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11
votes
2answers
340 views

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 ...
1
vote
0answers
154 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 ...
19
votes
1answer
862 views

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 ...
4
votes
1answer
480 views

Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?

Theorem 4.4.4.7 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 a given symmetric monoidal ...
5
votes
1answer
392 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 ...
2
votes
1answer
137 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 ...
5
votes
1answer
107 views

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 ...
7
votes
2answers
286 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: $ ...
1
vote
0answers
87 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 $R$ ...
6
votes
1answer
539 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 ...
11
votes
1answer
274 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 ...
3
votes
1answer
208 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 ...
6
votes
1answer
227 views

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 ...
2
votes
2answers
349 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 ...