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/Kmodule of rank one and [M] = z[cokcr p] in
C£ R/K for some integer 2, then M is a CohenMacaulay 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{zl\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 EagonNorthcott 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
0R^F^G^R,