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Results tagged with ag.algebraic-geometry
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user 12858
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
2
votes
Accepted
Let G be an affine connected algebraic group. When a subvariety of G with codimension one...
It is perhaps easiest to express this in terms of Hopf algebras. The coordinate ring $K[G]$ has the structure of a Hopf algebra; the subvariety $Y$ is a closed subgroup of $G$ if and only if the ideal …
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1
vote
affine and projective schemes over $\mathbf{F}_1$?
Your question sounds rather vague, but in any case, you might be interested in the paper "On the notion of geometry over $\mathbb{F}_1$" by Alain Connes and Caterina Consani, available on the arXiv (o …
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6
votes
Synthetic projective lines
Following up on Matthias Wendt's comment, the language of Moufang sets is indeed a suitable axiomatic approach to (generalizations of) projective lines.
Formally speaking, a Moufang set is a set $X$ …
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3
votes
Moufang identities and Moufang plane
A very accessible book for such connections between geometric and algebraic properties in general, is John Faulkner's "The Role of Nonassociative Algebra in Projective Geometry" (https://bookstore.ams …
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15
votes
3
answers
4k
views
Connectedness of the linear algebraic group SO_n
I apologize in advance if my question is too elementary for MO.
It is a well known fact that the linear algebraic group $G = \mathsf{SO}_n$ is connected, and there exist a few different proofs of thi …
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3
votes
Accepted
Maximal split torus of universal chevalley group
By Lemma 28(b), $H$ is an abelian group generated by the $h_i(t)$'s (where $h_i = h_{\alpha_i}$), and since each $h_i$ is multiplicative (by Lemma 28(a)), the existence follows.
To prove uniqueness, …
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2
votes
Algebraic Groups, Modules, and Comodules
I think that Waterhouse's "Introduction to Affine Group Schemes" (1979), section 3.2 "Comodules", might be what you're looking for.
I'm not sure why you have the constraint "non-zero characteristic" …
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9
votes
Spherical building of an exceptional group of Lie type
In the case of groups of rank 2, such as your examples $\mathrm{SL}_3(\mathbb{F}_2)$ or $\mathsf{G}_2(3)$, the building is rather easy to describe (either as an incidence geometry or as a bipartite gr …
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6
votes
Accepted
Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.
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