arXiv subjects

A. Salch

Publications and source records attributed to A. Salch.

18 recordsLinked to original sources

arXiv2024

An algebraic approach to asymptotics of the number of unlabelled bicolored graphs

We define and study two structures associated to permutation groups: Dirichlet characters on permutation groups, and the "cycle form," a bilinear form on the group algebras of permutation groups. We use Dirichlet characters and the cycle form to find a new upper bound on the number of unlabelled bicolored graphs with $p$ red vertices and $q$ blue vertices. We use this bound to calculate the asymptotic growth rate of the number of such graphs as $p,q\rightarrow\infty$, answering a 1973 question of Harrison in the case where $q-p$ is fixed. As an application, we show that, in an asymptotic sense, "most" elements of the power set $P(\{ 1, \dots ,p\} \times \{ 1, \dots ,q\})$ are in free $Σ_p\times Σ_q$-orbits.

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arXiv2023

Ravenel's May spectral sequence collapses immediately at large primes

At large primes, the height $n$ Ravenel-May spectral sequence takes as input the cohomology of a certain solvable Lie $\mathbb{F}_p$-algebra, and produces as output the mod $p$ cohomology of the height $n$ strict Morava stabilizer group scheme. We construct simultaneous integral deformations of the height $n$ Morava stabilizer algebras and related objects, and we use them to prove that, for fixed $n$, the height $n$ Ravenel-May spectral sequence collapses for all sufficiently large primes $p$. Consequently, for large $p$, the mod $p$ cohomology of the strict Morava stabilizer group scheme is the cohomology of a finite-dimensional solvable Lie algebra, and is computable algorithmically.

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arXiv2023

Derived functors of product and limit in the category of comodules over the dual Steenrod algebra

In the 2000s, Sadofsky constructed a spectral sequence which converges to the mod $p$ homology groups of a homotopy limit of a sequence of spectra. The input for this spectral sequence is the derived functors of sequential limit in the category of graded comodules over the dual Steenrod algebra. Since then, there has not been an identification of those derived functors in more familiar or computable terms. Consequently there have been no calculations using Sadofsky's spectral sequence except in cases where these derived functors are trivial in positive cohomological degrees. In this paper, we prove that the input for the Sadofsky spectral sequence is simply the local cohomology of the Steenrod algebra, taken with appropriate (quite computable) coefficients. This turns out to require both some formal results, like some general results on torsion theories and local cohomology of noncommutative non-Noetherian rings, and some decidedly non-formal results, like a 1985 theorem of Steve Mitchell on some very specific duality properties of the Steenrod algebra not shared by most finite-type Hopf algebras. Along the way there are a few results of independent interest, such as an identification of the category of graded $A_*$-comodules with the full subcategory of graded $A$-modules which are torsion in an appropriate sense.

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arXiv2023

KU-local zeta-functions of finite CW-complexes

Begin with the Hasse-Weil zeta-function of a smooth projective variety over the rational numbers. Replace the variety with a finite CW-complex, replace etale cohomology with complex K-theory $KU^*$, and replace the $p$-Frobenius operator with the $p$th Adams operation on $K$-theory. This simple idea yields a kind of "$KU$-local zeta-function" of a finite CW-complex. For a wide range of finite CW-complexes $X$ with torsion-free $K$-theory, we show that this zeta-function admits analytic continuation to a meromorphic function on the complex plane, with a nice functional equation, and whose special values in the left half-plane recover the $KU$-local stable homotopy groups of $X$ away from $2$. We then consider a more general and sophisticated version of the $KU$-local zeta-function, one which is suited to finite CW-complexes $X$ with nontrivial torsion in their $K$-theory. This more sophisticated $KU$-local zeta-function involves a product of $L$-functions of complex representations of the torsion subgroup of $KU^0(X)$, similar to how the Dedekind zeta-function of a number field factors as a product of Artin $L$-functions of complex representations of the Galois group. For a wide range of such finite CW-complexes $X$, we prove analytic continuation, and we show that the special values in the left half-plane recover the $KU$-local stable homotopy groups of $X$ away from $2$ if and only if the skeletal filtration on the torsion subgroup of $KU^0(X)$ splits completely.

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arXiv2023

The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective

It is well-known that the Steenrod algebra $A$ is self-injective as a graded ring. We make the observation that simply changing the grading on $A$ can make it cease to be self-injective. We see also that $A$ is not self-injective as an ungraded ring. These observations follow from the failure of certain coproducts of injective $A$-modules to be injective. Hence it is natural to ask: which coproducts of graded-injective modules, over a general graded ring, remain graded-injective? We give a complete solution to that question by proving a graded generalization of Carl Faith's characterization of $Σ$-injective modules. Specializing again to the Steenrod algebra, we use our graded generalization of Faith's theorem to prove that the covariant embedding of graded $A_*$-comodules into graded $A$-modules preserves injectivity of bounded-above objects, but does not preserve injectivity in general.

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arXiv2023

Graded comodule categories with enough projectives

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. Applications to algebraic topology are given: 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. We also prove that categories of connective graded comodules over appropriate Hopf algebras fail to be equivalent to categories of graded connective modules over a ring.

math.RA↗

arXiv2023

Kunneth formulas for Cotor

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e., Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the $0$th Cotor group, i.e., the cotensor product. We give topological applications in the form of consequences for the $E_2$-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

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arXiv2023

Denominators of special values of zeta-functions count KU-local homotopy groups of mod p Moore spectra

In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.

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arXiv2016

Moduli of formal A-modules under change of A

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an $A$-height $1$-formal $A$-module with the ramified part of the abelianization of the absolute Galois group of $K$, yielding an action of $Gal(K^{ab}/K^{nr})$ on the Lubin-Tate/Morava $E$-theory spectrum $E_2$ for each quadratic extension $K/\mathbb{Q}_p$. Finally, we run the associated descent spectral sequence to compute the $V(1)$-homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the $K(2)$-local homotopy groups of the Smith-Toda complex $V(1)$, either that element or its dual is detected in the Galois cohomology of the abelian closure of some quadratic extension of $\mathbb{Q}_p$.

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arXiv2016

Height four formal groups with quadratic complex multiplication

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with complex multiplication by the ring of integers in the field $\mathbb{Q}_p(\sqrt{p})$, for primes $p>5$. This is a large subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K(4)$-local $E_4$-Adams spectral sequence to compute the homotopy groups of the homotopy fixed-point spectrum of this group's action on the Lubin-Tate/Morava spectrum $E_4$.

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arXiv2016

The Bousfield localizations and colocalizations of the discrete model structure

We compute the Bousfield localizations and Bousfield colocalizations of discrete model categories, including the homotopy categories and the algebraic $K$-groups of these localizations and colocalizations. We prove necessary and sufficient conditions for a subcategory of a category to appear as the subcategory of fibrant objects for some such model structure. We also prove necessary and sufficient conditions for a monad to be the fibrant replacement monad of some such model structure.

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arXiv2016

Ravenel's algebraic extensions of the sphere spectrum do not exist

In this paper we prove a topological nonrealizability theorem: certain classes of graded $BP_*$-modules are shown to never occur as the $BP$-homology of a spectrum. Many of these $BP_*$-modules admit the structure of $BP_*BP$-comodules, meaning that their topological nonrealizability does not follow from earlier results like Landweber's filtration theorem. As a consequence we solve Ravenel's 1983 problem on the existence of "algebraic extensions of the sphere spectrum": algebraic extensions of the sphere spectrum do not exist, except in trivial cases.

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arXiv2016

The cohomology of the height four Morava stabilizer group at large primes

This is an announcement of some new computational methods in stable homotopy theory, in particular, methods for using the cohomology of small-height Morava stabilizer groups to compute the cohomology of large-height Morava stabilizer groups. As an application, the cohomology of the height four Morava stabilizer group is computed at large primes (its rank turns out to be $3440$). Consequently we are able to formulate a plausible conjecture on the rank of the large-primary cohomology of the Morava stabilizer groups at all heights.

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arXiv2015

Ravenel's Global Conjecture is true

I prove Ravenel's 1983 "Global Conjecture" on $\Ext^1$ over the classifying Hopf algebroid of formal $A$-modules, equivalently, the first flat cohomology group $H^1_{fl}$ of the moduli stack $\mathcal{M}_{fmA}$ of formal $A$-modules. I then show that the Hecke $L$-functions of certain Großencharakters of Galois extensions $K/\mathbb{Q}$ can be computed from $H^1_{fl} (\mathcal{M}_{fmA})$, and vice versa; as a consequence I show that, for a large class of Galois extensions of $\mathbb{Q}$, two extensions $K,L$ are arithmetically equivalent (i.e., they have the same Dedekind zeta-function) if and only if the flat cohomology groups $H^1_{fl}(\mathcal{M}_{fm\mathcal{O}_K})$ and $H^1_{fl}(\mathcal{M}_{fm\mathcal{O}_L})$ agree.

math.AT↗

arXiv2015

Relative homological algebra, Waldhausen $K$-theory, and quasi-Frobenius conditions

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated $K$-theory groups are "stable algebraic $G$-theory," which in degree zero form a certain stable representation group. We prove both some existence and nonexistence results about such Waldhausen category structures, including the fact that, while it was known that the category of $R$-modules admits a model category structure if $R$ is quasi-Frobenius, that assumption is required even to get a Waldhausen category structure with cylinder functor---i.e., Waldhausen categories do not offer a more general framework than model categories for studying stable representation theory of rings. We study multiplicative structures on these Waldhausen categories, and we relate stable algebraic $G$-theory to algebraic $K$-theory and we compute stable algebraic $G$-theory for finite-dimensional quasi-Frobenius nilpotent extensions of finite fields. Finally, we show that the connective stable $G$-theory spectrum of $\mathbb{F}_{p^n}[x]/x^{p^n}$ is a complex oriented ring spectrum, partially answering a question of J. Morava about complex orientations on algebraic $K$-theory spectra.

math.KT↗

arXiv2015

A recognition principle for the existence of descent data

Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twisted forms lets one compute, given an $R$-module $M$, how many isomorphism classes of $R$-modules $M^{\prime}$ satisfy $S\otimes_R M\cong S\otimes_R M^{\prime}$. This is really a uniqueness problem. But this theory does not help one to solve the corresponding existence problem: given an $S$-module $N$, does there exists {\em some} $R$-module $M$ such that $S\otimes_R M\cong N$? In this paper we work out (as a special case of a general theorem about existence of coalgebra structures over a comonad) a criterion for the existence of such an $R$-module $M$, under some reasonable hypotheses on the map $R\rightarrow S$.

math.CT↗

arXiv2013

Homotopy colimits in stable representation theory

We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). Along the way, we obtain relative-homological-algebraic generalizations of classical theorems of Hilton-Rees and Oort. We describe some applications to Waldhausen $K$-theory and to deformation-theoretic methods in stable representation theory.

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arXiv2010

Grothendieck duality under Spec Z

We define the derived category of a concrete category in a way which extends the usual definition of the derived category of a ring, and we prove that the bounded-below derived category of $\Spec \mathbb{M}_0$ (an approximation, used by e.g. Connes and Consani, to "$\Spec$ of the field with one element") is the stable homotopy category of connective spectra. We also describe some basic features of Grothendieck duality for the map from $\Spec \mathbb{Z}$ to $\Spec \mathbb{M}_0$, or, what comes to the same thing, the map from $\Spec \mathbb{Z}$ to $\Spec$ of the sphere spectrum; these basic features include a computation of the homology of the dualizing complex $f^!(S)$ of abelian groups associated to the sphere spectrum.

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