All Questions
Tagged with ring-spectra higher-algebra
20 questions
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 ...
- 448
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
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 ...
- 1,969
4
votes
1
answer
169
views
Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?
Let $A$ be an $\mathbb{E}_\infty$-ring spectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras.
We assume there is a homotopy fibre sequence
$$
R_1\to R_2 \to R_3
$$
in the stable ...
- 519
0
votes
0
answers
185
views
Stable homotopy group of K(1)-local spectra
Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$,...
- 127
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
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
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
6
votes
3
answers
465
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 ...
- 661
5
votes
0
answers
231
views
Two Hattori-Stallings trace questions
$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...
- 15.9k
4
votes
1
answer
297
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 ...
5
votes
1
answer
321
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_\...
- 448
7
votes
1
answer
307
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 ...
- 15.9k
3
votes
1
answer
127
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} \...
- 661
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
6
votes
0
answers
135
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 =...
- 15.9k
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
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
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
4
votes
1
answer
1k
views
Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?
Theorem 4.5.4.7 (4.4.4.7 in the old version) in Lurie's Higher Algebra (or Theorem 4.3.22 in DAG III) states (roughly speaking) that under certain conditions
the ∞-category of commutative ∞-monoids in ...
- 37.8k