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Noelle Wright 2022-05-14 Answered
Let's fix the notation, V = i 0 V i is a graded vector space and Λ V is the free commutative graded algebra on V. I have been struggling to understand this example:
Consider a graded vector space V with basis { a , b } such that a V 2 and b V 5 . Now define a linear map d (of degree 1) by d a = 0 and d b = a 3 . It follows that d extends uniquely to a derivation d : Λ  V Λ  V.
The point of the example is to show that the derivation on Λ V is completely determined by its values on V. So if i understand well, he considers a linear map d : V Λ V of degree one defined by
(here Λ k V is the set of elements of word length k) and
d 5 : V 5 Λ 6 V ; b a 3
The first question that i'm stuck on is for d 2 ( b ) = a 3 , i mean a 3 is of length 3, how it can be in Λ 6 V.
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Answers (1)

percolarse2rzd
Answered 2022-05-15 Author has 17 answers
Λ k V is not necessarily the set of elements of word length k. Rather than just word length, you need to consider the grading as well. Because a lies in Λ 2 V, a 3 lies in Λ 6 V ..
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