Exact Sequences In the study of modules, the exact sequence plays a central role. We relate it to the kernel and image, the direct sum and direct product. We introduce diagram chasing, and prove the Snake Lemma, which is a fundamental result in homological algebra. We define projective modules, and characterize them in four ways. Finally, we prove Schanuel's Lemma, which relates two arbitrary presentations of a module. In an appendix, we use deteminants to study free modules. DEFINITION $(5\mathrm{.}1).-$ A $($finite or infinite$)$ sequence of module homomorphisms cdots$->$M$\_($i$-1)->$alpha$\_($i$-1)$M$\_($i$)->$alpha$\_($i$)$M$\_($i$+1)->$cdots is said to be exact at M$\_($i$)$ if Ker$(\backslash$alpha $\_($i$))=$Im$(\backslash$alpha $\_($i$-1)).$ The sequence is said to be exact if it is exact at every M$\_($i$),$ except an initial sou