to hold: a B-module M admits an B/A-descent datum if and only if the base-change of M to B \tensor_A k is a free (B \tensor_A k)-module.This version of the paper also includes an example computation at the end which did not appear in the journal version (the referee thought it was redundant; I see the point, but I still prefer to give example computations). - How many adjunctions give rise to the same monad?, 2014.

Given an adjoint pair of functors F,G, the composite GF naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. This paper works out some methods for computing the collection of adjunctions which give rise a given monad. This includes a Beck-like criterion for uniqueness of such adjunctions, and also some explicit example computations. - Obstructions to compatible splittings, 2014.

If one has a map of split short exact sequences in a category of modules, or more generally, in any abelian category, then it is not always the case that the short exact sequences split*compatibly.*In this paper we define and prove basic properties of a group of obstructions to the existence of compatible splittings. - Failure of flat descent of model structures on module categories, 2014.

This one isn’t published or submitted for publication; it’s just a short note with one small interesting result: if you define a “presheaf of model structures on module categories” which sends a scheme (or stack) X to the collection of model structures on the category of quasicoherent X-modules, then this presheaf fails to satisfy fppf descent. - The Hochschild homology of A(1), 2015.

This paper solves an old “folk” problem, the computation of the Hochschild homology of A(1), the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares, Sq^1,Sq^2. The main tool in the computation is a number of May-type spectral sequences. - The Bousfield localizations and colocalizations of the discrete model structure, 2016. (The earliest version of this paper was another one of the first things I wrote, but in 2016 I went back and rewrote the paper from scratch.)

This short paper works out all the Bousfield localizations and colocalizations of the discrete model structure on any reasonable category, including the homotopy category and Waldhausen K-theory of each of those (co)localizations. As an application, it shows that every replete reflective subcategory is the subcategory of fibrant objects of a model structure.(The results in this particular paper aren’t difficult to prove: most of the work is done by a theorem of Cassidy, Hebert, and Kelly. But these results apparently aren’t already in the literature, and several other homotopy theorists have told me they found the results novel, so I went ahead and submitted the paper.) - Graded comodule categories with enough projectives, 2016.

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. In particular, the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson homology, and classical mod p homology all have the property that their categories of connective graded comodules have enough projectives.