# Questions tagged [ring-spectra]

For questions about ring spectra (in homotopy theory).

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### Is the $\infty$-category of spectra “convenient”?

A 1991 paper of Lewis, title “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**

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**1**answer

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### How is topological André-Quillen homology (TAQ) a “stabilization”, exactly?

Let $S\to A\to B$ be cofibrations of commutative $S$-algebras. Then the topological André-Quillen $B$-module $TAQ(B|A)$ can be computed as a stabilization. Precisely, I think it means the following: ...

**8**

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**0**answers

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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A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant.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 ...

**5**

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**2**answers

367 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 ...

**16**

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**1**answer

505 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 ...

**3**

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**0**answers

130 views

### 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 $...

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**1**answer

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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 ...

**16**

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**1**answer

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### A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.
However when reading ...

**9**

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**3**answers

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### 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 ...

**15**

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**0**answers

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### What to expect from spectral algebraic geometry

So I've been trying to learn some derived algebraic geometry, and I've chosen to approach the subject from the perspective of spectral or "brave new" algebraic geometry. Without having to go through ...

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**0**answers

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### Stable Dold-Kan correspondence

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $...

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**0**answers

120 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 $Y$?
...

**10**

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**2**answers

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### 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 ...

**23**

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**2**answers

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### 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 ...

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**2**answers

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### 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{...

**15**

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**1**answer

611 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) $R\{t\}\simeq \bigoplus_{n\ge 0} R_{\mathrm h\Sigma_n}$ ...

**3**

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**1**answer

223 views

### 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 $X_{\le 0}$ ...

**3**

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**1**answer

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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}^{...

**5**

votes

**2**answers

321 views

### Group of units of a ring spectrum vs of its connective cover

Let $R$ be a commutative ring spectrum (interpret this as you will; as an $E_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}_1(R)$ as usual denote its space of units. If $\...

**3**

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**0**answers

177 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 ...

**14**

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379 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 ...

**5**

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**0**answers

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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
$...

**17**

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**1**answer

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### 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 ...

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383 views

### $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...

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220 views

### Schwede-Shipley theorem for monoidal categories?

The Schwede-Shipley theorem gives a criterion for a presentable stable $\infty$-category to be the category of modules over an $\mathcal{E}_1$-algebra. Is there any similar criterion for a monoidal ...

**5**

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**1**answer

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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 $T$ ...

**18**

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**1**answer

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### What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$.
Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ (...

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**1**answer

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### Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...

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**1**answer

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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 ...

**12**

votes

**1**answer

574 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 ...

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**1**answer

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### Isomorphic quotient of a Module over Noetherian commutative algebra [closed]

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...

**24**

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**1**answer

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### 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 ...

**3**

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**0**answers

258 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....

**5**

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**1**answer

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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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**2**answers

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### 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 $E_\infty$-...

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316 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 ...

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**1**answer

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### 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 ...

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**1**answer

867 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 ...

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**1**answer

2k 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\...

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**1**answer

187 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 ...

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**1**answer

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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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votes

**2**answers

464 views

### Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes

I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes:
$ ...

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**0**answers

169 views

### 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 $R$ ...

**12**

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**1**answer

1k 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 ...

**12**

votes

**1**answer

634 views

### On the stable splitting of loops on a suspension

Let $X$ be a connected, based CW complex. Then the James splitting
of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra
$$
\Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad ...

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votes

**1**answer

342 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 ...

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**1**answer

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### State of knowledge on the Commutative W-spaces which appear in “Model Categories of Diagram Spectra”

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ...

**3**

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**2**answers

575 views

### 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 ...