Exercise 3, § 5

Let M be a finitely generated projective A-module. For every endomorphism u of M, let ũ denote the tensor θ_M^{-1}(u) ∈ M* ⊗_A M corresponding to it.

(a) For every element x ∈ M, show that u(x) = c_2^1(x ⊗ ũ), where x ⊗ ũ belongs to M ⊗ M* ⊗ M = T_{\{2\}}^{(1, 3)}(M).

(b) Let u, v be two endomorphisms of M and w = u ◦ v; show that ũw = c_3^2(ũ ⊗ ũ), where ũ ⊗ ũ belongs to T_{\{1, 3\}}^{(2, 4)}(M).

(c) For every automorphism s of M and every tensor z ∈ T_J^I(M), a tensor s . z ∈ T_J^I(M) is defined by the condition that if $z = \bigotimes_{k \in I \cup J} z_k$ where z_k ∈ M for k ∈ I and z_k ∈ M* for k ∈ J, then s . z = $\bigotimes_{k \in I \cup J} z'_k$ with z'_k = s(z_k) if k ∈ I, z'_k = t_s^{-1}(z_k) \text{ if } k \in J. Show that the group $\mathbf{GL}(M)$ operates by the law $(s, z) \mapsto s . z$ on the A-module $T_J^1(M)$, the mappings $z \mapsto s . z$ being automorphisms of this A-module. Moreover, for every ordered pair of indices $i \in I$, $j \in J$,

If M is projective and finitely generated, show, in the notation of Exercise 2, that $s . \tilde{u} = (s \circ u \circ s^{-1})^\sim$.