Grading on Tor modules coming from different constructions

Question

Let $M, N$ be finitely generated modules over a commutative Noetherian ring $R$. There are at least three ways to compute the Tor-modules $\operatorname{Tor}^R_i(M, N)$ all of which gives isomorphic modules hence the construction is unambiguous:

(1) Take a free resolution of $M$, tensor with $N$ and compute homologies.

(2) Take a free resolution of $N$, tensor with $M$ and compute homologies.

(3) Take free resolutions of $M$ and $N$, build the total tensor product complex and compute homologies.

Now assume $R$ is $\mathbb Z$-graded (or more generally, graded by an abelian group) and $M, N$ are graded $R$-modules. In each of the above three constructions, we can take resolution by finitely generated graded free modules which in turn produce gradings on the Tor modules. My question is: Are all these grading on the Tor modules coming from the above constructions are same? I.e., do all these constructions produce graded isomorphic modules?

I feel like there should be some standard literature discussing these, but I have been unable to locate any (perhaps I don't know the correct sources). Any pointers to literature would also be very appreciated. Thank you

Answer 1

I believe this is for the straightforward reason pointed out by Dave Benson in the comments, which I will elaborate upon. Noetherian and finite generation hypotheses are not relevant.

Let $R$ be a graded commutative ring and $P_\bullet\to M$, $Q_\bullet\to N$ free resolutions of the modules $M,N$ (by graded module homomorphisms and graded differentials). In the ungraded category, the canonical disambiguation of $\mathsf{Tor}$ comes from the quasiisomorphisms: $P_\bullet\otimes_RN\leftarrow P_\bullet\otimes^{\mathsf{total}}_R Q_\bullet\rightarrow M\otimes_RQ_\bullet$, where the maps involved are quite straightforward - e.g. the map $Q_0\to N$ makes a map $\oplus_{j+k=n}P_j\otimes Q_k\to P_n\otimes Q_0\to P_n\otimes N$ which defines the $n$th component of $P_\bullet\otimes Q_\bullet\to P_\bullet\otimes N$.

Any sensible convention of graded tensor product will have that if $Q_0\to N$ is graded, then so is $P_n\otimes Q_0\to P_n\otimes N$. And so on. Therefore, we see these quasiisomorphisms are all graded maps of complexes of graded $R$-modules, so that the resulting identification (on taking homology and inverting) is a graded isomorphism.