(CO)HOMOLOGY AND CRITICAL POINT THEORY - Math - IPM

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450 CHAPTER 3. (CO)HOMOLOGY AND CRITICAL POINT THEORY Exercise 3.5.7 Let R = Z[Z/m] be the integral group algebra of the cyclic group Z/m (i.e., all polynomials m−1 i=0 aiT i where multiplication is formal subject to the relation T m = 1 and ai ∈ Z). Regard Z as an R-modules by defining T.a = a for all a ∈ Z. Let Fj = R, f2j be multiplication by 1 + T + · · · + T m−1 for j ≥ 1, and f2j+1 be multiplication by T − 1. Show that the sequence · · · −→ Fj fj −→ Fj−1 −→ · · · −→ F◦Z −→ 0, where f−1 : R → Z is given by f−1( aiT i ) = ai, is a free resolution of Z as an R-module. Let C be an R-module and C1 = (T − 1)(C), C2 = (1 + T + · · · + T m−1 )(C). Show that Ext ◦ R(Z, C) = {c ∈ C|T c = c} and Ext j {c ∈ C|T c = c}/C2 if j is even and positive R (Z, C) = {c ∈ C|(1 + T + · · · + T m−1 )c = 0}/C1 if j is odd Finally in this subsection we consider the issue of independence of extorsion from the choice of free resolution. The argument is simple and instructive. An important property of free modules (motivated by the splitting homomorphism) is that if F is a free R-module and C → C ′ → 0 is an exact sequence of R-modules, then for every homomorphism F → C ′ there is a homomorphism F → C such that the diagram F ↙ ↓ C −→ C ′ −→ 0 commutes. That free modules have this property is trivial. This property is the defining property of projective modules, but for our purposes, free modules will suffice. We refer to this property as the universal mapping property of projective modules. Now assume we have two free resolutions · · · · · · fn+1 −→ Fn gn+1 −→ Gn fn −→ · · · −→ F1 −→ F◦ −→ E −→ 0 ↓ βn ↓ β1 ↓ β◦ ↓ id. gn −→ · · · −→ G1 −→ G◦ −→ E −→ 0 where the maps βj remain to be specified such that the diagram commutes. The fact that β◦ exists follows from the universal mapping property of projective modules. Now from exactness and existence of β◦ it follows that Im(β◦f1) ⊂ Im(g1). Therefore the universal mapping property is applicable to give the existence of β1. Proceeding inductively in the obvious manner we obtain βj’s such the above diagram commutes. Reversing the roles of the top and bottom free resolutions we similarly obtain γj : Gj → Fj such the corresponding diagram commutes diagram. Let Φj = γjβj to obtain the commutative row exact diagram · · · · · · fn+1 −→ Fn fn −→ · · · −→ F1 −→ F◦ −→ E −→ 0 ↙ ↓ Φn ↙ ↙ ↓ Φ1 ↙ ↓ Φ◦ ↓ id. fn+1 −→ Fn fn −→ · · · −→ F1 −→ F◦ −→ E −→ 0 (3.5.10) where the maps sk : Fk → Fk+1 (chain homotopy) will be now determined. Here sk’s are constructed to enable us to compare the endomorphisms of Tork(E, C)’s induced by Φk’s and the identity map. Clearly
3.5. SOME ALGEBRAIC CONSIDERATIONS 451 Φ◦(x) − x ∈ ker f◦ = Imf1, and therefore by the universal mapping property there is s◦ such that Φ◦(x) − x = s◦f1(x). (3.5.11) From commutatvity of the squares, it follows that Φ1(x) − x − s◦f1(x) ∈ ker f1 for x ∈ F1. Therefore, by the universal mapping property, there is s1 : F1 → F2 such that Φ1(x) − x = s◦f1(x) + f2s1(x). Proceeding inductively in the obvious manner we obtain sj such that Φj(x) − x = sj−1fj(x) + fj+1sj(x) for j ≥ 1 (3.5.12) and (3.5.11) for j = 0. The relations (3.5.11) and (3.5.12) remain valid after tensoring with C and where the maps Φj, fj etc. are replaced with Φj ⊗ id., fj etc. It follows from (3.5.11) and (3.5.12) that the endomorphisms induced by Φj’s on the Torj(E, C)’s are the identity maps. Since Φj = γjβj we immediately obtain independence of Torj(E, C)’s from the choice of free resolution. Similar argument applies to the Ext j (E, C), and we can simply say that extorsion is independent of the choice of free resolution. Therefore we have shown the first statement of Theorem 3.5.1 Extorsion is independent of the choice of free resolution. Given a short exact sequence 0 → C ′ → C → C ′′ → 0 of R-modules, there are long exact sequences · · · → Torn(E, C ′ ) → Torn(E, C) → Torn(E, C ′′ ) → Torn−1(E, C ′ ) → · · · → Tor◦(E, C ′′ ) → 0 0 → Ext ◦ (E, C ′′ ) → · · · → Ext n−1 (E, C ′ ) → Ext n (E, C ′′ ) → Ext n (E, C) → Ext n (E, C ′ ) → · · · (The homomorphisms δn : Torn(E, C ′′ ) → Torn−1(E, C ′ ) and δ n : Ext n (E, C ′ ) → Ext n+1 (E, C ′′ ) are called connecting homomorphisms.) Proof - It remains to prove the the second assertion. Tensoring the free resolution of E (3.5.1) with the short exact sequence we obtain the following row exact commutative diagram ↓ ↓ ↓ 0 −→ Fn ⊗ C ′ −→ Fn ⊗ C −→ Fn ⊗ C ′′ −→ 0 ↓ ↓ ↓ 0 −→ Fn−1 ⊗ C ′ −→ Fn−1 ⊗ C −→ Fn−1 ⊗ C ′′ −→ 0 ↓ ↓ ↓ This is exactly the same situation as where we constructed the connecting homomorphism for homology. The same argument is applicable to both Tor and Ext modules. ♣ Exercise 3.5.8 Assume the following diagram of free R-modules is row exact and commututative: ↓ ↓ ↓ 0 −→ Zn −→ Cn −→ Bn−1 −→ 0 ↓ ↓ ↓ 0 −→ Zn−1 −→ Cn−1 −→ Bn−2 −→ 0 ↓ ↓ ↓
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