
Operadic Homological Algebra and Gröbner Bases 217
= T − r
g
(T ) + (r
g
(T ) − r
g
(T )) = T − r
g
(T ) = T − T ,
since for a Gröbner basis the residue does not depend on a choice of reductions.
Assume that k > 0, and that our statement is true for all l < k. We have
D
k
D
k+1
(m) = 0 since
D
k
D
k+1
(m) = D
k
(d
k+1
(m) −H
k−1
D
k
d
k+1
(m)) =
= D
k
d
k+1
(m) −D
k
H
k−1
D
k
d
k+1
(m) = D
k
d
k+1
(m) −D
k
d
k+1
(m) = 0,
because D
k
d
k+1
k ∈ ker D
k−1
, and so D
k
H
k−1
(D
k
(y)) = D
k
(y) by induction.
Also, for u ∈ ker D
k
we have
D
k+1
H
k
(u) = D
k+1
h
k
(lt(u)) + D
k+1
H
k
(u −D
k+1
h
k
(lt(u))),
and by the induction on lt(u) we may assume that
D
k+1
H
k
(u −D
k+1
h
k
(lt(u))) = u − D
k+1
h
k
(lt(u))
(on elements of positive syzygy degree, ...