isomorphism between vector spaces and modules - Commutative Algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:12:50Z http://mathoverflow.net/feeds/question/59420 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59420/isomorphism-between-vector-spaces-and-modules-commutative-algebra isomorphism between vector spaces and modules - Commutative Algebra Andrei 2011-03-24T11:32:03Z 2011-03-24T22:23:31Z <p>Hi, Let $M_i$ be A modules. Then we know that $Ass (\oplus M_i) = \bigcup Ass(M_i)$. We consider here isomorphisms between modules. </p> <p>Now consider a stanley decomposition so $M=\oplus ^r_{i=1} u_iK[Z_i]$ where $Z_i \subseteq \left\lbrace x_1,...,x_n \right\rbrace$, $u_i$ is a monomial in $S=K[x_1,...,x_n]$ . M is a $K[x_1,...,x_n]$ module $Z^n$ graded and $u_iK[Z_i]$ is $K[Z_i]$ free . In this direct sum the above equality is not true because here we consider isomorphism between vector spaces. I mean by this that Ass M is not $\bigcup Ass(u_iK[Z_i])$. This happens because in the direct sum we have a vector spaces isomorphism but I don't understand the difference between the module isomorphisms and the vetor space isomorphisms.</p>