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Algebraic Operads
book

Algebraic Operads

by Murray R. Bremner, Vladimir Dotsenko
April 2016
Intermediate to advanced content levelIntermediate to advanced
383 pages
11h 46m
English
Chapman and Hall/CRC
Content preview from Algebraic Operads
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
N1
) 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
N1
).
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
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Publisher Resources

ISBN: 9781482248579