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\Materia{Teoría de Álgebras}
\Cuatrimestre{Segundo Cuatrimestre --- 2012}
\Practica{2}
\Titulo{Producto tensorial}

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\begin{document}

\maketitle

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\begin{ejercicios}

%%%%%%%%%%
\item Muestre que $\QQ\ot_\ZZ\QQ\cong\QQ$.

%%%%%%%%%%
\item Sea $A$ un anillo y $M_A$ y ${}_AM$ $A$-módulos. Muestre que $M\ot_AN$ es
un $\End_A(M)$-$\End_A(N)$-bimódulo.

%%%%%%%%%%
\item Sean $A$ y $B$ anillos y $M{}_A$, ${}_AN{}_B$ y ${}_BP$ módulos. Muestre
que hay un isomorfismo natural
  \[
  M\ot_A(N\ot_BP)\cong(M\ot_AN)\ot_BP.
  \]

%%%%%%%%%%
\item Sea $A$ un anillo conmutativo, $\a\subset A$ un ideal y $M$ un $A$-módulo.
Muestre que hay un isomorfismo natural $A/\a\ot_AM\cong M/\a M$.

%%%%%%%%%%
\item Sea $A$ un anillo, $M{}_A$ y ${}_AN$ módulos y supongamos que
$M=\sum_{i\in I}M_i$ es suma de una familia de submódulos $\{M_i\}_{i\in I}$.
Si $M_i\ot_AN=0$ ara todo $i\in I$, entonces $M\ot_AN=0$.

%%%%%%%%%%
\item Sea $A$ un anillo y $M$ un $A$-módulo playo. Si $N\subset M$ es un sumando
directo, entonces $N$ es playo.

%%%%%%%%%%
\item Si $A$ es un anillo conmutativo y $M$, $N$ son $A$-módulos playos,
entonces $M\ot_AN$ es un $A$-módulo playo.

%%%%%%%%%%
\item Sea $A$ un anillo y $S\subset A$ un subconjunto multiplicativamente
cerrado.
\begin{ejitems}
\item Si $M$ es un $A$-módulo izquierdo, entonces hay un isomorfismo
$A_S\ot_AM\cong M_S$.
\item El $A$-módulo derecho $A_S$ es playo.
\end{ejitems}

%%%%%%%%%%
\item Sea $A$ un anillo. Las siguientes afirmaciones son equivalentes:
\begin{enumerate}[label=(\roman*)]
\item Todo $A$-módulo a izquierda es playo.
\item Todo $A$-módulo a derecha es playo.
\item Para todo $a\in A$, existe $x\in A$ tal que $a=axa$.
\item Todo ideal izquierdo principal está generado por un idempotente.
\item Todo ideal derecho principal está generado por un idempotente.
\end{enumerate}

%%%%%%%%%%
\item\header{Producto tensorial de álgebras} Sea $k$ un cuerpo y sean $A$ y $B$
$k$-álgebras. Muestre que $A\ot_kB$ es un álgebra de forma tal que el producto
está dado por
  \[
  a\ot b\cdot a'\ot b=(aa')\ot (bb').
  \]

%%%%%%%%%%
\item Sea $k$ un cuerpo, $A$ una $k$-álgebra y $n$, $m\in\NN$. Muestre que hay
isomorfismos naturales de álgebras
$A[X]\cong k[X]\ot_kA$, $\M_n(A)\cong\M_n(k)\ot_kA$ y
$\M_{nm}(A)\cong\M_n(A)\ot_k\M_m(A)$.

\end{ejercicios}

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\clearpage
 
\foto{Whitney}{Hassler Whitney\\1907--1989, Estados Unidos-Suiza.}{%
Whitney fue el primero en introducir explícitamente el producto tensorial de
grupos abelianos, en \emph{Tensor Products of Abelian Groups}, Duke
Math.~J.~\textbf{4}, 495--528 (1938).}


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