By Samuel Eilenberg, J. C. Moore
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Extra resources for Memoirs of the American Mathematical Society 1965 No. 55 Foundations of Relative Homological Algebra
Every left ideal I of the algebra A can be generated p, s0+slb , with P,S0,Sle~(a), and generates the ideal where p is a symmetric the corresponding Since ~(a) Consider element is a principal a fixed element of I. Here element IA~(a). P r o o f . E v e r y e l e m e n t o f A c a n be e x p r e s s e d Consider all the elements x=to+tlb of the left over I, by X]eL], element ideal t 1 ranges ring, x0=s0+slbel in the form f+gb; f,ge~(a). i d e a l I . Then x r a n g e s o v e r an i d e a l Sle~(a).
H(a) +r(a) , = 0 or not uniquely I r(a) I, but the elements h(a) and r(a) are determined. We set ~ = f(a-1). -- va-mf(a) the elements [g(a) I > vam The element for some meT7 and v ~ . f(a)elR(a) is called symmetric The units of ~(a) if are exactly (~,IR ; me2Z). LEMMA 1 . 1 . h(a)+rCa) , = 0 or Ir(a)[ Pr_oo_f_f. h(a)[. +a0, g(a) a 0 ~ O, ~0 ~ 0. 1) < deg g(a). V0÷X0 . hl(a) ~0 ~ 0 . Consider the I. element ~0 = h 1 Ca) $o Then the e q u a l i t y f(a) holds. h(a) clear that x0 ~oo g ( a ) ] + [r](a) the element x0 r(a) = r l(a) - ~0 g(a) has no constant term, so Ir(a) l < deg r(a).
Fl, where ct a n d 13 are square matrices o f h o m o g e n e o u s p o l y n o m i a l s , correspond bijectively t o i s o m o r p h i s m classes o f g r a d e d M C M - m o d u l e s over R . Under this correspondence one associates with f = a/~ the m o d u l e M = Coker¢, where ¢: R m ~ R m is t h e m a p corresponding to a (with respect t o t h e canonical basis of R m ) . Moreover, if ¢: R m --* R m is t h e m a p corresponding to fl, then t h e periodic sequence ... ¢~ R m ¢~ R m ~ Rm --~ M ,0 is exact.
Memoirs of the American Mathematical Society 1965 No. 55 Foundations of Relative Homological Algebra by Samuel Eilenberg, J. C. Moore