I am looking for a reference for the following basic fact:

Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. Then, $Tor_i^R(M,N)$ is artinian.

(I know that it is easy to prove. I guess nevertheless that this is written down somewhere in the standard literature about homological algebra, and I would like to know where.)

Remark 1: The above conclusion also holds if $R$ is coherent, $M$ is artinian, and $N$ is of finite presentation. A reference for this generalisation would also be fine.

Remark 2: Leamer proves in his PhD Thesis (2010) a generalisation to the case where $R$ is noetherian, $M$ is artinian, and $N$ is minimax. But I guess there must be an earlier reference for the less general case.

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    $\begingroup$ I doubt that there is a precise reference. As you point out, this is a straightforward consequence of the definitions; if I had to use it, I'd just say that, I wouldn't bother to write a lemma. $\endgroup$ – abx Nov 24 '19 at 13:05
  • $\begingroup$ Ok, it seems there is no such reference. Thanks, @abx, for your suggestion, to which I will probably adhere. $\endgroup$ – Fred Rohrer Nov 27 '19 at 12:26

This follows immediately from the slightly more general Lemma 2.2 in M. Brodmann, S. Fumasoli, R. Tajarod, Local cohomology over homogeneous rings with one-dimensional local base ring, Proc. Amer. Math. Soc. 131 (2003), 2977-2985, which considers additionally a change of rings.


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