By Andrew O Lindstrum

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22), we have proved that pdS −1 R S −1 M ≤ p. 34) Exercise: Projective dimension and special over-lining. Let a = (x) be a proper principal ideal in R and consider the corresponding over-lining. ϕ ψ (a) Let O → K −→ L −→ M → O be an exact sequence, and assume that x ∈ / ZeroR M , cf. 18). Prove the exactness of the induced sequence ϕ ψ O → K −→ L −→ M → O . (b) Prove that pdR M ≤ pdR M if x ∈ / ZeroR M ∪ ZeroR R. REFERENCES: [2AL] Anders Thorup: Matematik 2AL: Algebra, 2. udgave (1998). Index n R , 10 Mod(R), category of R–modules, 17 Zero = zero-divisors, 34 abelian group, 1 addition, 1 additive functor and direct sum, 29 additive functor and splitting sequence, 27 additive functor and zero, 27 additive functor, 26, 27 additive notation, 1 algebraic structures, 1 annihilator and linear independence, 13 annihilator, 9 associativity, 1–3 basis, 14 category of R–modules, 17 commutative group, 1 commutative ring, 2 constant functor, 17 contra-variant Hom functor, 19 contra-variant Hom is left-exact, 25 contra-variant Hom is not exact, 25 contravariant functor, 19 covariant Hom functor, 17 covariant Hom is left-exact, 24 covariant Hom is not exact, 25 covariant functor, 17 cyclic module, 5, 9 direct sum of projectives, 33 direct sum, finite, 7 direct sum, inner, 8 distributivity, 2, 3 domain = integral domain, 2 endomorphism ring, 2 exact functor of exact sequence, 30 exact functor, 22 exact sequence, 20 exact tilted sequences, 29 exactness of short sequence, 20 exactness of special functors, 23 field of fractions, 19 field, 2 finite direct sum, 7 finite generation theorem, 10 finitely generated free module over PID, 11, 12 finitely generated free module over field, 11 finitely generated free module over non-commutative ring, 11 finitely generated free module, 10, 11 finitely generated module, 5 five lemma, 25 fraction functor is exact, 24 fraction functor, 19 fractions and projectivity, 34 fractions, 18, 19 free is projective, 32 37 free module and isomorphism, 11, 14 free module, finitely generated, 10, 11 free modules over integral domain, 14 free module, 14 free resolution, 34 functor and direct sum, 29 functor of isomorphism, 17, 19 gamma functor, 29 generated submodule, 5 group, 1 half-exact functor of O, 22 half-exact functor, 22 half-exact is additive, 29 homomorphism theorem, 9 homomorphism, 6 homomorphic image of free, 14 homomorphism functor Hom(–,H ), 19 homomorphism functor Hom(H,–), 17 homomorphism modules, 15 homomorphisms from the ring, 17 ideal, 3 image of submodule, 7 image, 6 indentity element, 2 induced homomorphism (contra-variant case), 16 induced homomorphism (covariant case), 16 inner direct sum, 8 integral domain, 2 intersection of submodules, 4 inverse image of submodule, 7 invertible element, 2 isomorphism theorem, 9 isomorphism, 6 isomorphic short sequences and functors, 21 isomorphic short sequences, 21 kernel–cokernel sequence, 20 kernel, 6 kronecker delta function, 13 left-exact functor of left exact sequence, 22 left-exact functor of right exact sequence, 22 left-exact functor, 22 linear combination, 5 linear independence, 13 linear map, 6 module of fractions, 19 module of functions, 12 module, 3 multiplication homomorphism, 6 multiplication, 2, 3 multiplicative system, 18 natural basis, 14 noetherian module, 10 non–trivial ring, 2 non-zero-divisor, 2 opposite element, 1 over-lining and freeness, 14 38 HANS–BJØRN FOXBY over-lining and projectivity, 34 over-lining functor, 17 over-lining is not exact, 24 over-lining is right-exact, 23 over-lining, 9 pid = principal ideal domain, 3 principal ideal domain, 3 principal ideal, 3 projective but not free, 32 projective dimension and fractions, 36 projective dimension and over-lining, 36 projective dimension over a PID, 35 projective dimension over a field, 35 projective dimension, 35 projective is direct summand in free, 33 projective module, 31 projective resolution, 34 projectivity diagram, 31 proper ideal, 3 regular element, 2 residue map, 8 residue module, 8 residue ring, 8 residue, 8 right-exact functor of left exact sequence, 22 right-exact functor of right exact sequence, 22 right-exact functor, 22 ring of fractions, 18 ring of functions, 12 ring, 2 sequence, 20 set of fractions, 18 short sequence, exact, 20 short sequences, isomorphic, 21 short sequence, 20 spanW , 5 spanning homomorphism, injectivity, 13 spanning homomorphism, surjectivity, 12 spanning homomorphism, 12 special multiplicative system, 18 special short-exact sequence, 20, 21 split-exact sequence, 27–29 splitting sequence, additive functor, 27 splitting sequence, 27–29 submodule rule, 4 submodule, 4 subsets of the ring, 13 sum of cyclic submodules, 5 sum of submodules, 4, 5 surjection onto projective, 32 transcendence, 13 trivial ideals, 3 unitarity, 3 unit, 2 zero element 0, 1 zero homomorphism o, 6 zero ideal, 3 zero module O, 1 AUGUST 11, 2004 zero-divisor, 2, 34 HOMOLOGICAL ALGEBRA August 11, 2004 Matematisk Afdeling, Universitetsparken 5, DK–2100 København Ø, Denmark.

P β M { ϕ / 1P P / O. HOMOLOGICAL ALGEBRA 33 August 11, 2004 Choose β such that the triangle commutes, that is, ϕβ = 1P . 45) is fulfilled, and hence the sequence (†) splits. The last assertion results from (v ) in that theorem. 12) Lemma: Direct sum of projectives 1. If P and Q are R–modules, then the following holds. P ⊕ Q is projective ⇐⇒ P and Q are projective. 13) Proof. Set S = P ⊕ Q and consider the next split-exact sequence: O / ι P / π S / Q / O. Let ϕ : M → N be any R–homomorphism, and consider the induced commutative diagram: O HomR (Q, M ) / HomR (π,M ) HomR (S, M ) / HomR (Q,ϕ) O / HomR (P, M ) / HomR (S,ϕ) HomR (Q, N ) HomR (ι,M ) HomR (π,N ) / HomR (S, N ) HomR (ι,N ) / O / HomR (P,ϕ) HomR (P, N ) / O The rows are exact as they result from application of an additive functor to a split-exact sequence, cf.

48). Proof of “⇒”. Assume that S is projective. By symmetry it suffices to prove that P is projective. Let ϕ : M → N be surjective [ and we want to prove that HomR (P, ϕ) is surjective ]. HomR (S, ϕ) is surjective by assumption, and HomR (ι, N ) is surjective by the exactness of the bottom row in the diagram. The commutativity of the diagram yields the equality HomR (ι, N ) HomR (S, ϕ) = HomR (P, ϕ) HomR (ι, M ) . We already know that the left hand side is surjective. Thus, so is the left factor of the right hand side, and the desired assertion has been established.

### Abstract Algebra (Holden-Day Series in Mathematics) by Andrew O Lindstrum

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