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Math 724

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Spring 2014

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Homework Set 5

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\begin{exer}
Set $R=A[X_1,\ldots,X_4]$ and find an irredundant m-irreducible decomposition of the ideal
$J=(X_1^2X_2^2X_3^2,X_1^3X_2^3X_4^3,X_1^2X_3^2X_4^2,X_2^4X_3^4X_4^4)R$ using our decomposition result for weighted facet ideals.
Justify your answer.
\end{exer}

\begin{exer}
Let $\Delta$ be a simplicial complex on $V=\{v_1,\ldots,v_d\}$.
Let $\Delta''$ denote the set of non-faces of $\Delta$, and let $\psi\colon\Delta''\to\mathbb N_+$.
Set $R=A[X_1,\ldots,X_d]$ and consider the face ideal $J_{\Delta}$ and the weighted face ideal
$J_{\Delta}^{\psi}$. Prove that $\dim(R/J_{\Delta}^{\psi})=\dim(R/J_{\Delta})$ where $\dim$ is the Krull dimension.
\end{exer}

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