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arXiv · 1508.04732

Cable algebras and rings of $G_a$-invariants

Gene Freudenburg·Shigeru Kuroda
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Abstract

For a field $k$, the ring of invariants of an action of the unipotent $k$-group $G_a$ on an affine $k$-variety is quasi-affine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the $G_a$-action on $A^5_k$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known $G_a$-action on $A^7_k$ due to Roberts is a cable algebra.

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Gene Freudenburg, Shigeru Kuroda. 2015-08-19. Cable algebras and rings of $G_a$-invariants. https://doi.org/10.1215/21562261-3821828

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