All Questions
Tagged with ring-spectra homotopy-theory
46 questions
41
votes
1
answer
10k
views
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 ...
- 1,162
26
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 ...
- 118k
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 ...
- 433
18
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 ...
- 11.8k
18
votes
1
answer
930
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) $R\{t\}\simeq \bigoplus_{n\ge 0} R_{\mathrm h\Sigma_n}$ ...
- 2,011
16
votes
1
answer
608
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 ...
- 64k
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 ...
- 562
13
votes
1
answer
998
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 ...
- 18.9k
12
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 ...
- 3,019
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 ...
- 9,863
12
votes
1
answer
703
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 ...
12
votes
1
answer
661
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 ...
- 19.9k
10
votes
1
answer
655
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 ...
- 7,637
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\...
- 743
9
votes
0
answers
223
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 ...
- 15.9k
9
votes
0
answers
317
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 ...
- 64k
8
votes
2
answers
294
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{...
- 2,011
8
votes
1
answer
557
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: ...
- 3,004
8
votes
0
answers
450
views
Descent vs effective descent for morphisms of ring spectra
Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...
- 448
8
votes
0
answers
328
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 ...
- 30.3k
7
votes
1
answer
247
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 \...
- 151
6
votes
1
answer
575
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 ...
- 317
6
votes
1
answer
257
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 ...
- 1,349
6
votes
1
answer
390
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
...
- 2,356
5
votes
2
answers
598
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 $\...
- 2,011
5
votes
1
answer
776
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 ...
- 37.8k
5
votes
1
answer
296
views
Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv}...
- 11.8k
5
votes
1
answer
444
views
endomorphisms of modules over symmetric ring spectra
I have a probably very basic question about modules over symmetric ring spectra:
Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let $\...
- 7,637
5
votes
1
answer
322
views
Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?
In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...
- 30.3k
5
votes
1
answer
480
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$ ...
- 371
5
votes
0
answers
210
views
Dependence of completion on the base ring
Let $M$ be a module over an $E_\infty$ ring, $A$. Let $I$ be an $A$-non unital commutative algebra together with an associative map $I \wedge_A M \to M$.
Define ${_A}(M/I^n)$ as the cofiber of $I^{\...
- 579
5
votes
0
answers
120
views
Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories
There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):
Symmetric monoidal categories model all connective ...
- 11.8k
5
votes
0
answers
239
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 ...
- 15.9k
5
votes
0
answers
550
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$?
...
- 9,863
4
votes
1
answer
206
views
Does the forgetful functor $F:\mathrm{CAlg}\to\mathrm{Alg}^{(1)}$ sending $E_\infty$-ring spectra to $E_1$-ring spectra preserve limits and colimits?
In remark 7.1.0.4. of Lurie's Higher Algebra, the sequence $(E_n^{\otimes})_{0\leq n\leq\infty}$ of $\infty$-operads induces forgetful functors for the sequence of categories $(\operatorname{Alg}^{(n)}...
- 71
4
votes
0
answers
196
views
Valuations and (semi)norms on ring spectra
Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let'...
- 11.8k
4
votes
0
answers
294
views
Can one define fields in stable homotopy theory via invertibility?
In Nilpotence in Stable Homotopy Theory II, Hopkins–Smith define a field spectrum to be a ring spectrum $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$. They then show that any field ...
- 11.8k
4
votes
0
answers
153
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 ...
- 2,011
4
votes
0
answers
477
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 ...
4
votes
0
answers
376
views
matrix ring spectra
I am trying to understand matrix ring spectra. Apparently, I have two different definitions of those and I did not manage to show that they are equivalent - maybe they even are not in the general case....
- 7,637
3
votes
0
answers
145
views
What is the group completion of the underlying multiplicative $\mathbb{E}_\infty$-monoid of the sphere spectrum?
I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$:
We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and —...
- 11.8k
2
votes
0
answers
169
views
Formal power series over the sphere spectrum?
In Section 11 of their paper https://arxiv.org/pdf/1802.03261, Bhatt-Morrow-Scholze discuss the polynomial algebra over the sphere spectrum. I'm wondering whether its possible to define a notion of ...
- 959
2
votes
0
answers
116
views
Quasicompact quasiaffine classical schemes are nonconnectively-affine
In this answer to What is the relationship between connective and nonconnective derived algebraic geometry? I learned that any quasicompact open subscheme of an affine scheme is affine in the sense of ...
- 448
1
vote
1
answer
169
views
[M,N]≅ [M,R] ⊗ N for E-infinity modules
Let $\texttt{R}$ be an $\texttt{E}$-infinity ring and let $\texttt{M,N}$ be $\texttt{E}$-infinity modules. Under what conditions do we have
$$ \texttt{[M, N] ≅ [M,R] ⊗ N}$$
Under ordinary ...
user30211
1
vote
0
answers
171
views
A question on $BP$ and $E_\infty$ models for ring spectrums
I am a beginner in this field. My question is
(1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra?
(2) If (1) is true, what is the risk of replacing a ...
- 71
0
votes
1
answer
458
views
Homotopical interpretation of Langlands correspondence
Recently I began learning about homotopy theory, I am very far from being familiar with all the basic notions and constructions, however I heard of the notion of topological modular forms. I also ...
- 2,907