# One question about homological algebra

Let $A$ be an Artin algebra, and $0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$ a short exact sequence with $pd N<\infty$ in $\text{mod}-A$, where $pd N$ is projective dimension of $N$.

Then why $\Omega^{pd N}(L) \cong \Omega^{pd N}(M)$? Where $\Omega^{i}$ is the $i-\text{th}$ syzygy.

• If you have two different questions, ask them separately, please. Especially if they are not related at all like here! – Najib Idrissi Nov 29 '16 at 14:39
• 1: To preserve exact sequences, $F$ should be an exact functor. Maybe it's not worth posting as a separate question. – user144221 Nov 29 '16 at 16:32
• To question 2, you can look at the Horseshoe lemma: en.wikipedia.org/wiki/Horseshoe_lemma – Xiaosong Peng Nov 30 '16 at 7:06
• @jin zhang Any long exact sequence may be decomposed into short exact sequences. A quick search shows that this question has been answered various times on this site: see e.g. math.stackexchange.com/questions/207551 and math.stackexchange.com/questions/803578 – user144221 Nov 30 '16 at 7:06
• @jin zhang An equivalence of categories is both left adjoint and right adjoint, hence preserves all colimits and limits, in particular cokernels and kernels, so it is exact. – user144221 Nov 30 '16 at 15:18