By Jensen C.U.

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21, we have σ in n−1 = σ nj i−1 if j < i. The term σ in n−1 j j occurs in the first sum (over all j < i) with sign (−1)i+ j , and the term n−1 σ nj i−1 occurs in the second sum (the first index j is now the larger index), and with opposite sign (−1) j+i−1 . Thus, all terms in ∂∂(σ ) cancel, and ∂∂ = 0. • We can now define singular cycles and singular boundaries. Definition. For each n ≥ 0, the group of singular n-cycles is Z n (X ) = ker ∂n , and the group of singular n-boundaries is Bn (X ) = im ∂n+1 .

A left R-module M is simple (or irreducible) if M = {0} and M has no proper nonzero submodules; that is, {0} and M are the only submodules of M. 17. A left R-module M is simple if and only if M = I is a maximal left ideal. Proof. This follows from the correspondence theorem. • For example, an abelian group G is simple if and only if G is cyclic of order p for some prime p. The existence of maximal left ideals guarantees the existence of simple modules. 46 Hom and Tens or Definition. Ch. 2 A finite or infinite sequence of R-maps and left R-modules f n+1 fn · · · → Mn+1 −→ Mn −→ Mn−1 → · · · is called an exact sequence2 if im f n+1 = ker f n for all n.

A contravariant functor T : C → D, where C and D are categories, is a function such that (i) if C ∈ obj(C), then T (C) ∈ obj(D), (ii) if f : C → C in C, then T ( f ) : T (C ) → T (C) in D (note the reversal of arrows), f g T (g) T( f ) (iii) if C → C → C in C, then T (C ) → T (C ) → T (C) in D and T (g f ) = T ( f )T (g), (iv) T (1 A ) = 1T (A) for every A ∈ obj(C). To distinguish them from contravariant functors, the functors defined earlier are called covariant functors. 20 Introduction Ch. 10.