# Download e-book for kindle: Abelian Groups, Rings and Modules: Agram 2000 Conference by Andrei V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz,

G1 @Po ~ Go --+ O, 26 1. THE CLASSICAL GORENSTEIN DIMENSION which we now 2 show is exact. • T h e m a p H,~ -~ Kn @ P,,-1 is given by h ~ (Tn(h), - z r n ( h ) ) , and this m a p is injective because 7rn is so. This proves exactness in Hn. • T h e m a p K~ (9 Pn-1 --+ G n - 1 (9 Pn--2 is given by (k,p) , ) Or~(k) +%~-l(p),-Trn-l(p)), so an element ('Tn(h),-~rn(h)) is m a p p e d to (zc~')'n(h)- ')'n-lrn(h), 7rn-l~rn(h)) = (0,0).

The last assertion is immediate by (a), (b), and (c). 10)(a). We now assume that G - d i m R M " _< n and n E N. Let . - . - - + P ~ P ~ _ I - ~ ' ' " ~ P g ~ 0 and 28 1. T H E CLASSICAL GORENSTEIN DIMENSION "'" --4 P~' -+ P~-I -'4 "'" --~ P0' --4 0 be resolutions by finite projective modules of, respectively, M ' and M " , then we have a commutative diagram 0 > 0 0 0 T T T M > M" M' > T 0 0 > Pg T > Pg@Pg' T > Pg' T T T T T T T T T ~ K~ > > 0 K. > K~ } } T 0 0 0 > 0 >0 with exact rows and columns.

But this is easy: if X • P(f)(R), then X is equivalent to a complex P • C~(R), cf. 2), and RHomn(X, R) is represented by Homn(P, R), which is a bounded complex of finite projective modules, so RHomn(X, R) • P(f)(R). 8). 11) Observation. Let p • SpecR and X • c((f~(R). Let I • C~(R) and L L • C~(R) be resolutions of, respectively, R and X, then Ip • C~(Rp) and Lp • C~(Rp) are resolutions of Rp and Xp. The commutative diagram Homn(Homn(L,I),I)p ~- > Homn,(Homn,(Lp,Ip),Ip) Lp Lp shows that 6~p is a quasi-isomorphism if and only if (6/)p is so.