Theorem 0.3. Retain the hypotheses of Theorem 0.2. with Ro a normal domain.
u _ h(v) + K _ (ui)+7_i (columns 2 to / of X) + K _ //_i (rows 1 to f-l of X) + K
D3 - ^ , p
- x
2 ~ K
_ IS-i{X) + K , _ (ufl)+//_1(rows 1 to f-l ofX) + K
X2 ^ , ana pi j?
represent various ideals of R/K.
(a) If f g, then b$, 02, and p2 all are height one prime ideals of R/K. Further-
more, R/K = Ro © Z, where the summand Z is generated by the class [63]
and the equations
[b3] = [a2] = -[p2]
hold in R/K.
(6) If f g, then bs, p2, t2, and PL all are height one prime ideals of R/K.
Furthermore, R/K = Ro © I* B Z where one summand Z is generated by the
class [63], £/ie o£/ier summand Z zs generated by [12], arcd ^ e equations
N = -[P2] ™d N - -[pi] - [P2]
hold in Ci R/K.
(c) If UJR0 is the canonical module of Ro, then the class of the canonical module
of R/K in C£R/K is [coRoR/K] + ( # - / ) [ b
] .
(d) If P is a prime ideal of R, then [R/K)p is a regular local ring if and only if
(Ro)R0np is a regular local ring and Ij_i(X) -f Ii(u)I\(v) £ P.
Section 1 is devoted to collecting the relevant facts; especially from the theory of
multilinear algebra. In 2, we define
prove that it is a complex, give examples,
and establish the duality between fl(2) and 1^-/-^) \n 3^
w e
identify the zeroth
homology of the complex 1 ^ ; we establish homomorphisms from Ho(I^) to ideals
of Ho(I^) = R/K (these homomorphisms are shown to be isomorphisms in section
8); and we record the short exact sequence of complexes (0.1) for the I^ZK In 4, we
split off a split exact summand of 1 ^ in order to produce the complex M^z\ which
is minimal whenever the data is local or homogeneous of positive degree. This
section concludes with a list of examples. The modules M(p, q, r), which comprise
the complex M^z\ are defined and shown to be free in 5. Section 6 is a calculation
about binomial coefficients which is used to find the rank of M(p, ?, r). In 7, we
prove the results which are stated in section 4; thereby completing the proof that
M ^ is homologically equivalent to l^zK In sections 8 and 9 we prove that the
complex I^) is acyclic. The proof is by induction on g and uses the short exact
sequence (0.1). The inductive step is in 8 and the base case, g f 1, is in 9.
1. Preliminary results.
In this paper "ring" means commutative noetherian ring with one. The grade
of a proper ideal J in a ring R is the length of the longest regular sequence on
R in I. An R—module M is called perfect if the grade of the annihilator of M is
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