All Questions
Tagged with ring-spectra homotopy-theory
17 questions with no upvoted or accepted answers
9
votes
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223
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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
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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
450
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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
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328
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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A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant.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
5
votes
0
answers
210
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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
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$\mathbb Z \otimes_\mathbb S \mathbb Z$ 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
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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
0
answers
196
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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
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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
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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
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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
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376
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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
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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
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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
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0
answers
116
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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
0
answers
171
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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