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This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced "k" -algebras. en.wikipedia.org

These groups are finitely generated, but not finitely presented. en.wikipedia.org

This follows from the classification of finitely generated abelian groups. en.wikipedia.org

His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated. en.wikipedia.org

Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. en.wikipedia.org

If this is required only for finitely generated submodules, it is called semihereditary. en.wikipedia.org

For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated. en.wikipedia.org

In the case of finitely generated free groups, the proof is easier, and leads to a more precise result. en.wikipedia.org

One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. en.wikipedia.org

It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). en.wikipedia.org

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