Theorem 11.34 in Rotman's Homological Algebra, 1st edition says that given two positively graded complexes $A$ and $C$ such that $A$ is flat, there exists a certain spectral sequence which converges to the homology of the tensor complex $A\otimes C$. My question is: Can we say something similar when either of both of the complexes are negatively graded?
I care about this because I want to compute the Hochschild cohomology of a certain algebra with coefficients. So, I have the Hochschild cochain complex which I tensor with a certain module, and then I want to compute the cohomology of the complex. So, in this case, one of the complexes is negatively graded and the other is concentrated in degree zero. If the above spectral sequence doesn't generalise to negative complexes, is there another way to compute the cohomology in the example that I have in mind?