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



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> 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.
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(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.



Artinian Tor modules (Reference request)
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.


