Betti numbers of modules over Noetherian rings with ... - IPM

Chapter 1: Preliminaries 11
of S/I using the differentials:
d(Ti1...ip) =
p
(−1)
j=1
j−1 mi1...ip
m i1...îj...ip
where mi1...ip denotes the least common multiple of the monomials mi1, . . . , mip andˆ
denotes omission. This resolution is usually far from being minimal. An obstruction
to minimality occurs every time mi1...ip = m . Let J be the indexed set of the
i1...îj...ip
minimal monomial generating set of I. If K is any subset of J we use the notation
mk to denote the least common multiple of the monomials indexed by K. Of course
when K ′ ⊆ K, mK ′ divides mK. It is also known that the Taylor resolution is minimal
if and only if mK = mK ′ for all subsets K′ ⊆ K; see [26, pp. 439] and [18] for more
details.
Example 1.3.2. Let R = K[x, y] and I = (x 2 , xy, y 2 ). The Koszul complex of R/I
is as follows:
⎛
⎜
⎝
x 2
−xy
⎞
⎟
⎠
⎛
⎜
⎝
0 y 2 xy
y 2 0 −x 2
y
0 −→ R(−6)
2
−−−−−−−→ R(−4) 3
−xy −x2 0
−−−−−−−−−−−−−−−−→ R(−2) 3
x2 xy y2 −−−−−−−−−−−→ R −→ 0,
and the Taylor resolution of R/I:
where
0 −→ R(−4) f2
→ R(−3) 2 ⊕ R(−4) f1 3 f0
→ R(−2) → R → R/I → 0,
⎛
⎞
⎛
⎜ −y ⎟ ⎜ y
⎜ ⎟ ⎜
⎜ ⎟ ⎜
f2 = ⎜
1 ⎟ , f1 = ⎜
⎟ ⎜
⎝ ⎠ ⎝
y
−x
2 −x 0
0
y
0 −x2 ⎟
⎟ , f0 =
⎟
⎠
−x
⎞
⎞
⎟
⎠
,
x 2 xy y 2
.