Lie groups: Positive characteristic is weird

[@lukasjuhrich503]

„Positive characteristic is weird“ sums up every encounter with positive characteristic I've had so far.

[@vladkovalchuk8299]

In what sense are projective varieties "almost opposite" of affine varieties? (6:02)

[@AsvinGothandaraman]

Lots of reasons, here's one: it's really easy to map to projective space and pretty hard to map put of it. Conversely, it's really easy to map out of affine space and really hard to map to it. For instance, all maps from projective varieties to affine varieties are constant

[@AsvinGothandaraman]

Here's another related reason: affine things are really local, for instance all higher sheaf cohomology vanishes. Projective things are very global - higher sheaf cohomology is almost always very non trivial.

[@vladkovalchuk8299]

@@AsvinGothandaraman Thanks for your answer! Do you know of any more general result exemplifying your intuition? E.g. something along the lines(within an appropriate category) of "of all abstract varieties, those that have most morphisms out of them are affine, and dually, those that have most morphisms into them are projective"? Algebraic geometry has never been my strong suit, feel free to ignore the question if it is greatly uninformed.

[@AsvinGothandaraman]

@@vladkovalchuk8299 Well, I am not sure I can pin down something so exactly but it's not hard to classify maps to either affine space or projective space. In the first case, it's given by find a map from the corresponding ring which is just picking some number of global sections. In the second case, you are allowed the freedom of picking a lime bundle and then some global sections. So clearly there is more freedom in the projective case.

[@AsvinGothandaraman]

Here's another sense in which affine spaces are small: under niceness assumptions, the complement of any open affine subvariety is always codimension one (that is, pretty large!).

[@anthonymurphy5689]

18:50 - small typo I think - RHS should have powers m+1 and n+1 in the calculation of [Lm,Ln]. following line should have m+n+1 in the power of x. Final result is however correct.

[@samhughes8017]

If you were willing to do it I would love to see a lecture series on algebraic groups

[@migarsormrapophis2755]

yeeee

[@fischergriess6321]

Yeeeee

[@atomiccompiler9495]

yEEEEE

[@ZetaCarinae]

The best kind of click-bait.