All Questions
Tagged with ring-spectra stable-homotopy
40 questions
3
votes
0
answers
145
views
What is the group completion of the underlying multiplicative E∞-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 —...
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 ...
5
votes
1
answer
335
views
Morita equivalence and connectivity
Let A, B be Morita equivalent \mathbb{E}_1-ring spectra. Fix an an (A, B)-bimodule P and a (B, A)-bimodule Q such that P \otimes_B Q \cong A and Q \otimes_A P \cong B. If A is ...
3
votes
1
answer
624
views
Homotopy colimit commutes with homotopy groups
I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem
Let (E, \mu, \iota) be a ring spectrum.
For \beta \in \pi_n(E) an element of the nth stable ...
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 ...
5
votes
0
answers
138
views
Examples of comonoids (coalgebras) in the stable homotopy category \mathbf{SH}
My question is both for the topological and for the algebraic/motivic version of the stable homotopy category \mathbf{SH}.
It is well known that most cohomologies are represented in \mathbf{SH} by ...
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 ...
2
votes
1
answer
173
views
Monoidal colimit-preserving functor from spaces to A-modules
I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an \mathbb{E}_\infty-ring A "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \...
8
votes
0
answers
300
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 ...
5
votes
0
answers
173
views
Uniqueness of complex topological K-theory as an S-algebra
This might be well-known or trivial, but I could not figure out how to fill in the details: For an S-algebra K denote its associated multiplicative cohomology theory by h^*_K. Suppose that I ...
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}...
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 ...
4
votes
1
answer
241
views
Is there essentially unique notion of module over monoidal stable \infty-categories?
There is this (folklore?) fact: for a commutative ring R, the category of R-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...
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
...
4
votes
0
answers
176
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 ...
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 \...
6
votes
1
answer
578
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 ...
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 ...
5
votes
1
answer
348
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 ...
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 ...
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 ...
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 ...
6
votes
2
answers
1k
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 ...
17
votes
1
answer
777
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 ...
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 ...
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?
...
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 ...
28
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 ...
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{...
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} ...
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 ...
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 ...
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 $\...
14
votes
2
answers
738
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-...
3
votes
0
answers
434
views
Units of a ring spectrum
Is there a good notion of the spectrum of units R^\ast in a (possibly non-connective) E_\infty-ring spectrum R?
A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...
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 ...
3
votes
1
answer
217
views
Does Wolbert's derived equivalence between E_*^R-local R-modules and R_E-modules come from a Quillen equivalence?
Let R be a ring spectrum (in the world of EKMM S-modules) and let E be a smashing R-module. Denote by R_E the E_*-localization of R. By a theorem of Wolbert (Theorem 2 in Classifying ...
5
votes
1
answer
187
views
Is this a description of the \aleph_1-localizing subcategory generated by a compact generator?
This should be obvious but I'm not seeing it:
The \mathfrak T be a triangulated category with coproducts and with a compact generator A (that is, the functor \mathfrak T(A,\_) preserves ...
14
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 ...
4
votes
1
answer
438
views
If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?
Let S be the unit object in a monoidal stable homotopy category SH (we demand that the multiplication S\times S\to S is commutative and associative on the level of spectra, and not just up to ...