Hey, I'm worried that the video is mathematically misleading! So, let me summarize here once again: T-T paper provides a (relatively) explicit construction of K-theory for every scheme, but we are not explaining that construction, and instead we are telling an implicit way to define it. Also we are saying "algebraic variety" when we actually mean "scheme" (a more general notion), but we did not want to talk here about schemes in more detail. Sorry about that!
But why is K(-) a sheaf and K_i(-) not a sheaf? Intuitively they are all some kind of functions defined locally (that is on affine varieties), so what is the difference?
@@pupfer K_i(-) is the homotopy group of K(-). The local parts of K(-) altogether determine how the various K_i(-) look, but the latter are still global objects.
Welcome to wonderland! CERN ALICE detector, the white rabbit timing ToF. Root OS and trees. Alice strings and alice rings, and boojums. Snark graph theory and color theory Q.C.D., tweedle sets, quantum cats, and the particle zoo. Mad hatter an anagram for mathed art. 😮 The memetic history of wonderland is rich.
Hey, I'm worried that the video is mathematically misleading! So, let me summarize here once again: T-T paper provides a (relatively) explicit construction of K-theory for every scheme, but we are not explaining that construction, and instead we are telling an implicit way to define it. Also we are saying "algebraic variety" when we actually mean "scheme" (a more general notion), but we did not want to talk here about schemes in more detail. Sorry about that!
But why is K(-) a sheaf and K_i(-) not a sheaf? Intuitively they are all some kind of functions defined locally (that is on affine varieties), so what is the difference?
@@pupfer K_i(-) is the homotopy group of K(-). The local parts of K(-) altogether determine how the various K_i(-) look, but the latter are still global objects.
okay, I'm slightly confused.. for a scheme X, how is K_* of X_red related to that of X? I mean.. I see there's a one-way map, but... 🤔
I love k-theory. It’s excellent topic.
Welcome to wonderland! CERN ALICE detector, the white rabbit timing ToF. Root OS and trees. Alice strings and alice rings, and boojums. Snark graph theory and color theory Q.C.D., tweedle sets, quantum cats, and the particle zoo. Mad hatter an anagram for mathed art. 😮 The memetic history of wonderland is rich.