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1. (华中科技大学, Tao 供题)
Let \((A, \mathfrak{m})\) be a local Noetherian ring (with element 1) and \(M\) be a finitely generated \(A - \) module. A free resolution of $M$ is an exact sequence
$$
\cdots \stackrel{\varphi_{k+2}}{\longrightarrow} F_{k+1} \stackrel{\varphi_{k+1}}{\longrightarrow} F_k \stackrel{\varphi_k}{\longrightarrow} \cdots \stackrel{\varphi_2}{\longrightarrow} F_1 \stackrel{\varphi_1}{\longrightarrow} F_0 \stackrel{\varphi_0}{\longrightarrow} M \rightarrow 0
$$
with finitely generated free \(A-\) {modules} \( F_i\) for \(i \geq 0\).
(a) Consider \(M / \mathfrak{m} M\) as a \(A / \mathfrak{m}- \) vector space. Let \(\left\{x_1, \ldots, x_n\right\}\) be a minimal set of generators of \(M\). Prove that \(\left\{\overline{x_1}, \ldots, \overline{x_n}\right\}\) is a basis of the vector space \(M / \mathfrak{m} M\), where \( \overline{x_i}=x_i+\mathfrak{m} M \in M / \mathfrak{m} M\).
(b) A free resolution as above is called minimal free resolution if \(\varphi_k\left(F_k\right) \subseteq \mathfrak{m} F_{k-1}\) for \(k \geq 1\). Prove that \(M\) has a minimal free resolution.
(c) If \( M\) has two minimal free resolutions
$$
\begin{array}{l}
\cdots \stackrel{\varphi_{k+2}}{\longrightarrow} F_{k+1} \stackrel{\varphi_{k+1}}{\longrightarrow} F_k \stackrel{\varphi_k}{\longrightarrow} \cdots \stackrel{\varphi_2}{\longrightarrow} F_1 \stackrel{\varphi_1}{\longrightarrow} F_0 \stackrel{\varphi_0}{\longrightarrow} M \rightarrow 0, \\
\cdots \stackrel{\psi_{k+2}}{\longrightarrow} G_{k+1} \stackrel{\psi_{k+1}}{\longrightarrow} G_k \stackrel{\psi_k}{\longrightarrow} \cdots \stackrel{\psi_2}{\longrightarrow} G_1 \stackrel{\psi_1}{\longrightarrow} G_0 \stackrel{\psi_0}{\longrightarrow} M \rightarrow 0,
\end{array}
$$
prove that \(\operatorname{rank}\left(F_k\right)=\operatorname{rank}\left(G_k\right)\) for \( k \geq 0\).
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2. (华中科技大学, Tao 供题)
\( k\) is a field. Let \( E\) be an algebraic extension of \(k\), and let \(\sigma: E \rightarrow E\) be an embedding of \(E\) into itself over \(k\). Then \(\sigma\) is an automorphism. \[ \]
3. (华中科技大学, Tao 供题)
\(k\) is a field. Let \(V\) be a \(k -\) vector space. \(W\) is a subspace of \(V, T: V \rightarrow V\) is a injective \(k-\) linear map such that \(T(W) \subseteq W\). Suppose \(V / W\) and \(W / T(W)\) are finite-dimensional \(k -\) vector spaces. Prove that as \(k -\) vector spaces, \(T(V) /(W \cap T(V))\) and \( (W+T(V)) / W\)
have same dimension. \(W /(W \cap T(V))\) and \((W+T(V)) / T(V)\) have same dimension. \(V /(W+T(V))\) and \((W \cap T(V)) / T(W)\) have same dimension. \(V / T(V)\) and \(W / T(W)\) have same dimension.
