# Wild Spectral Sequences Ep. 4: Schanuel’s Lemma

It’s time for another installment of Wild Spectral Sequences! We shall start our investigations with a classic theorem useful in many applications of homological algebra called Schanuel’s lemma, named after Stephen Hoel Schanuel who first proved it.

Consider for a ring $R$ the category of left $R$-modules, and let $A$ be any $R$-module. Schanuel’s lemma states: if $0\to K_1\to P_1\to A\to 0$ and $0\to K_2\to P_2\to A\to 0$ are exact sequences of $R$-modules with $P$ projective, then $K_1\oplus P_2\cong K_2\oplus P_1$.

We shall prove this using spectral sequences. I came up with this proof while trying to remember the “usual” proof of Schanuel’s lemma and I thought that this would be a good illustration of how spectral sequences can be used to eliminate the dearth of clarity in the dangerous world of diagram chasing.

Before I start, I’d like to review a pretty cool fact I which I think of as expanding the kernel, which is pretty useful More »