Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.
Incoherence and fibering of many free-by-free groups
A group is called free-by-free if it is the semi-direct product of two finitely generated free groups. A group is coherent if any finitely generated subgroup is finitely presented, and incoherent otherwise. In this paper, the authors provide evidence towards the conjecture (due independently to the authors and Dani Wise) that every free-by-free group is incoherent. To do this, they give a homological condition which lets them conclude that the free-by-free group has a finite index subgroup which surjects onto ℤ with finitely generated kernel; standard arguments imply that this kernel cannot be finitely presented. As an important special case, they show that if the free-by-free group is hyperbolic and virtually special, then it is incoherent.
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- Award ID(s):
- 2005353
- PAR ID:
- 10554957
- Publisher / Repository:
- Annales de l'Institut Fourier
- Date Published:
- Journal Name:
- Annales de l'Institut Fourier
- Volume:
- 72
- Issue:
- 6
- ISSN:
- 1777-5310
- Page Range / eLocation ID:
- 2385 to 2397
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.5802/aif.3494
