2
votes
0answers
23 views

On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
1
vote
0answers
67 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
1
vote
1answer
55 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
1
vote
0answers
20 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
0
votes
0answers
36 views

Show $\mathbb F_{p}[x]/(x^{p}-1)$ is indecomposable as a representation of $\mathbb Z/ p\mathbb Z$

Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant ...
1
vote
0answers
28 views

Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there ...
0
votes
0answers
58 views

The annihilator of finitely generated modules over PID

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Suppose there exists prime ideal $p$ and integer $i$ such that $p^i=\operatorname{Ann}(M)$. Then prove: (1) there ...
0
votes
0answers
49 views

Conjecture about some rings and roots of unity. [duplicate]

Let $\Bbb R_{\geqslant 0}[X_n]$ be a polynomial semiring. More precisely $\Bbb R_{\geqslant 0}[X_n]$ are the polynomials of $X_n$ with positive real coefficients with $(X_n)^n = 1$. Let $F(n)$ be ...
1
vote
1answer
48 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
2
votes
1answer
131 views

Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
1
vote
1answer
38 views

Four term exact sequence for Artin algebras

Suppose $\Lambda$ is an Artin algebra, i.e. an algebra finitely generated as $R$-module for a commutative Artin ring $R$, $A$ is a finitely generated left $\Lambda$-module, and $X$ a finitely ...
1
vote
0answers
104 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
2
votes
2answers
139 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
2
votes
0answers
52 views

If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
2
votes
1answer
34 views

What are modules in $\operatorname{add} T$ explicitly?

Let $A$ be a $K$-algebra and $T$ an $A$-module. The category $\operatorname{add} T$ is defined as the smallest additive subcategory of the category $\operatorname{mod} A$ (the category of all finite ...
1
vote
1answer
62 views

How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
1
vote
1answer
38 views

How to show that ${}_{B}T_{A} \otimes DM \in \operatorname{Gen}({}_{B} T)$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
0
votes
1answer
51 views

How to show that $DA\cong D\operatorname{Hom}_{B}(T, T) \cong DT \otimes_{B} T$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
3
votes
2answers
98 views

When a group algebra (semigroup algebra) is an Artinian algebra?

When a group algebra (semigroup algebra) is an Artinian algebra? We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group ...
4
votes
1answer
125 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
109 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 ...
5
votes
0answers
81 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} ...
3
votes
2answers
104 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 ...
5
votes
1answer
85 views

For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, ...
1
vote
1answer
104 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 ...
0
votes
1answer
37 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
75 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
50 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, ...
12
votes
2answers
895 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 $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
0
votes
1answer
30 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
127 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
61 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
56 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
0answers
52 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
36 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
133 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.
1
vote
2answers
65 views

What are maximal ideals of $K[t]$?

Let $K[t]$ be the algebra of all polynomials in $t$. What are maximal ideals of $K[t]$? I know that $\langle t \rangle = \{tf \mid f \in K[t]\}$ is a maximal ideal. Are there other maximal ideals? ...
1
vote
2answers
78 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 ...
1
vote
0answers
39 views

Isomorphisms betweenVerma modules over a semisimple Lie algebra

Fix a finite dimensional, semisimple Lie algebra $L$ and denote the Verma $L$-modules by $V(\lambda ')$ where $\lambda '$ are corresponding weights. Assume that there is an isomorphism between two ...
1
vote
1answer
47 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 ...
1
vote
1answer
24 views

Why $e_1A=M_1$?

Let A be a ring with identity $1$ and $M_1, M_2$ submodules of $A$. We have $1=e_1+e_2$, where $e_i\in M_i$, $i=1, 2$. We can show that $e_i$ are idempotent and $e_1e_2 = e_2e_1 = 0$. We have ...
0
votes
1answer
42 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
48 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
76 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
89 views

When are all irreps left ideals in the group algebra and generated by idempotents?

Let $A$ be an associative algebra. I am wondering under what conditions we can get all irreducible representations of $A$ as left ideals $A\cdot e$ with $e\in A$ an idempotent. This is certainly the ...
2
votes
1answer
76 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 ...
0
votes
1answer
116 views

Question about the socle of a finite-dimensional algebra

Let $n\in \mathbb{N}$ and $k$ be an arbitrary field. Is the socle of the algebra $k[x,y]/\langle x^2,y^{n+2}\rangle$ isomorphic to $k$? Is $k[x,y]/\langle x^2,y^{n+2}\rangle$ a symmetric algebra or ...
2
votes
0answers
61 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
1
vote
0answers
45 views

$GL_2$-Invariants of $\mathbb{C}[X,Y]$

One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: ...
4
votes
0answers
119 views

An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...