# ac.commutative algebra – Filtration over tensor product retort

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## ac.commutative algebra – Filtration over tensor product

Let
$$M supset M_1 supset ldots supset M_n supset ldots textual content{ and } N supset N_1 supset ldots supset N_n supset ldots$$
breathe exhaustive lowering filtrations of modules over a commutative integral ring $$R$$. I’d affection to know if there’s any path to outline a filtration over $$M otimes_R N$$ such that the commencement $$Gr_k(M otimes_r N)$$ verifies
$$Gr_k(M otimes_r N)=oplus_{i+j=okay} M_i/M_{i+1}otimes_RN_j/N_{j+1}.$$
I do know it’s workable to assassinate so within the illustration of finite dimensional vector areas by defining
$$(M otimes_R N)_k=sum_{i+j=okay}M_i otimes N_j.$$
So if anybody has an thought or a reference the place one thing affection might breathe written, I’d worth. Thanks for studying me.

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