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I added in the definition of algebraic group the requirement ”field” into ”algebraically closed field”. Alternatively one could omit ”field” in the definition at all since this is implicit in ”variety”.
It is the usual terminology about variety, but I am not sure that it is entirely universal when in very general contexts. Sometimes, following Mumford, one considers “varieties” over somewhat more general schemes (including over $Spec k$, where $k$ is not a field. In that case, it is required that both $S$ and $S$-variety as a $\mathbf{Z}$-scheme are reduced irreducible and separated. There is also a term “infinite-dimensional variety” for subvarieties in this sense of infinite-dimensional projective space.
While I still agree that the standard notion of algebraic group is for varieties (in the old fashioned sense) over a field, I would disagree with algebraic closedness. For example, the algebraic groups over finite fields are a pretty standard object, predating scheme theory. See
I made some minor changes to algebraic group.
In this case we could -as I said above- omit the word ”field” in the definition and take ”k-variety” as an atom since the notion in discussion is not the ”group”-part of the definition, but the ”k-variety”-part. In particular the definition of algebraic variety which redirects variety says that the field is algebraically closed, but I see that the prefix ”algebraic” shall indicate the algebraically closedness.
At least I think of a ”variety” $X$ (without prefix) as the space of maximal ideals in a polynomial ring over an (arbitrary) field. But the nlab indicated otherwise - mostly since the entry variety was lacking.
So, I made a short note variety.
I think I am OK with the idea section of algebraic group.
I am not familiar with the prefixing “algebraic” for variety to denote algebraic closedness, it is rather just to make it unambiguous not to mean analytic variety, or even rigid analytic variety. I think that, apart from that, variety and algebraic variety are synonyms, unless one allows also analytic varieties; the algebraic closedness is just a matter of the usual convention, as otherwise the geometry becomes highly irregular: one fails Nullstellensatz to start with. I should mention it was me who wrote the condition of algebraic closedness in the entry algebraic variety; though as I say most but not all literature restricts this way.
There is of course Mumford’s notion of variety. It is partly based on the fact that the category of varieties embeds into the category of reduced schemes. But this embedding is of course not identical on objects as topological spaces: for any variety the corresponding scheme has more points (as it has non-closed points).
Finally, entry variety might, as it is often referred as variety with links from the $n$Lab) be eventually just a disambiguation entry: namely in addition to analytic variety there is a notion of variety of algebras in universal algebra).
I made variety into a disambiguation entry. Entry algebraic variety has a discussion about the relation of traditional algebraic varieties over an algebraically closed field to the category of integral schemes of finite type.
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