Exercise 8, § 8

One calls tor-dimension of A and denotes by $\mathrm{td}(A)$ the upper bound in $\overline{\mathbf{Z}}$ of the set of integers n for which there exists a left a-module N and a right a-module M such that $\mathrm{Tor}_n^A(M, N) \neq 0$. a) Let n be an integer $\geqslant 0$. Prove that the following conditions are equivalent: α) $\mathrm{td}(A) \leqslant n$. β) For every a-module M, one has $\mathrm{dpl}_A(M) \leqslant n$ (exercise 7). γ) For every monogeneous a-module M, one has $\mathrm{dpl}_A(M) \leqslant n$. b) Prove that one has $\mathrm{td}(A) = \mathrm{td}(A^\circ)$ and $\mathrm{td}(A) \leqslant \mathrm{dh}(A)$; the equality holds if A is left Noetherian.