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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.

2 added 140 characters in body; added 154 characters in body

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 $Z_i \subseteq \left{x_1,...,x_n left\lbrace x_1,...,x_n \right} right\rbrace$and , $M$ 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. Can anyone explain me I mean by this that Ass M is not $\bigcup Ass(u_iK[Z_i])$. This happens because in the differences and why direct sum we have a vector spaces isomorphism but I can't apply don't understand the above equality? Thanksdifference between the module isomorphisms and the vetor space isomorphisms.

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isomorphism between vector spaces and modules - Commutative Algebra

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 $Z_i \subseteq \left{x_1,...,x_n \right}$ and $M$ is a $K[x_1,...,x_n]$ module. In this direct sum the above equality is not true because here we consider isomorphism between vector spaces. Can anyone explain me the differences and why I can't apply the above equality? Thanks