All Questions
Tagged with ra.rings-and-algebras at.algebraic-topology
53 questions
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
0
votes
1
answer
219
views
Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 2,837
2
votes
0
answers
122
views
Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
- 215
0
votes
1
answer
170
views
Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
4
votes
0
answers
451
views
Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
3
votes
1
answer
358
views
Geometric interpretation of shuffle product
Let $A=k\mathbb \Pi$ be the group algebra of an abelian group $\Pi$ and let $B(A)=\bigoplus_{k=0}^\infty\,B^k(A)$ be the unnormalized bar complex of $A$ with generators $[a_0,\dots,a_k] \in B^k(A)=A^{\...
- 1,813
6
votes
1
answer
457
views
Hochschild cohomology of group ring of a free group
Let $G$ be a free group of finite rank. Consider any commutative ring $R$ containing $\mathbb{Z}$. Consider the group ring $RG$.
Q) What can we say about the Hochschild cohomology groups of $RG$ with ...
- 1,713
5
votes
0
answers
114
views
Conilpotent coalgebras as pushouts of trivial coalgebras
Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$
whose underlying $K$-vector space is finite dimensional.
Question: Can one obtain $C$ by iterately taking ...
- 1,361
10
votes
0
answers
317
views
Near-ring spaces
$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
- 11.8k
7
votes
2
answers
746
views
Simplicial set construction of the classifying space
What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?:
Regard $G$ as a category with ...
- 275
3
votes
0
answers
278
views
What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?
This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms
\begin{align*}
\mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...
- 11.8k
29
votes
3
answers
2k
views
Categorification of determinant
The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See Ponto & Shulman for a nice description.)
Is there a ...
- 290
1
vote
1
answer
111
views
Properties of filtrations preserved by a DG-algebra homomorphism
Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration
\begin{align*}
A^{\bullet} \supseteq F^0A^{\...
- 113
5
votes
0
answers
139
views
How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
- 765
8
votes
1
answer
429
views
Koszulness of some DG-algebras and a paper by Kohno and Oda
This is a follow-up of my previous question Formality of the 2nd ordered configuration space of a closed Riemann surface.
At page 131 of [B], R. Bezrukavnikov states Proposition 4.1, in which he ...
- 66.3k
4
votes
0
answers
108
views
Mysterious identity in cosimplicial $R$-module with Lie brackets
I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
- 1,623
3
votes
0
answers
160
views
Ring spectra whose homotopy is the ring of ordinary differential operators
Consider $R=\mathbb{C}[[x]][\frac{d}{dx}]$ as a graded ring (assign $x$ zero weight, and $\frac{d}{dx}$ some non-zero weight). Are there some $A_n$-ring spectra naturally arising in homotopy theory ...
- 31
4
votes
1
answer
186
views
Is the category of rational Lie algebras monoidal?
I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes_{\mathbb{Q}} B$ is a rational ...
- 30.3k
3
votes
0
answers
53
views
Quotient of quasi-isomorphic nonpositively graded cdga's
I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) concentrated in nonpositive degree, and $\mathfrak ...
- 151
4
votes
1
answer
193
views
Quotient of quasi-isomorphic cdga's
I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, ...
- 151
5
votes
2
answers
266
views
Naturality of PD model of a CDGA
In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...
- 745
6
votes
0
answers
128
views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
- 52.6k
11
votes
3
answers
327
views
Unbiased Hopf algebras
In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
- 43.2k
10
votes
1
answer
274
views
A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
15
votes
1
answer
557
views
Defining Massey products as transgressions
Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps
$$ A \to A \oplus A \to A$$
given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as ...
- 40.2k
15
votes
3
answers
3k
views
What are Homotopy rings good for?
In his paper, Note on quasi-Lie rings, P. J. Hilton defines the (non-associative) Homotopy ring of a pointed space $X$ as$$\bigoplus_{n>1}\pi_n(X)$$ where the Whitehead product $\pi_m(X)\times\pi_n(...
- 175
3
votes
1
answer
277
views
What is the relation between cobar duality and Feynman transform
If $O$ is a cyclic operad, it can be regared as a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any ...
- 31
8
votes
1
answer
716
views
Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
4
votes
0
answers
185
views
ring structure of $KK_*(A,A)$ for a separable $C^*$-algebra $A$
Motivation:
For a topological space $X$ one can consider under certain circumstances the cohomology ring of suitable cohomology theories, for example:
1) The cohomology ring $H^*(X;R)=\oplus_{i\ge ...
- 1,188
2
votes
0
answers
1k
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 721
15
votes
2
answers
2k
views
When is bar-cobar duality an equivalence?
Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
- 4,133
7
votes
0
answers
178
views
Associated graded of double Koszul dual
Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...
- 4,133
12
votes
1
answer
458
views
Algebraic K-theory of a ring
I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is: What is the list of rings such that all their algebraic $K$-theory groups are known?
I ...
- 433
23
votes
0
answers
463
views
Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
- 6,069
5
votes
1
answer
136
views
connected stable rank
There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to
that of the matrix algebra M_n(A). Is there something ...
- 687
3
votes
1
answer
216
views
Global dimension of graded Lie algebra
The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm Ext}^...
- 189
1
vote
1
answer
375
views
configuration spaces of real projective space
Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any ...
- 1,990
1
vote
0
answers
279
views
Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras
Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the
lower central series of $G$. For each $k\geq 1$, set
$\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and
$$\mathrm{gr}_*(G):=\...
- 1,108
8
votes
0
answers
4k
views
Kunneth spectral sequence
In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also hold ...
- 1,589
19
votes
2
answers
702
views
Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
user1437
3
votes
1
answer
236
views
degree of polynomial in Gröbner basis
Let $f(x, y) = \sum_{m=0}^{M-1}\sum_{n=0}^{N-1} a_{m,n} x^m y^n$ and $g(x, y) = \sum_{m=0}^{M-1}\sum_{n=0}^{N-1} b_{m,n} x^m y^n$. Computing the Gröbner basis, we get an univariate polynomial $h_1(x)...
- 59
3
votes
1
answer
640
views
Resolutions chain homotopic to projective ones
Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...
- 38.6k
8
votes
2
answers
2k
views
Algebraic Morse theory
In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
- 1,589
14
votes
1
answer
2k
views
Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,550
2
votes
0
answers
252
views
Invariant Ideals in Split Hopf Algebroids
Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group $\mathrm{...
- 10.4k
5
votes
0
answers
241
views
Roots of unity in algebraic K-theory
For any commutative ring $R$, the tensor product of (finitely generated, projective) $R$-modules equips the algebraic K-theory $K(R) = K_0(R)$ with the structure of a commutative ring with unit.
For $...
- 4,133
6
votes
1
answer
1k
views
Solid rings and Tor
A solid ring is a ring $R$ such that the multiplication
$R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism.
These were classified by Bousfield and Kan; they are
subrings of $\mathbb{Q}$,
$\mathbb{Z}/...
- 12.5k
7
votes
1
answer
457
views
The geometric meaning of the higher quotient by the commutant ideal
The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal.
For any dg-algebra $A$ let $A_{Ab}$ denote the derived ...
- 743
3
votes
3
answers
757
views
Are these Two Definitions of Quadratic Form (Algebraic, Topological) Related to Each Other?
Hi, All:
I am trying to see if there is a nice relation between two different definitions of quadratic form q; a topological definition $q_T$, and an algebraic definition $q_A$, and, if there is, how ...
- 105
2
votes
0
answers
148
views
Is the homotopy of a primitively generated Hopf algebra still primitively generated?
Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$.
Question: is the graded Hopf algebra ...
- 681