2 ANDREW R. KUSTIN (c) If —1 2, then C^ resolves a representative of the class zfcokerp] from ce R/K. (d) The canonical class in the Ci R/K is equal to m[cokerp]. (e) Cz * (Cm"z))* [s]. (f) If M is a reflexive R/K-module of rank one and [M] = z[cokcr p] in C£ R/K for some integer 2, then M is a Cohen-Macaulay module if and only if— 1 2 m -f 1. (g) If p — [ P X), where X is the submatrix of X which consists of columns 1 to / — 1, then, for each integer 2, there is a short exact sequence of complexes 0 - C{z\p) - &z) - C{z-l\p)[-l] — 0. Indeed, if / is a grade two perfect ideal, then n = g — 1, P is the g x g — 1 matrix of indeterminates whose 7 — 1 x g — 1 minors generate /, p is the # x {f + g — 1) matrix of indeterminates [PX], /^ is generated by the g x p minors of p (see [11, Thm. 4.1] or [12, pg. 4]), and C^ is the Eagon-Northcott type complex D i G ^ A 2 " ^ 1 E - D0G*®/\2*S E — 50GOA2 £ - SiGfcA*""1 # - . . . , with 5ZG 0 A E in position 0 see, for example, [6, Sect. 2C]. If / is a grade three Gorenstein ideal, then n — g, P: G* = Rn — • Rg — G is the # x # alternating matrix of indeterminates whose g — 1 order pfafnans generate J, E = G* 0 F, /" is generated by the pfaffians of all principal sub matrices of ( _x%. J which contain P, and C^ is the complex •.. - (siGoA™"*"1^* - (s0G®Am_2£)* - Qz S0G®/\ZE^ SiG^h^E- with SZG S /\ E in position 0, where 9 (^cx A * F - s.G®A*s3 ^ • ° ^ A -& ~ (SnGSbfii'G*. (S 0 G®A ^*^)' proj and r/ is the element of G®E which corresponds to E* = G 0 F * G under the natural identification of Hom(E*, G) and G&E. See [17]. If I is a grade # complete intersection, then n = (2), a is a 1 xg matrix of indeterminates, P: f\ R9 — * R9 is the Koszul complex map, K is equal to Ji(aX) + Ig(X), the complex C^0) is given in [5], and the entire family {C^} is given in [13]. There is a second starting point which produces an analogous family of com- plexes. In this case, there is no ideal /, there is no presentation map P of /, and there is no interpretation in terms of residual intersection. The best examples of this second starting point come from the theory of varieties of complexes. Start with the data 0-R^F-^G^R,

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