(c) If —1 2, then C^ resolves a representative of the class zfcokerp] from
(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
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 -
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* =

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
•.. -
- Qz
with SZG S /\ E in position 0, where
9 (^cx A * F - s.G®A*s3
^ ° ^ A -& ~ (SnGSbfii'G*.
G®A ^*^)'
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\
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
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