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Exactness via kernels and images

A sequence is exact at a term precisely when the incoming image equals the outgoing kernel.
Exactness via kernels and images

Exactness via kernels and images: A sequence of RR-modules and homomorphisms

⋯⟶Mi−1→di−1Mi→diMi+1⟶⋯ \cdots \longrightarrow M_{i-1}\xrightarrow{d_{i-1}} M_i \xrightarrow{d_i} M_{i+1}\longrightarrow \cdots

is exact at MiM_i if and only if im⁡(di−1)=ker⁡(di)\operatorname{im}(d_{i-1})=\ker(d_i).

This rewrites purely in terms of and , and is used constantly for .