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An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology

I am currently reading Künneth spectral sequence, which is given below. Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
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