Exercise 11, § 4

Let $A$ be the ring $\mathbf{Z}[X]$, where $X = (X_i)_{i \in I}$ is a finite family of indeterminates. Let $J$ be an ideal of $A$ generated by monomials. a) Let $i \in I$; let $P_1, \ldots, P_r ; Q_1, \ldots, Q_s$ be monomials generating $J$, where the $P_j$ are divisible by $X_i$ whereas the $Q_k$ are not. Prove that the transporter $J : X_i$ (ideal of $A$ formed by the elements $P$ such that $X_iP \in J$) is generated by the monomials $X_i^{-1}P_1, \ldots, X_i^{-1}P_r ; Q_1, \ldots, Q_s$. b) Deduce that if $J : X_i$ is equal to $J$ or $A$ for every $i \in I$, $J$ is the ideal generated by the variables $X_i$ which belong to $J$. c) Let $M$ be an $A$-module, and let $p$ be an integer. Suppose that $\mathrm{Tor}_p^A(A/J', M)$ (resp. $\mathrm{Ext}_A^p(A/J', M)$) is zero for every ideal $J'$ containing $J$ and generated by certain of the $X_i$. Prove that $\mathrm{Tor}_p^A(A/J, M)$ (resp. $\mathrm{Ext}_A^p(A/J, M)$) is zero (let $J'$ be an ideal containing $J$, generated by monomials, such that $\mathrm{Tor}_p^A(A/J', M) \neq 0$, and maximal for these properties; deduce from the exact sequence

and from b) that $J'$ is generated by the elements $X_i$ which it contains, which contradicts the assumption). d) Deduce the inequality $\mathrm{dp}_A(A/J) \leqslant \mathrm{Card}(I)$. e) Let $A$ be a ring, $x = (x_i)_{i \in I}$ a finite family of elements of $A$; suppose that for every subset $I'$ of $I$ the family $(x_i)_{i \in I'}$ is completely secant for $A$. Let $J$ be an ideal of $A$ generated by elements $x^\alpha$, where $\alpha$ runs through a finite subset $F$ of $\mathbf{N}^I$. Prove the inequality $\mathrm{dp}_A(A/J) \leqslant \mathrm{Card}(I)$ (let $J_0$ be the ideal of the ring $A_0 = \mathbf{Z}[(X_i)_{i \in I}]$ generated by the $X^\alpha$ for $\alpha \in F$; deduce from c) that $\mathrm{Tor}_p^{A_0}(A_0/J_0, A)$ is zero, and conclude by considering a projective resolution of the $A_0$-module $A_0/J_0$ of length $\leqslant n$).