Questions tagged [ring-spectra]

For questions about ring spectra (in homotopy theory).

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Why not a Roadmap for Homotopy Theory and Spectra?

MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
John Salvatierrez's user avatar
26 votes
2 answers
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 ...
მამუკა ჯიბლაძე's user avatar
25 votes
1 answer
1k views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
Qiaochu Yuan's user avatar
21 votes
1 answer
3k 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 ...
JDou9's user avatar
  • 433
19 votes
1 answer
1k views

What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$. Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ (...
Catherine Ray's user avatar
18 votes
1 answer
1k 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 ...
ExceptionallyCluelessGrad's user avatar
18 votes
1 answer
869 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) $R\{t\}\simeq \bigoplus_{n\ge 0} R_{\mathrm h\Sigma_n}$ ...
A Rock and a Hard Place's user avatar
17 votes
1 answer
730 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 ...
Simon Henry's user avatar
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17 votes
1 answer
2k views

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

A 1991 paper of Lewis, titled “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 ...
Emily's user avatar
  • 10.3k
16 votes
1 answer
575 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 ...
Tim Campion's user avatar
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15 votes
1 answer
2k views

Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum. In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO). On the other hand, MU and ...
Dmitri Pavlov's user avatar
15 votes
0 answers
3k 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 ...
Exit path's user avatar
  • 2,969
14 votes
2 answers
686 views

Truncations of E_infinity algebras

In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an $E_\infty$-...
user49450's user avatar
  • 143
14 votes
0 answers
410 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 ...
Vivek Shende's user avatar
  • 8,663
13 votes
1 answer
2k 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 ...
Mattia Coloma's user avatar
12 votes
2 answers
2k 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 ...
Matthias Ludewig's user avatar
12 votes
1 answer
952 views

On the stable splitting of loops on a suspension

Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad ...
John Klein's user avatar
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12 votes
1 answer
628 views

Nonunital $E_\infty$-rings

An elementary fact of algebra is that the category of nonunital commutative rings is equivalent to that of $\mathbb{Z}$-augmented unital commutative rings, the equivalence being given by forming ...
Urs Schreiber's user avatar
12 votes
0 answers
834 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
user40276's user avatar
  • 2,199
11 votes
3 answers
2k 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 ...
Exit path's user avatar
  • 2,969
11 votes
1 answer
632 views

Does the spectrum of Morava E-theory depend only on height?

I almost expect the answer to this question to be no, but I can't find it explicitly said anywhere. Given a formal group law $f$ of height $n$ over a perfect field $k$ of characteristic $p$, we can ...
Sofía Marlasca Aparicio's user avatar
10 votes
2 answers
631 views

Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes

I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes: $ ...
Samuel Monnier's user avatar
10 votes
1 answer
621 views

Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...
Ulrich Pennig's user avatar
9 votes
1 answer
3k views

Topological Hochschild cohomology?

Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as $A\wedge_{A\...
nikitamarkarian's user avatar
9 votes
1 answer
439 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 ...
Nikhil Sahoo's user avatar
  • 1,175
9 votes
0 answers
217 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 ...
Maxime Ramzi's user avatar
  • 13.3k
9 votes
0 answers
302 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 ...
Tim Campion's user avatar
  • 60.6k
8 votes
3 answers
1k views

Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?

In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...
Doron Grossman-Naples's user avatar
8 votes
2 answers
269 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{...
A Rock and a Hard Place's user avatar
8 votes
1 answer
507 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: ...
Bruno Stonek's user avatar
  • 2,914
8 votes
1 answer
589 views

Schwede-Shipley theorem for monoidal categories?

The Schwede-Shipley theorem gives a criterion for a presentable stable $\infty$-category to be the category of modules over an $\mathcal{E}_1$-algebra. Is there any similar criterion for a monoidal ...
abc.xyz's user avatar
  • 131
8 votes
1 answer
374 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_{\...
domenico fiorenza's user avatar
8 votes
0 answers
260 views

What can I say about an $E_\infty$ ring spectrum with an odd invertible element?

I have an $E_\infty$ ring spectrum $R$. I suspect it is trivial, but I'm not sure. What I do know is that there is an $R$-linear equivalence $R \simeq \Sigma R$. Unless I am very confused, this ...
Theo Johnson-Freyd's user avatar
8 votes
0 answers
313 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 ...
David White's user avatar
  • 29.4k
7 votes
1 answer
232 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 \...
Ann's user avatar
  • 151
7 votes
1 answer
275 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 ...
Maxime Ramzi's user avatar
  • 13.3k
7 votes
0 answers
215 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 ...
Dmitry Vaintrob's user avatar
6 votes
3 answers
432 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 ...
Bryan Shih's user avatar
6 votes
2 answers
841 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 ...
xir's user avatar
  • 1,964
6 votes
1 answer
502 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 ...
Mr. Palomar's user avatar
6 votes
1 answer
248 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 ...
Stahl's user avatar
  • 1,049
6 votes
1 answer
356 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 ...
davik's user avatar
  • 2,035
6 votes
1 answer
200 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 ...
domenico fiorenza's user avatar
6 votes
1 answer
380 views

State of knowledge on the Commutative W-spaces which appear in "Model Categories of Diagram Spectra"

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ...
David White's user avatar
  • 29.4k
6 votes
0 answers
145 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 ...
Surojit Ghosh's user avatar
6 votes
0 answers
131 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 =...
Maxime Ramzi's user avatar
  • 13.3k
6 votes
0 answers
189 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 ...
მამუკა ჯიბლაძე's user avatar
5 votes
2 answers
559 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 $\...
A Rock and a Hard Place's user avatar
5 votes
1 answer
318 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 ...
Excalibur's user avatar
  • 301
5 votes
1 answer
753 views

Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?

The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an E∞-operad; algebras in spaces over the Barratt-Eccles operad model E∞-spaces, i.e., homotopy ...
Dmitri Pavlov's user avatar