Questions tagged [ring-spectra]

For questions about ring spectra (in homotopy theory).

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4
votes
1answer
135 views

Is the rank of free module spectra unique?

Given a commutative ring, the rank of a free module is unique. This is the well known statement that commutative rings have invariant basis numbers. Does an analogue of this property hold for free ...
15
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1answer
462 views

Multiplicative Brown representability?

The Brown representability theorem can be convenient way to construct a spectrum. But to get a ring spectrum of even a very unstructured form seems to be harder. There's even currently a statement on ...
6
votes
1answer
255 views

What is the topological Hochschild cohomology of $\mathbb{F}_p$?

Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus. We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute ...
4
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0answers
129 views

Preorientation of additive formal group

In "A Survey of Elliptic Cohomology", Section 3.2, Lurie asserts that the preorientations of the additive formal group $\widehat{\mathbf G}_a$ over $\mathbf Z$ are classified by the $\mathbb ...
4
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0answers
77 views

Have mod $p^k$ Dyer Lashof operations been studied?

Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it ...
3
votes
1answer
84 views

Vanishing tate of a $p$-complete spectra

I was told: if $X$ is bdd below and $p$-complete spectra then $X^{tC_q}$ vanishes for primes $q \not= p$. I do not see how this holds. I am aware from I.2.9 that if $X$ is bdd. below, then $X^{tC_q} \...
6
votes
3answers
293 views

How does one rigorously lift a map $Sp \rightarrow Sp$ of spectra to equivariant spectra?

This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my ...
8
votes
1answer
210 views

K-theory on finite-dimensional (possibly not finite) CW complexes

I am trying to understand why (at least my most elementary understanding of) topological K-theory breaks down for non-compact things (which I have seen asserted in various places). In particular, as ...
5
votes
1answer
227 views

Is there a definition of reduced $E_\infty$ ring?

[Edit: I have completely changed the question in response to the replies given] I am curious if there is well defined notion of reduced $E_\infty$-ring. Let $CAlg$ denote the $\infty$-category of $E_\...
6
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0answers
103 views

Homotopy groups of certain geometric fixed point spectrum

Let $G$ be a finite group and $E$ be a genuine $H$-spectrum for $H\leq G.$ Then for any subgroup $K$ of $G$, consider the $K$-spectrum $X=Res^G_K Ind^G_H(E).$ Is there any reference for computing the ...
5
votes
1answer
166 views

Interesting “epimorphisms” of $E_\infty$-ring spectra

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...
7
votes
1answer
182 views

Module spectrum maps up to stable homotopy

Let $R$ be a commutative ring spectrum, $M$ and $N$ be a $R$-module spectra. Let us consider $R$-module maps from $M$ to $N$ up to stable homotopy, that is maps $M \to N$ such that the composites $R \...
6
votes
1answer
380 views

Two definitions of power operations — how do they relate?

Below are two different stories about power operations for $\mathbb{E}_\infty$-ring spectra, and I am struggling to see how they relate. In the following we let $R$ be an $\mathbb{E}_\infty$-ring ...
9
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0answers
176 views

Two $\mathbb Z$-algebra structures on $\mathbb Z\otimes_{\mathbb S} R$

$\newcommand{\Sph}{\mathbb S} \newcommand{\Z}{\mathbb Z} \newcommand{\F}{\mathbb F}$ In this question I will abuse notation by writing $A$ for the (generalized) Eilenberg-MacLane spectrum associated ...
6
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0answers
123 views

$E_\infty$-maps of diagrams

I'm asking my question for a general symmetric monoidal $\infty$-category $ V$ and a general indexing simplicial set $I$, but my specific interest is for $ V = Sp$ with the usual smash product and $I =...
5
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0answers
209 views

$\mathbb Z \otimes_\mathbb S \mathbb Z$ is concentrated in degree $0$ : mistake in the argument

I'm not sure this is research level so if this is not appropriate, feel free to move the question to StackExchange. However, I post it here since my "fake proof" is based on a (recent) paper and I'm ...
4
votes
1answer
205 views

$KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I posted this question some days ago at math.stackexchange, but didn't receive an answer. I have two questions: I wonder whether anyone has taken the time to compute $KO_*(\mathbb{R}P^\infty)$? The ...
7
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0answers
169 views

Duality of Hopf algebras and duality of spectra

Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...
8
votes
1answer
251 views

A Thom spectrum from “doubled” tautological bundles?

Let us consider real vector bundles, and denote by $V_k$ the tautological bundle $V_k\to BO(k)$. From $$ Thom(V\oplus 1_{\mathbb{R}}\to X)=\Sigma Thom(V\to X) $$ and from $j^*V_{k+1}=V_k\oplus1_{\...
6
votes
1answer
163 views

Morphisms from $bstring$ to $X\otimes \mathbb{Q}$ and sequences $s_n\in\pi_n(X)\otimes \mathbb{Q}$

I'm currently studying Ando-Hopkins-Rezk's work Multiplicative orientations of KO-theory and of the spectrum of TMFs. At a point a presumably obvious isomorphism is mentioned, which I'm however not ...
4
votes
1answer
379 views

Does Spec functor sends pushouts of rings into pullbacks of sets?

This question was posted here on StackExchange. Let $A$ be a commutative ring and $B,C$ be two commutative $A$-algebras. Consider the pushout square of ring homomorphism $\require{AMScd}$ \begin{CD} ...
5
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0answers
124 views

Which ring spectra are homotopy limits of simpler ones?

Most surely I will tag this by reference request: I am sure very much is known about this question, I am just too ignorant to even guess where to look. What makes me feel especially foolish is the ...
12
votes
1answer
739 views

Equivalent definitions of Thom spectra

Background and notations: Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
4
votes
1answer
102 views

Kuenneth short exact sequence for K-homology

Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one? Using general spectra stuff, one gets a ...
9
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0answers
275 views

Are there non-obvious finite $E_\infty$ ring spectra?

I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$: $R = \Sigma^\infty_+ (S^1)^{\times n}$ $R = D\Sigma^\infty_+ X$ ($X$ a finite space) Questions: Are there any others? In ...
3
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0answers
344 views

DAG applied to homotopy theory: how to reach research level?

It is my dream to do research on applications of spectral algebraic geometry in homotopy theory one day. Specifically, giving a more uniform treatment for the results proved via scary computations (of ...
6
votes
1answer
197 views

$p$-adic equivalence of spectra with $G$-action

In Krause and Nikolaus' "Lectures on topological Hochschild homology and cyclotomic spectra" (which can be found here), there is a lemma I'm trying to understand, but one line in the proof ...
14
votes
1answer
1k views

Is the $\infty$-category of spectra “convenient”?

A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$: There is a symmetric monoidal smash ...
8
votes
1answer
390 views

How is topological André-Quillen homology (TAQ) a “stabilization”, exactly?

Let $S\to A\to B$ be cofibrations of commutative $S$-algebras. Then the topological André-Quillen $B$-module $TAQ(B|A)$ can be computed as a stabilization. Precisely, I think it means the following: ...
8
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0answers
280 views

Did the Goerss-Hopkins manuscript “Multiplicative stable homotopy theory” ever appear?

A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...
6
votes
2answers
534 views

Ring structures on algebraic K-theory spectrum, and its non-connective counterpart

I have a few naive questions on the algebraic K-theory spectrum construction, but whose answers I couldn't figure out using the internet. I'm mostly interested in the case of a commutative ring, but I ...
16
votes
1answer
584 views

Idempotent ring spectrum

Is there a lot of ring spectrum which are idempotent in the sense that the multiplication map $R \wedge R \rightarrow R$ is an equivalence ? The sphere spectrum $\mathbb{S}$ and the $0$ spectrum are ...
3
votes
0answers
252 views

On rings for which given an ideal , over it every minimal prime ideal is finitely generated

Let $R$ be a commutative ring with unity. If for every ideal of $R$, the minimal prime ideals over it are all finitely generated, then there are finitely many minimal prime ideals over every ideal of $...
1
vote
1answer
141 views

Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ? If $R$ is normal (integrally ...
18
votes
1answer
861 views

A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough. However when reading ...
10
votes
3answers
1k views

What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in ...
13
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0answers
2k views

What to expect from spectral algebraic geometry

So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...
5
votes
1answer
369 views

Stable Dold-Kan correspondence and symmetric group actions

There exists a Quillen equivalence between $HRModSpectra$ (model category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $Ch$ ...
4
votes
0answers
168 views

Truncation and connected cover of spectra

Let $X$ be an $n$-connective spectrum for some $n\in \mathbb{Z}$. Is then $[X, Y] = [X, Y\langle n\rangle]$ for all spectra $Y$, where $Y\langle n\rangle$ denotes the $n$-connective cover of $Y$? ...
11
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2answers
1k views

Connective spectra and infinite loop spaces

It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective. For me, an infinite loop ...
23
votes
2answers
2k views

Has anyone seen a nice map of multiplicative cohomology theories?

I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere. I could try to do it myself but I really lack expertise, hence am afraid to miss something or ...
8
votes
2answers
224 views

Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$

Given two commutative rings $A$ and $B$, any map of rings $A\to B$ will automatically preserve the commutative structure. This is to say, the forgetful functor $\operatorname{CRing}\to \operatorname{...
18
votes
1answer
727 views

When do the polynomial algebra and free algebra coincide in brave new algebra?

Given an $\mathbb E_\infty$-ring (highly structured commutative ring spectrum if you want) $R$, we have the free $R$-algebra (on one generation) $R\{t\}\simeq \bigoplus_{n\ge 0} R_{\mathrm h\Sigma_n}$ ...
3
votes
1answer
282 views

Monoidality of truncation of spectra

If $X$ is a spectrum, we have a notion of its connective part $X_{\le 0}$ and the corresponding notion of truncation $X_{[i:j]} = X_{\le j}/X_{\le i-1}$, where $X_{\le j}$ is deduced from $X_{\le 0}$ ...
3
votes
1answer
145 views

Concrete pull-back calculation along H-space map

I am trying to calculate the pull-back of a cohomology class on the loopspace of the algebraic $K$-theory space $\Omega K(\mathbb{C})$ along the H-space map of $K(\mathbb{C}).$ Let $b_k\in \tilde{H}^{...
5
votes
2answers
419 views

Group of units of a ring spectrum vs of its connective cover

Let $R$ be a commutative ring spectrum (interpret this as you will; as an $E_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}_1(R)$ as usual denote its space of units. If $\...
3
votes
0answers
206 views

Can ring spectra be thought of as some sort of operad in $Top$?

It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of ...
14
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0answers
393 views

Does the category of G-spectra know G?

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
5
votes
0answers
189 views

Bockstein morphism of spectral sequences

Given an omega spectrum $E$, there is a type of chern character map given by its rationalization $$r:E\to E\wedge M\mathbb{R}\;,$$ where $M\mathbb{R}$ denotes a Moore spectrum. The cofiber of the map $...
17
votes
1answer
2k views

Motivation and potential applications of spectral algebraic geometry

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry. Now I'm curious what future is there for spectral algebraic ...