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There is a natural isomorphism of $kG$-modules: $U \otimes_k (Ind_H^G(V) ) \cong Ind_H^G(Res_H^G(U) \otimes_k V)$ sending $u \otimes (x\otimes v)$ to $x \otimes (x^{-1}u \otimes v)$ where $u \in U, x \ …

If R is a PID, then every finitely generated R-module M is a direct sum of cyclic modules. If M is torsion-free, then it is a direct sum of copies of R. …

I was checking this for the following example, but it doesn't seem to line up with the definitions since there are obviously simple modules of the form $(0,1,0)^T$ whereas the socle of the example is only …

Let $A, B$ be $k$-algebras. $M$ is $A$-$B$-bimodule that is finitely generated projective as a left $A$-module and as a right $B$-module, $N, U$ are $B$-$A$-bimodules that is finitely generated projec …

For any subgroup $H$ of a fnite group $G$ and any $x ∈ G$, the right $kH$-module $k[xH]$ is also a left $k^{x}H$-module because $xH = xHx^{−1}x = {^xH}x$ is also a left $^xH$-coset. Thus $k[xH]$ is in …

Let $G$ be a finite group. Suppose that $k$ is a splitting field for all subgroups of $G$ and that $|G|$ is invertible in $k$. Let $N$ be a normal subgroup of $G$. Let $χ ∈ \operatorname{Irr}(kG)$ and …

Let $U,V$ be $A$-modules having composition series. Let $W$ be a submodule of $U \oplus V$. Denote by $\alpha: W \to U$ and $\beta: W \to V$ the components of the inclusion map $W \to U\oplus V$. …

Let $A, B$ be $k$-algebras, and let $M, M'$ be $A-B$-bimodules. There is a bijection $Nat(Hom_A(M, −), Hom_A(M', −)) \cong Hom_{A⊗_kB^{op}} (M', M)$ sending a natural transformation $η : Hom_A(M, −) \ …

$A,B$ are $k$-algebras and finitely generated as $k$-modules. Let $X$ be a simple $A \otimes_k B$ module. …

Is there any relation between f.g. projective $k$-modules and f.g.projective $A$-modules? Thank you! …

Suppose one has a surjective algebra homomorphism $f: A \to B$. It induces a surjective algebra homomorphism $f': A/J(A) \to B/J(B)$. Then the author goes on to prove that this homomorphism $f'$ split …

Let $A$ be a symmetric $k$-algebra with a symmetrising form $s : A → k$. For any $k$-subspace $U$ of $A$ we consider the space $U^⊥$ = {$a ∈ A | s(aU) = 0$}. Then $(U^⊥)^⊥ = U$ and if $k$ is not a fie …

In that case, $A$ has, up to isomorphism, a unique simple module $S$, and moreover the following hold: (i) We have an isomorphism of left $A$-modules $A \cong S_n$; in particular, $A$ is semisimple. …

$A$ a finite-dimensional $k$-algebra, $X$ projective $A ⊗_k A^{op}$-mdoule. The projectivity of $X$ as a bimodule implies that $X⊗_A$ − sends finite dimensional module to a projective module. I am not …

Suppose that $k$ is a field of prime characteristic $p$. Let $P$ be a finite $p$-group, and let $U$ be a finitely generated $kP$-module. Then: (i) We have $soc(U) = U^P$ ($P$ fixed points in $U$). (ii …