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2018-04-28 17:17:41 -0500 asked a question How do I define a homomorphism of a graded commutative algebra?

Why does the following throw a TypeError: images do not define a valid homomorphism?

E = GradedCommutativeAlgebra(QQ,'x,y',degrees=(1,1))
f = E.hom([x,y])

I expected it to define $f$ to be the identity homomorphism of $E$. What is the right way to define a homomorphism of $E$? I'm more interested in the one that switches $x$ and $y$ than the identity homomorphism, but this seemed a more obvious version of the question.

2017-06-22 15:20:38 -0500 asked a question How do I find the image of an element of a differential algebra in the cohomlogy?

Following the documentation on Commutative Differential Graded Algebras, I have defined a differential graded algebra $C$. I have some element $x \in C$, in degree $4$. I can get a basis for the cohomology at degree 4 by


and generators for cocycles and coboundaries by


How do I check if $x$ is a cocycle, and if it is, what it is in terms of the basis of the cohomology above?

I'm not sure I used the right tags, feel free to edit.