All Questions
Tagged with homology or homology-cohomology
5,680 questions
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Trouble verifying details in Theorem 4.18 in Rotman [closed]
Theorem: Let $X = \bigcup_{p = 1}^{\infty} X^p$ with $X^p \subset X^{p + 1}$ for all $p$ (call the inclusion maps $\lambda^p: X^p \to X$ and $\varphi^p: X^p \to X^{p + 1}$). If every compact subspace $...
- 23
3
votes
2
answers
75
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Integrating classes in Čech cohomology
In a seminar I attended there was a discussion regarding the first Chern class of line bundles on compact complex manifolds. The exponential sequence $$0 \longrightarrow \mathbb Z \longrightarrow \...
- 10.4k
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0
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21
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Quasi-isomorphism that is not a weak homotopy equivalence [duplicate]
I'm looking for an example of a continuous map $f:X\to Y$ between topological spaces that induces isomorphisms on all homology groups $f_\ast:H_n(X)\overset{\cong}{\longrightarrow} H_n(Y)\ \forall n\...
2
votes
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47
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does an exact sequence of filtered complexes induce an exact sequence of spectral sequences
Let $(A_i, d_i, F_i)$ be a filtered differential complex (with bounded filtration) for $i=1,2,3$. Assume we have filtered chain maps $i:A_1\to A_2$ and $j:A_2\to A_3$ such that
$$0\to A_1 \overset{i}{\...
- 3,332
2
votes
1
answer
111
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Hodge-Riemann form is non-degenerate
I'm studying Griffiths and Harris Principles of Algebraic Geometry. In particular I'm studying the Lefschetz Decomposition of a Kahler Manifold and its applications (pages 118-126).
After the proof of ...
- 673
1
vote
0
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57
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Homology groups of the non orientable genus g surface
I've been asked to compute the homology groups of $\Sigma_g^-$, the non-orientable genus g surface defined by the following $2g$-gon. To do so, I've thought of the surface as a CW-complex with $1$ $0$-...
3
votes
1
answer
95
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Mayer-Vietoris sequence on chain complexes of modules
I am trying to solve the following exercise:
The following is a commutative square of chain complexes of modules, where $\ker(f)=0$
$\require{AMScd}
\def\coker{\operatorname{coker}}$
\begin{CD}
A_\...
- 85
3
votes
1
answer
69
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Closed simply connected rationally acyclic manifold
Let $M$ be a closed rationally acyclic manifold of dimension $n$. This means that $M$ is compact without boundary and that $M$ does not have rational cohomology in any positive degree.
Examples of ...
- 183
1
vote
1
answer
34
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The natural map $H^{1,0}(M)\to H^1(M)$ is injective for $M$ a compact complex surface
Let $M$ be a compact complex surface. Then On a compact complex surface, every holomorphic 1-form is closed, so the natural map $H^{1,0}(M)\to H^1(M)$ is well-defined. But why is this map injective? ...
- 1,682
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2
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42
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Homology of complex projective plane with a ball glued to it
Let $\mathbb{C}P^n$ be the complex projective plane, let $B^3$ be a $3$ ball, and suppose we attach $B^3$ to $\mathbb{C}P^1 \subset \mathbb{C}P^n$ using a map $f : \partial B^3 \rightarrow \mathbb{C}P^...
- 157
1
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0
answers
114
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Homology and Cobordism maps
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few days. Any good references on this direction would be very helpful as well.]
Suppose $f : X \rightarrow Y$ is a (...
- 11
3
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0
answers
44
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Restriction map in Borel Moore Homology
Assume $\mathrm{L}\to \mathrm{ M}$ a closed embedding of someth oriented manifold and $M$ is smooth oriented manifold. I wonder if the following diagram commute.
$$
\matrix{
H^*(\mathrm{M}) ...
- 201
1
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29
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Determinant of Symmetrized Seifert Matrix
Exercice 3.4 of Livingston's Knot Theory: For a Seifert matrix $V$, $\text{det}(V+V^t) \neq 0.$ (Why?)
I know that $V-V^t$ will return the intersection form of the Seifert surface, which is unimodular....
- 1,476
2
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0
answers
44
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Generalization of Koszul Cohomology for Wedge Product with a Fixed q-Form: Literature and References?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $0 \leq p \leq q-1$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \...
- 21
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Computation of Hochschild cohomology for a graded algebra
Let $k$ be a field and $A= k[x]/(x^2)$ with $|x|=2\ell+1$ is a graded algebra, consider the module $M=k$ over $A$ with the module structure induced by the map $\epsilon: A \to M$ given by $\epsilon (x)...
- 881
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1
answer
23
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How can we paste together some finite long exact sequences (LES's) at the top page 14 of Weibel (in the SES of complexes to infinite LES proof)?
The Snake Lemma starting diagram at the bottom of page 13 in this proof is:
Here is a link to the above diagram.
Now at the top of page 14, it states "The kernel of the left vertical is $H_n(A)$ ...
- 21.6k
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2
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24
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Page 13 of Weibel's Intro to Hom. Alg. - How do we get from one diagram to the other (question in the SES to Long Exact Sequence proof)
The goal theorem the book is trying to explain is:
Theorem 1.3.1. Let $0\to A_{\bullet} \xrightarrow{f} B_{\bullet} \xrightarrow{g} C_{\bullet} \to 0$ be a short exact sequence (SES) of chain ...
- 21.6k
2
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answers
53
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Corollary I.3.5.3 in Tamme's Introduction to Etale Cohomology
Corollary 3.5.3, page 62 in Tamme's Introduction to Etale Cohomology, says that an abelian sheaf $F$, on a topology $(\mathcal C, \tau)$, is flabby iff ($*$) $H^q(U,F)=0$ for all objects $U$ in $\...
- 35
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85
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Inclusion of Grassmannians Induces Isomorphism on Cohomology
Let $\text{Gr}_2^+(\mathbb{R}^n)$ denote the the Grassmannian of oriented $2$-planes in $\mathbb{R}^n$, i.e. the orientation-double-cover of the usual Grassmannian $\text{Gr}_2(\mathbb{R}^n)$. Let $i :...
- 2,681
1
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1
answer
82
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Tensor product $\mathbb{Z}^n\otimes \mathbb{Z}^m$
For a problem in my PhD I'm using Leray–Hirsch Theorem (as presented in the Hatcher) to compute a cohomology ring. For a particular group $G$, I have the s.e.s.
$$
\def\Inn{\operatorname{Inn}}
\def\...
- 111
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0
answers
36
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Sheaf-structure of the functor of small cochain in singular cohomology
Let me briefly define some notation:
Let $X$ be a topological space, let $G$ be an abelian gruop, let $p\in\mathbb{N}$, let $S_p(X)=F_{ab}(\{\phi\in C(\Delta^p, X)\})$, let $S^p(X, G) = Hom(S_p(X), G)$...
- 365
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1
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42
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Are 2 chain complexes isomorphic if their homology groups are isomorphic?
Given 2 chain complexes $(C^k, \partial^k)$ and $(\bar C^k, \bar \partial^k)$ with $\dim(C^k) = \dim(\bar C^k)$, does the isomorphism of the homology groups $H^k \cong \bar H^k$ imply the chain ...
- 161
2
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0
answers
92
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Homotopy type of wedge of spheres via cohomology
It is known to me that given a simply connected finite dimensional which is also level-wise finite CW-complex $X$ such that $\tilde{H}_k(X,\mathbb{Z})=\mathbb{Z}^m$ and all other homologies with ...
- 181
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61
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Adding a boundary to an ideal doesn't change homology
Suppose that we have a differential graded associative algebra $(A,d)$, a differential graded ideal $I \subseteq A$ and a boundary $dx \in A$ (we can assume $dx \notin I$). $A$ can be assumed to be ...
- 452
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2
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84
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Betti Numbers over Z/2
My understanding is that if we have some nth-homology group H_n(X), then the rank of H_n(X) is the Betti number B_n. This is like saying the Betti number is the same as the number of linearly ...
- 51
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1
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37
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Explicit formula for $\check{H}^1(\mathcal{U}, F) \rightarrow coker(G(X) \rightarrow H(X))$ for Cech cohomology.
$\newcommand{\ra}{\rightarrow}$
Let $X$ be a topological space, and $\mathcal{U} = (U_i)_{i \in I}$ be an open cover. Let $F$ be a sheaf of abelian groups, and embed it into a flasque sheaf $G$. Let $...
- 6,468
4
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0
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86
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Lifting a mod-n class to an integral class and generalized Stiefel-Whitney classes
Suppose $M$ is a smooth, compact, oriented Riemannian 4-manifold, and let $n$ be a positive integer greater than $2$.
Does the obstruction to lifting a class $x \in H^2(M, \mathbb{Z}/n\mathbb{Z})$ to ...
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0
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70
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How to prove that a Lie algebra-valued differential form is exact for the covariant derivative
Given a differential $p$-form $\omega^A$ over a smooth manifold with values on some Lie algebra, I wanted to know how could one prove that it can be written as an exact form for the exterior covariant ...
- 227
2
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1
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69
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If $n$-th Homology is isomorphic to $\mathbb{Z}$, then $M$ is Orientable
I apologize if this has been asked before, but I could not find a reference. It is known if $M$ is an orientable connected $n$-manifold (without boundary), then $H_n(M;\mathbb{Z})\cong \mathbb{Z}$. Is ...
- 1,011
1
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1
answer
45
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Generators for homology groups of projective spaces
Let $H_k(\mathbb{R}P^n; \mathbb{Z})$ be the homology groups of the real projective plane.
We know that this groups are either $\mathbb{Z}$ if $k = 0$ or if $n$ is even and $k = n$, $\mathbb{Z}/2\...
- 157
2
votes
1
answer
40
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Homology of Genus 2 Surface, writing an element as linear combination of basis [closed]
Let $M_g$ be the orientable surface of genus $g$ (the "$g$-handles Torus"). In class we saw that $H_1(M_2) \cong \Bbb Z ^4$ and, on the internet, I found the following generators
My ...
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vote
0
answers
31
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Compute relative homology $H_k \left(E_+^n, E_+^n - \left\{0\right\} \right)$, where $E_+^n$ is the upper half space
I am asked to compute the relative homology pair $H_k \left(E_+^n, E_+^n - \left\{0\right\} \right)$ where $E_+^n = \left\{ \left(x_1, \ldots, x_n \right) : x_n \geqslant 0 \right\}$.
and find out ...
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0
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51
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Topological/ (Co)homological Interpretation of Cyclic Covers
I have a question about topological interpretation of concept of cyclic covers as suggested in Birational Geometry of Algebraic Varieties by Kollár and Mori in Definition 2.49 (p 63):
Let $X$ be a ...
- 8,449
1
vote
1
answer
97
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Where's my missing cokernel? (Cech complex computations in algebraic geometry)
Let $X=\Bbb P^2_k$, let $Y=V(f)$ for $f$ a homogeneous polynomial of degree $2$, let $g$ be a linear form, and let $L=V(g)$. Then the ideal sheaf of $L\cap C$ is isomorphic to $\mathcal{O}_X(-1)|_Y\...
- 2,819
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1
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50
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Meaning of "quasi-isomorphism" of diagram indexed by the negative integers
I've been given this exercise:
Fix a commutative unital ring $R$ and a diagram $$A^\bullet = \cdots \to A_{-2} \to A_{-1}\to A_{0}$$ of
objects and maps in $\mathrm{Ch}(R \mathrm{-Mod})$ indexed by ...
- 35
0
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0
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39
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Local homology groups of set of points with integer points
Let $X_1, X_2, X_3 \subset \mathbb{R}^3$ be the set of points in $\mathbb{R}^3$ with at least $i$ integer coordinates, with $i \in \{1,2,3\}$, so $X_3 \subset X_2 \subset X_1$.
I am interested in ...
- 157
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61
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Formalizing the boundary function $\partial_n$ and proving $\partial_n\partial_{n+1}=0$
I'm trying to formalize homologies in Mizar, following Allen Hatcher's Algebraic Topology.
I start by defining a Simplex as a one-to-one tuple $(v_0,\cdots,v_n)$ ...
- 3,223
0
votes
1
answer
54
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Question about excision axiom
I have read about the excision axiom in Hatcher's book about topology where it is stated as follows:
Theorem 2.20. Given subspaces $Z \subseteq A \subseteq X$ such that the closure of $Z$ is contained
...
- 99
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1
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38
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Understanding $C_n(X)$
I have this quote from Allen Hatcher's book Algebraic Topology:
A singular $n$-simplex in a space $X$ is by definition just a map $\sigma:\Delta^n\rightarrow X$. [...] All that is required is that $\...
- 3,223
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0
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45
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Intersection of product of cycles with graph equals intersection of pushforward
I'm trying to work out the proof of Lemma 2.5 in these notes: fix $A, B \in H_\bullet (X)$ of degrees adding up to $n$, the dimension of $X$. Let $f : X \to X$ be a continuous map, and let $\Gamma(f) \...
- 71
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0
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73
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A simple example of perverse sheaf
I'm calculating a simple example of the perverse sheaf:
Let $i:\mathbb{C}-\{0\} \to \mathbb{C}$ be the embedding, then I want show that shifted constant sheaf $Ri_{*}\mathbb{C}_{\mathbb{C}-\{0\}}[1]$ ...
1
vote
1
answer
65
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How to obtain a map $BU(n-1) \to BU(n)$ in the calculation of $H^*(BU(n), \mathbb Z)$?
I'm trying to follow the proof that $H^*(BU(n), \mathbb Z) \cong \mathbb Z[c_1, \dotsc, c_n]$, from Wikipedia.
It contains the following paragraph:
There are homotopy fiber sequences
$$S^{2n-1} \to ...
- 10.4k
1
vote
0
answers
77
views
Homotopy equivalent spaces have the same Homology groups
I'm reading "Algebraic Topology" by Hatcher, and in page 111 he writes the theorem:
Theorem 2.10: if two maps $f$,$g$: $X \rightarrow Y$ are homotopic, then they induce the same ...
- 136
1
vote
1
answer
67
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Interpretation of Homology Groups as holes
I'm trying to understand how the interpretations of $k$-dimensional holes comes from the quotient group definition of Homology Groups.
Taking $\partial_k$ as the $k$-dimensional boundary operator on a ...
- 1,095
1
vote
0
answers
106
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Identification of relative long exact sequence with Gysin sequence for de Rham cohomology
For a smooth manifold $M$ and a submanifold $S$ with inclusion $i:S\to M$, the relative de Rham cohomology is defined from the complex $\Omega^q(M,S):=\Omega^q(M)\oplus \Omega^{q-1}(S)$ with ...
- 1,682
0
votes
1
answer
37
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About an Argument for Relative Homology Groups
I am working on showing that $H_0(X,A)=0$ iff $A$ meets every path component of $X$. I know that this question has been answered, I just think an argument that is commonly used for one direction does ...
3
votes
0
answers
73
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Lifting problem in group cohomology (coefficient change)
Disclaimer: I do not know much about group cohomology except from the topological point of view.
Let $G$ be a group. We can consider the group ring $\mathbb{Z}[G]$ and the canonical composition $\phi: ...
- 163
1
vote
0
answers
89
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Homotopy invariance axiom, singular homology.
I am trying to prove the homotopy axiom defining the chain homotopy:
$$P_n:S_n(X)\rightarrow S_{n+1}(X), \quad P_n(\sigma)=F\circ(\sigma\times Id)\circ c,$$ where
$F:X\times I\rightarrow Y$ is the ...
- 59
2
votes
1
answer
93
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Computing cohomology ring $H^{\ast}(\mathbb{R} P^{\infty};\mathbb{Z}_{2k})$
As the title suggests we want to compute the cohomology ring $H^{\ast}(\mathbb{R} P^{\infty};\mathbb{Z}_{2k})$. For the rest of my question I will denote $\mathbb{R} P^{\infty}= X$.
To do this we have ...
- 33
2
votes
1
answer
32
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Cocycles of an Algebra with linear maps between two specific vector spaces as coefficients
I encountered the concept of “cocycle” in Problem 2.21 of Introduction to Representation Theory by Etingof et al. In the book, every algebra considered is associative with a unit, and the field is ...
