arXiv subjects

Daniel G. Davis

Publications and source records attributed to Daniel G. Davis.

At least 19 recordsLinked to original sources

arXiv2023

A homotopy orbit spectrum for profinite groups

For a profinite group $G$, we define an $S[[G]]$-module to be a certain type of $G$-spectrum $X$ built from an inverse system $\{X_i\}_i$ of $G$-spectra, with each $X_i$ naturally a $G/N_i$-spectrum, where $N_i$ is an open normal subgroup and $G \cong \lim_i G/N_i$. We define the homotopy orbit spectrum $X_{hG}$ and its homotopy orbit spectral sequence. We give results about when its $E_2$-term satisfies $E_2^{p,q} \cong \lim_i H_p(G/N_i, π_q(X_i))$. Our main result is that this occurs if $\{π_\ast(X_i)\}_i$ degreewise consists of compact Hausdorff abelian groups and continuous homomorphisms, with each $G/N_i$ acting continuously on $π_q(X_i)$ for all $q$. If $π_q(X_i)$ is additionally always profinite, then the $E_2$-term is the continuous homology of $G$ with coefficients in the graded profinite $\widehat{\mathbb{Z}}[[G]]$-module $π_\ast(X)$. Other results include theorems about Eilenberg-Mac Lane spectra and about when homotopy orbits preserve weak equivalences.

math.AT↗

arXiv2023

A continuous $p$-adic action on the $K(2)$-local algebraic $K$-theory of $p$-adic complex $K$-theory

Let $p$ be a prime, let $KU_p$ be $p$-complete complex $K$-theory, and let $\mathbb{Z}_p^\times$ denote the group of units in the $p$-adic integers. The $p$-adic Adams operations induce an action of the profinite group $\mathbb{Z}_p^\times$ on $KU_p$, and hence, on the algebraic $K$-theory spectrum $K(KU_p)$. For $p \geq 5$, we give an elementary construction of the continuous homotopy fixed point spectrum $(L_{K(2)}K(KU_p))^{hG}$, where $K(2)$ is the second Morava $K$-theory and $G$ is any closed subgroup of $\mathbb{Z}_p^\times$. Also, for each $G$, we show that there is an associated strongly convergent homotopy fixed point spectral sequence whose $E_2$-term is given by Jannsen's continuous group cohomology, with $E_2^{s,\ast} = 0$, for all $s > 2$. This work is related to a conjecture of Ausoni and Rognes.

math.AT↗

arXiv2023

Several homotopy fixed point spectral sequences in telescopically localized algebraic $K$-theory

Let $n \geq 1$, $p$ a prime, and $T(n)$ any representative of the Bousfield class of the telescope $v_n^{-1}F(n)$ of a finite type $n$ complex. Also, let $E_n$ be the Lubin-Tate spectrum, $K(E_n)$ its algebraic $K$-theory spectrum, and $G_n$ the extended Morava stabilizer group, a profinite group. Motivated by an Ausoni-Rognes conjecture, we show that there are two spectral sequences \[{^{I}}\mspace{-3mu}E_2^{s,t} \Longrightarrow π_{t-s}((L_{T(n+1)}K(E_n))^{hG_n}) \Longleftarrow {^{II}}\mspace{-2mu}E_2^{s,t}\] with common abutment $π_\ast(-)$ of the continuous homotopy fixed points of $L_{T(n+1)}K(E_n)$, where ${^{I}}\mspace{-3mu}E_2^{s,t}$ is continuous cohomology with coefficients in a certain tower of discrete $G_n$-modules. If the tower satisfies the Mittag-Leffler condition, then there are continuous cochain cohomology groups \[{^{I}}\mspace{-3mu}E_2^{\ast,\ast} \cong H^\ast_\mathrm{cts}(G_n, π_\ast(L_{T(n+1)}K(E_n))) \cong {^{II}}\mspace{-2mu}E_2^{\ast,\ast}.\] We isolate two hypotheses, the first of which is true when $(n,p) = (1,2)$, that imply $(L_{T(n+1)}K(E_n))^{hG_n} \simeq L_{T(n+1)}K(L_{K(n)}S^0)$. Also, we show that there is a spectral sequence \[H^s_\mathrm{cts}(G_n, π_t(K(E_n) \otimes T(n+1))) \Longrightarrow π_{t-s}((K(E_n) \otimes T(n+1))^{hG_n}).\]

math.AT↗

arXiv2021

A computational reduction for many base cases in profinite telescopic algebraic $K$-theory

For primes $p\geq 5 $, $K(KU_p)$ -- the algebraic $K$-theory spectrum of $(KU)^{\wedge}_p$, Morava $K$-theory $K(1)$, and Smith-Toda complex $V(1)$, Ausoni and Rognes conjectured (alongside related conjectures) that $L_{K(1)}S^0 \mspace{-1.5mu}\xrightarrow{\mspace{-2mu}\text{unit} \, i}~\mspace{-7mu}(KU)^{\wedge}_p$ induces a map $K(L_{K(1)}S^0) \wedge v_2^{-1}V(1) \to K(KU_p)^{h\mathbb{Z}^\times_p} \wedge v_2^{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}S^0) \wedge v_2^{-1}V(1) \to (K(KU_p) \wedge v_2^{-1}V(1))^{h\mathbb{Z}^\times_p}$, which is induced by $i$ and also should be an equivalence. We show that for any closed $G < \mathbb{Z}^\times_p$, $π_\ast((K(KU_p) \wedge v_2^{-1}V(1))^{hG})$ is a direct sum of two pieces given by (co)invariants and a coinduced module, for $K(KU_p)_\ast(V(1))[v_2^{-1}]$. When $G = \mathbb{Z}^\times_p$, the direct sum is, conjecturally, $K(L_{K(1)}S^0)_\ast(V(1))[v_2^{-1}]$ and, by using $K(L_p)_\ast(V(1))[v_2^{-1}]$, where $L_p = ((KU)^{\wedge}_p)^{h\mathbb{Z}/((p-1)\mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in \[(-)^{h\mathbb{Z}^\times_p} \wedge v_2^{-1}V(1) \simeq (K(KU_p) \wedge v_2^{-1}V(1))^{h\mathbb{Z}^\times_p},\] $K(KU_p)$ fills in the blank; we show that for any $G$, the blank can be filled by $(K(KU_p))^\mathrm{dis}_\mathcal{O}$, a discrete $\mathbb{Z}^\times_p$-spectrum built out of $K(KU_p)$.

math.AT↗

arXiv2020

A construction of some objects in many base cases of an Ausoni-Rognes conjecture

Let $p$ be a prime, $n \geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $\mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) \to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map \[K(L_{K(n)}(S^0)) \wedge v_{n+1}^{-1}V_n \to (K(E_n))^{h\mathbb{G}_n} \wedge v_{n+1}^{-1}V_n\] that is a weak equivalence, where (b) since $\mathbb{G}_n$ is profinite, $(K(E_n))^{h\mathbb{G}_n}$ denotes a continuous homotopy fixed point spectrum, and (c) $π_\ast(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n = 1$, $p \geq 5$, and $V_1 = V(1)$, we give a way to realize the above map and (c), by proving that $i_1$ induces a map \[K(L_{K(1)}(S^0)) \wedge v_{2}^{-1}V_1 \to (K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1},\] where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence \[(K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1} \simeq (K(E_1))^{\widetilde{h}\mathbb{G}_1} \wedge v_2^{-1}V_1,\] where $(K(E_1))^{\widetilde{h}\mathbb{G}_1}$ is the homotopy fixed points with $\mathbb{G}_1$ regarded as a discrete group.

math.AT↗

arXiv2019

A lift from group cohomology to spectra for trivial profinite actions

Let $G$ be a profinite group, $X$ a discrete $G$-spectrum with trivial action, and $X^{hG}$ the continuous homotopy fixed points. For any $N \trianglelefteq_o G$ ("$o$" for open), $X = X^N$ is a $G/N$-spectrum with trivial action. We construct a zigzag $\text{colim}\,_N \,X^{hG/N} \buildrelΦ\over\longrightarrow \text{colim}\,_N \,(X^{hN})^{hG/N} \buildrelΨ\over\longleftarrow X^{hG}$, where $Ψ$ is a weak equivalence. When $Φ$ is a weak equivalence, this zigzag gives an interesting model for $X^{hG}$ (for example, its Spanier-Whitehead dual is $\text{holim}\,_N \,F(X^{hG/N}, S^0)$). We prove that this happens in the following cases: (1) $|G| < \infty$; (2) $X$ is bounded above; (3) there exists $\{U\}$ cofinal in $\{N\}$, such that for each $U$, $H^s_c(U, π_\ast(X)) = 0$, for $s > 0$. Given (3), for each $U$, there is a weak equivalence $X \buildrel\simeq\over\longrightarrow X^{hU}$ and $X^{hG} \simeq X^{hG/U}$. For case (3), we give a series of corollaries and examples. As one instance of a family of examples, if $p$ is a prime, $K(n_p,p)$ the $n_p$th Morava $K$-theory $K(n_p)$ at $p$ for some $n_p \geq 1$, and $\mathbb{Z}_p$ the $p$-adic integers, then for each $m \geq 2$, (3) is satisfied when $G \leqslant \prod_{p \leq m} \mathbb{Z}_p$ is closed, $X = \bigvee_{p > m} (H\mathbb{Q} \vee K(n_p,p))$, and $\{U\} := \{N_G \mid N_G \trianglelefteq_o G\}$.

math.AT↗

arXiv2015

Profinite and discrete G-spectra and iterated homotopy fixed points

For a profinite group $G$, let $(\text{-})^{hG}$, $(\text{-})^{h_dG}$, and $(\text{-})^{h'G}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K \vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G/K$-spectrum with $(E_n^{hK})^{hG/K} \simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{h'K})^{h'G/K} \simeq E_n^{h'G}$ or the Devinatz-Hopkins proof (which requires $|G/K| < \infty$) of $(E_n^{dhK})^{h_dG/K} \simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G/K$-homotopy fixed point spectral sequence for $π_\ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; π_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $π_\ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; π_t(E_n^{dhK}))$, and the descent spectral sequence for $π_\ast((E_n^{h'K})^{h'G/K})$.

math.AT↗

arXiv2013

Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups

If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum H_{G,X} is just "a profinite version" of Map_*(EK_+, Z): at each stage of its construction, H_{G,X} replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z) (up to a certain natural identification).

math.AT↗

arXiv2013

Commutative ring objects in pro-categories and generalized Moore spectra

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.

math.AT↗

arXiv2013

A descent spectral sequence for arbitrary K(n)-local spectra with explicit $E_2$-term

Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the continuous cohomology of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a continuous cohomology group.

math.AT↗

arXiv2011

Every K(n)-local spectrum is the homotopy fixed points of its Morava module

Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to π_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

math.AT↗

arXiv2010

Obtaining intermediate rings of a local profinite Galois extension without localization

Let E_n be the Lubin-Tate spectrum and let G_n be the nth extended Morava stabilizer group. Then there is a discrete G_n-spectrum F_n, with L_{K(n)}(F_n) \simeq E_n, that has the property that (F_n)^{hU} \simeq E_n^{hU}, for every open subgroup U of G_n. In particular, (F_n)^{hG_n} \simeq L_{K(n)}(S^0). More generally, for any closed subgroup H of G_n, there is a discrete H-spectrum Z_{n, H}, such that (Z_{n, H})^{hH} \simeq E_n^{hH}. These conclusions are obtained from results about consistent k-local profinite G-Galois extensions E of finite vcd, where L_k(-) is L_M(L_T(-)), with M a finite spectrum and T smashing. For example, we show that L_k(E^{hH}) \simeq E^{hH}, for every open subgroup H of G.

math.AT↗

arXiv2010

Delta-discrete $G$-spectra and iterated homotopy fixed points

Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K/H-spectra, there is always an iterated homotopy fixed point spectrum, denoted (X^{hH})^{h_δK/H}, and it is just X^{hK}. Additionally, we show that for any delta-discrete G-spectrum Y, (Y^{h_δH})^{h_δK/H} \simeq Y^{h_δK}. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum {X_δ} that is equivalent to X and, though X^{hH} is not even known in general to have a K/H-action, there is always an equivalence ((X_δ)^{h_δH})^{h_δK/H} \simeq (X_δ)^{h_δK}. Therefore, delta-discrete L-spectra, by letting L equal H, K, and K/H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G-spectra.

math.AT↗

arXiv2009

The homotopy fixed point spectra of profinite Galois extensions

Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points.

math.AT↗

arXiv2008

Iterated homotopy fixed points for the Lubin-Tate spectrum, with an Appendix: An example of a discrete G-spectrum that is not hyperfibrant

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (Z^{hH})^{hK/H}, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=G_n, the extended Morava stabilizer group, and Z=L_{K(n)}(E_n \wedge X), where L_{K(n)} is Bousfield localization with respect to Morava K-theory, E_n is the Lubin-Tate spectrum, and X is any spectrum with trivial G_n-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that (E_n^{hH})^{hK/H} is just E_n^{hK}, extending a result of Devinatz and Hopkins.

math.AT↗

arXiv2008

Explicit fibrant replacement for discrete G-spectra

If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.

math.AT↗

arXiv2006

Rognes's theory of Galois extensions and the continuous action of G_n on E_n

Let us take for granted that L_{K(n)}S^0 --> E_n is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that ``E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This preprint arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available.

math.AT↗

arXiv2006

The site R^+_G for a profinite group G

Let G be a non-finite profinite group and let G-Sets_{df} be the canonical site of finite discrete G-sets. Then the category R^+_G, defined by Devinatz and Hopkins, is the category obtained by considering G-Sets_{df} together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R^+_G is a site when equipped with the pretopology of epimorphic covers. Also, we explain why the associated topology on R^+_G is not subcanonical, and hence, not canonical. We note that, since R^+_G is a site, there is automatically a model category structure on the category of presheaves of spectra on the site. Finally, we point out that such presheaves of spectra are a nice way of organizing the data that is obtained by taking the homotopy fixed points of a continuous G-spectrum with respect to the open subgroups of G.

math.AT↗