Questions tagged [ring-spectra]
For questions about ring spectra (in homotopy theory).
102 questions
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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 ...
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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_\...
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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}...
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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 $\...
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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 ...
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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 ...
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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 ...
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solid commutative ring spectra
Let $R$ be a discrete (i.e. an ordinary) commutative ring and let $HR\rightarrow T$ be a map of $E_{\infty}$-ring spectra where $HR$ is the associated Eilenberg-Mac Lane ring spectrum. We say that $T$ ...
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Does Spec functor sends pushouts of rings into pullbacks of sets?
This question was posted here on StackExchange.
Let $A$ be a commutative ring and $B,C$ be two commutative $A$-algebras.
Consider the pushout square of ring homomorphism
$\require{AMScd}$
\begin{CD}
...
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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 $Ch$ ...
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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^{\...
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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 ...
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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}\...
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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 ...
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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 ...
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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 ...
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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 $Y$?
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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
$...
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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}/...
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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)}...
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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 ...
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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 ...
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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 ...
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Kuenneth short exact sequence for K-homology
Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?
Using general spectra stuff, one gets a ...
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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 ...
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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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Orientation of complex bordism spectrum
I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read ...
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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} \...
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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 $n$th stable ...
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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 ...
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Monoidality of truncation of spectra
If $X$ is a spectrum, we have a notion of its connective part $X_{\le 0}$ and the corresponding notion of truncation $X_{[i:j]} = X_{\le j}/X_{\le i-1}$, where $X_{\le j}$ is deduced from $X_{\le 0}$ ...
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Concrete pull-back calculation along H-space map
I am trying to calculate the pull-back of a cohomology class on the loopspace of the algebraic $K$-theory space $\Omega K(\mathbb{C})$ along the H-space map of $K(\mathbb{C}).$
Let $b_k\in \tilde{H}^{...
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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 —...
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On rings for which given an ideal , over it every minimal prime ideal is finitely generated
Let $R$ be a commutative ring with unity. If for every ideal of $R$, the minimal prime ideals over it are all finitely generated, then there are finitely many minimal prime ideals over every ideal of $...
user111524
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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Relating the cardinalities of rings and their spectra
Does there exist an uncountable (possibly commutative, unital) ring with a countably infinite spectrum?
More generally, given two cardinalities $\kappa$ and $\lambda$, does there exist a ring $R$ ...
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[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
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Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field
If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ?
If $R$ is normal (integrally ...
user111492
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Intuition behind formal neighborhood and local ring and formal power series
In The Geometry of Schemes by David Eisenbud and Joe Harris, on page 57, there is an explanation on "node" of a plane curve. The book says that, a curve $X\subseteq \mathbb A_{\mathbb C}^2$ ...
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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 ...
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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 ...
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