By Benjamin Fine

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic learn of Fuchsian teams to the extra normal context of one-relator items and comparable team theoretical issues. It presents a self-contained account of definite traditional generalizations of discrete teams.

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E. there is an m:2: 0 with a"'s(a") O. Assume without lOBS of generality that m n. Then alts(c) = ca(a") 0 for all c E am. Hence we have for all c E am: 8 for all:p E Ass(AjAnn 8(cl) ~ Ass M. a E :p n S This implies Ass(AIAnn 8(cl) ~ V(a), Le. there is a q 0 with aIl8(c) = O. Since a'" is finitely generated we can find an r 0 with a'8(c) = 0 for all c E am. e. I is injective. We now prove (vi). We see that the maps HO(M) ...... p: HO(M) ...... p, choose an 8 E Hom(a", M) representing 8(alt) (J.

Some foundations of commutative and homological algebra Assume first that H~Pli) =0 for i 1,2. 8 we have for each t'? (q~, M 2 )} -+ Homn(ql, a(Mb Mi»)' Therefore in this case U O is an isomorphism. l~,(M;) for z· Let now lVI' M2 be arbitrary modules. l~,(Mi) 1,2. l°(Mi) for i 1,2. Let;' denote the natural projection a(Mb M 2 ) commutative diagram: -'>- a(M~, M~). 1°(a(M}, M 2») V tirO) a(HO(M~), HO(M~») ~ HO(a(M~, ~V~») We also have an exact sequence (see Cartan-Eilenberg [1], Chap. IV, Prop. e.

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). Therefore we define the project~'ve and the znjective dimension of M: pdR M := inf{n E N I there is a projective resolution o inj dimR M -)0- pn -)0- .