By F. Albert Cotton

Keeps the easy-to-read structure and casual style of the former variations, and comprises new fabric at the symmetric homes of prolonged arrays (crystals), projection operators, LCAO molecular orbitals, and electron counting ideas. additionally includes many new workouts and illustrations.

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This exact sequence is called a free re8olution of M. "Ve note that we work in the category of graded R-modules with degree zero homomorphisms as Illorphisms. This is a complete (and cocomplete) Grothendieck category, (d. Schubert [1], Def. ). e. in the category of k-vector spaces. This category possesses enough projectives (as we have seen before) and therefore enough injectives. This fact we will use in Section 4 of this paragraph. For any graded R-module M we have projective and injective resolutions (even free resolutions).

1 This o. 32 Some foundations of commutative and homological algebra (v) Assume g is injective. For Ass M ~ V(a) let lJ E Ass M" V(a). Then there is a monomorphism A/lJ ...... M giving rise to the following commutative diagram Alp -4 HO(AllJ) 4 (Alp)s t t HO(M) 4 Ms where all homomorphisms are injective (HO( ) is left exact and H~(Alp) 0 since lJ ~V(a». lJ n 8 = O. Assume now :p n S = 0 for all :p E Ass M" V(a). Let a E Ker g and choose 8(ult ) . E Hom(alt ; M) representmg a. e. there is abE S with bs(a") = O.

Let M be a Noetherian graded R-module where R is a graded k-algebra. ) (note that for all n E Z[MJ. is a k-vector space of finite rank). HM is called the Habert junction of M. There is a numerical polynomial hM (E Q[TJ, T an indeterminate) such that HM(n) = hM(n) for all sufficiently large n. The polynomial hM is called the 3* 36 O. Some foundations of commutative and homological algebra Hilbert polynomial of ill. M) (~') + h (M) (a 1 '1' 1) + ... + hdUI1) with d := deg hM (considered as a polynomial in '1'), and where the integers ho(~~f) > 0, hl(JI), ...

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