
Operadic Homological Algebra and Gröbner Bases 207
• Suppose that P is generated by operations of the same arity N ≥ 2 and of
homological degree zero. In that case H
P
(t) = tf(t
N−1
) for some power
series f; if P is Koszul, then
H
Σ
P
¡
(
g
H
Σ
P
(t)) = t,
where the “sign modified series”
g
H
Σ
P
(t) is defined by the formula
g
H
Σ
P
(t) = tf(−t
N−1
).
Proof. The first part follows from the definition of the Koszul complex, Propo-
sition 6.3.4.2, and the observation that the homological degree in the Koszul
complex comes from the weight grading. The second part follows from the
first part after letting x = 1 and noticing that in case all the generators
are of the same arity N