## Example of a morphism between exterior algebras that is $\mathbb{Z}_2$ graded but not $\mathbb{Z}$ graded??

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The title pretty much states my problem. I consider only finitely generated exterior algebras $\bigwedge V$. It is known that any morphism between exterior algebras y determined by its action on generators, i.e. its action on $V$. Does anyone know a good example of this kind of morphisms?

By $\mathbb{Z}_2$ graded I mean a morphism of algebras such that the parity of the degree of a form $\eta$ is preserved by such a morphism; and by $\mathbb{Z}$ graded I mean a morphism that preserves the grade of $\eta$, i.e. if $\eta$ is a $k$-form, then so is $f(\eta)$ where $f$ is the morphism in question. Thanks in advance.

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Map the elements of a basis of $V$ to elements in the highest non-zero odd exterior power of $V$. – Mariano Suárez-Alvarez Nov 22 2011 at 2:53

## 1 Answer

There doesn't seem to be much more to say, so I'll just repeat Mariano's comment with an example. Let $V$ be a 3-dimensional vector space in degree 1, and consider any nonzero linear map from $V$ to $\wedge^3 V$. This induces an algebra endomorphism of $\bigwedge^\bullet V$ that preserves the $\mathbb{Z}/2\mathbb{Z}$-grading but not the $\mathbb{Z}$-grading.

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