All Questions
Tagged with at.algebraic-topology ring-spectra
15 questions with no upvoted or accepted answers
14
votes
0
answers
414
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 ...
- 8,723
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
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 ...
- 54.6k
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A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant.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
0
answers
223
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 ...
- 7,972
6
votes
0
answers
155
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 ...
- 745
6
votes
0
answers
201
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 ...
- 18.4k
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
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 ...
- 7,637
5
votes
0
answers
229
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
$...
- 928
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
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
284
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 ...
- 7,972
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 ...
- 31