By Julian Lowell Coolidge

An intensive creation to the speculation of algebraic airplane curves and their kinfolk to varied fields of geometry and research. nearly solely constrained to the homes of the final curve, and mainly employs algebraic strategy. Geometric equipment are a lot hired, even though, specifically these regarding the projective geometry of hyperspace. 1931 version. 17 illustrations.

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G. g. projectives. However, the opposite is true, as Bass [63] so skillfully pointed out. 62: Suppose there is some infinite cardinal X for which the projective module M is generated by X elements and contains a summand isomorphic to R“’. Then M is free. 4‘)) Thus to prove M is free we need merely find a “big” free summand R‘”. The first step, of course, is to find when R is isomorphic to a summand of M; since R is projective it suffices to find an epic f :M -+ R. This implies the trace ideal T ( M )= R, and we have the following converse.

By induction on m we now see that if P 0 R(m)z R(")then P is free. D. 49 (*) that local commutative rings have the unimodular vector property, since all projectives are free. 50: (Horrocks) Suppose C is a local commutative ring, and u = (fi, . If some 1;. is monic then u is strongly unimodular. Proof: Reordering the 1;. we may assume f l is monic. Suppose m = deg(f i ). We can use f l to cancel higher degree terms in the other entries; since this involves elementary transformations and thus affects neither unimodularity nor strong unimodularity, we may assume deg(f,) < m for all i > 1.

Since GF is naturally equivalent to 1 it suffices to show P z FGP (graded) for every P in Yr-fiwj(R). 6. Hence there is a graded monic f: Q 4 P. Now FGP = R ORoQ so we can define a graded map cp: FGP + P such that cp(r 0 x) = r f x for r in R and x in Q. 34’. But then cp is epic and thus split, so G(ker cp) = 0, likewise implying ker cp = 0. Thus cp is an isomorphism, proving claim 1. The next part of the proof of the theorem involves two clever tricks, the first of which is very cheap. Grade R[L] by putting R[IL], = RiL“-i.