Tagged Questions

1
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0answers
13 views

Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
7
votes
1answer
41 views

Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 ...
4
votes
1answer
58 views

How to compute Nakayama functor explicitly?

I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
3
votes
2answers
51 views

$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups

Nice to meet you folks! I'm currently a grad student reviewing some representation theory of finite groups for prelims next year, and I'm stuck proving a simple statement. Translating the question ...
2
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1answer
59 views

Question about minimal projective presentations of a module.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 108, line 11-14, there is a claim: If $P_0^{t}\to P_1^{t} \to TrM \to 0$ is not a minimal ...
3
votes
1answer
37 views

Does the projectively stable category have projective modules?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 109, the projectively stable category is defined by $$ \underline{mod} A = mod A/\mathcal{P}. $$ ...
1
vote
1answer
34 views

Image of the projection map onto an irreducible module

Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
0
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1answer
14 views

Submodules and $p$-adic numbers

I am a little bit confused about the terminology of simple $\mathbb{Q}_p[G]$ module. E.j.: If one take an $\mathbb{Z}_p[G]$ module $M$, then $pM$ is a submodule, so one can just look for ...
0
votes
1answer
31 views

$\operatorname{Res}(V+W)=\operatorname{Res}(V)+\operatorname{Res}(W)$?

if there are two $R[G]$ Modules $V,W$ and $R$ some ring, $S$ subgroup of $G$. Is the formula $$\operatorname{Res}_S (V \oplus W) = \operatorname{Res}_S (V) \oplus \operatorname{Res}_S (W) $$ true? I ...
4
votes
0answers
56 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
1
vote
1answer
39 views

Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$

Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
1
vote
1answer
22 views

Questions about maximal submodules.

Let $A$ be a $K$-algebra and $M$ a right $A$-module, where $K$ is a field. Suppose that $M=C\oplus D$, where $C, D$ are right $A$-modules. If $C', D'$ are maximal right $A$-submodules of $C, D$ ...
2
votes
1answer
42 views

Characters of elements under every representation equal implies conjugacy

If $G$ is a group, suppose that for every $G$-module $V$ we have $$\chi_V(g_1)=\chi_V(g_2).$$ How can I be sure $g_1$ and $g_2$ are conjugate in $G$? Its easy to the reverse implication; ...
3
votes
2answers
72 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
1
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0answers
26 views

Surjective map results in subrepresentation

I need to prove that a surjective homomorphism of finite $\mathbb{F}_p[\Delta]$-modules $$A \twoheadrightarrow B$$ results in $B$ being a subrepresenation of $A$ of the group $\Delta$ of order prime ...
1
vote
1answer
64 views

Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
1
vote
3answers
65 views

Show $U \otimes V$ is an irreducible G-module

Let $G$ is some group and $U$ is an irriducible $G$-module over the complex numbers. Now if $V$ is a $G$-module of dimension 1, I would like to prove $U \otimes V$ is an irriducible $G$-module. My ...
0
votes
1answer
32 views

Why $A/\operatorname{rad}A$ is generated by $e_a$?

Let $A$ be an algebra over an algebraically field $K$ and $(Q_A)_0$ be its ordinary quiver. Let $\{e_a \mid a \in (Q_A)_0\}$. Then $\{e_a \mid a \in (Q_A)_0\}$ is a complete set of primitive ...
0
votes
1answer
39 views

How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem?

Let $A$ be a $K$-algebra and $K$ an algebraically closed field. How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem? Thank you very much. ...
1
vote
1answer
22 views

Question about the connectivity of the ordinary quiver of a connected algebra.

I am reading the book Elements of Representation Theory of Associative Algebras, Volume 1. I have some questions about the connectivity of the ordinary quiver of a connected algebra. On page 61, ...
0
votes
0answers
30 views

Map between completions

Let $R$ be a ring with $1$ and let $T$ and $T'$ denote finite dimensional $R$ bimodules. Let $C(T)$ and $C(T')$ denote the completions of $T$ and $T'$ with respect the ideal generated by $R$. Now ...
7
votes
2answers
263 views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebra, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. Edit: ...
0
votes
1answer
21 views

How can we show that $X_A \cong T_e(Y_B)$?

I am reading the book Elements of representation theory of associative algebras 1. On Line 7 of page 37 (I attached this page below), it is said that $X_A \cong T_e(Y_B)$ since the diagram above line ...
1
vote
1answer
46 views

Is cyclic modules $=$ simple modules?

Let $A$ be an algebra with identity $1$ and $N$ be a right module of $A$ generated by $n_1 \in N$. That is $N=n_1A$. Is $N$ a simple module? I think that maybe this is not true. Let $N=A$ and suppose ...
2
votes
1answer
65 views

Is a surjective maps from a projective module to another projective module bijection?

Let $M$ and $N$ be projective $A$-modules. If we know that $f: M \to N$ is surjective and $g: N \to M$ is surjective, can we conclude that $M$ is isomorphic to $N$? More generally, if $M$ and $N$ are ...
2
votes
1answer
38 views

Length of modules.

Let $M, N$ be two $A$-modules. If there is a surjective $A$-map from $M$ to $N$, can we conculde that $\ell(M) \geq \ell(N)$. Here $\ell(M)$ is the number of modules in a composition series of $M$. ...
2
votes
0answers
49 views

Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
0
votes
1answer
58 views

Tilting modules

The theory of tilting modules seems to be a very fruitful field. I have some questions which seem natural to me, but can be trivial or stupid for people who works with this sort of theory. 1) Do all ...
0
votes
0answers
28 views

How to show that a local finite dimensional algebra is basic from definition?

A basis algebra is an algebra $A$ such that $e_i A \not\simeq e_j A$ for any $i \neq j$, where $e_1, \ldots, e_n$ is a complete set of primitive orthogonal idempotents. A local algebra is an algebra ...
2
votes
1answer
31 views

Questions about epimorphisms.

Let $P, M, N$ be $A$-modules over a field $K$. If we know that $h:P\to M$ is surjective, $g:N\to P$ is a A-homomorphism such that $hg$ is surjective, can we have $\operatorname{im} g + \ker h = P$? I ...
2
votes
2answers
49 views

Examples of projective modules which are not free modules. [duplicate]

Are there some examples of projective modules which are not free modules? Thank you very much.
3
votes
1answer
25 views

Question about maximal ideals of an algebra.

Let $A$ be a $K$-algebra. Let $\text{rad}A$ be the radical of $A$. That is the intersection of all maximal right ideals of $A$. Suppose that $A/\text{rad}A$ is isomorphic to $K$. How can we show that ...
1
vote
2answers
68 views

Question about idempotents.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $e$ be an idempotent of $A$ and ...
4
votes
1answer
33 views

A question about the quotient of a $K$-algebra by its radical.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $e$ be an idempotent of $A$ and ...
1
vote
1answer
38 views

Question about modules.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $I$ be an ideal of $B$ and $S$ be a ...
0
votes
1answer
41 views

Why $(M/M \operatorname{rad} A) \operatorname{rad}A=0$?

Let $A$ be a ring and $M$ a right $A$-module. Why we have $(M/M \operatorname{rad}A) \operatorname{rad}A=0$? Thank you very much.
3
votes
1answer
41 views

How to show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?

Let $M, N$ be right $A$-modules. How can we show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?
3
votes
1answer
33 views

Question about radical of a module.

Let $M$ be a right $A$-module. How to show that $m\in \operatorname{rad}(M)$ iff for any simple right $A$-module $S$ and any $f\in \operatorname{Hom}_A(M, S)$, $f(m)=0$? I think that if $m$ is ...
1
vote
1answer
50 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
8
votes
2answers
185 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
3
votes
1answer
36 views

Relating Modules and Representations

I am currently studying representation theory and am struggling with the concepts which relate modules and representations. The specific question I am looking at right now is this: but while ...
1
vote
1answer
70 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
4
votes
0answers
55 views

Character of $\text{Hom}_{\mathbb{C}}(\mathbb{C}G,\mathbb{C})$

I've been given the following two questions, and for both I'm really unsure what to do (I'm a beginner with the theory of characters). Let $G$ be a finite group, $\mathbb{C}G$ its group algebra over ...
5
votes
1answer
79 views

Endormorphism ring of quaternions isomorphic to the quaternion ring

I found the quoted question in this post interesting: Confusion regarding what kind of isomorphism is intended. I don't have commenting privileges just yet, and since the question already has an ...
2
votes
1answer
73 views

Confusion regarding what kind of isomorphism is intended.

For a class I'm taking this semester, I was given this question: (To guarantee clarity, I am quoting the full question even though my own question is only regarding the final sentence.) Let $G$ be ...
4
votes
3answers
323 views

Permutation module of $S_n$

Let $G=S_n$ and let $V$ be the permutation module of $G$ with basis $\{x_1,\cdots,x_n\}.$ Let $\lambda, \mu \in \mathbb{C}$ to allow one to define a $\mathbb{C}G$-homomorphism $\rho:V \to V$ by ...
1
vote
2answers
75 views

Minimal Right Ideals in $\Bbb{C}[G]$

Let $G$ be a finite group. Consider the group algebra $\Bbb{C}[G]$ as a right module over itself. By Maschke's Theorem, $\Bbb{C}[G]$ is semisimple. How can we identify all the minimal right ideals of ...
2
votes
1answer
50 views

Modules of a group over different fields.

Let $G$ be the cyclic group of order 2 and let $V$ be the regular $\mathbb{F}G$-module of $G$. Determine the $\mathbb{F}G$-submodules of $V$ where $\mathbb{F}$ has each of zero or positive ...
4
votes
1answer
95 views

Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
3
votes
2answers
89 views

questions about representation theory/structure theorem for finitely generated modules

I am reading a book in representation theory by James and Liebeck. They define an FG-module as: Let $V$ be a vector space over the field $F$ and let $G$ be a group. Then $V$ is an FG-module if a ...

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