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.

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.