on February 5, 2004 by lieven in geometry, Comments (0)
connected component coalgebra
non-commutative geometry
- Brauer’s forgotten group
- connected component coalgebra
- Galois and the Brauer group
- a noncommutative Grothendieck topology
- noncommutative geometry
- noncommutative geometry 2
- projects in noncommutative geometry
- points and lines
- more noncommutative manifolds
- the necklace Lie bialgebra
- the one quiver for GL(2,Z)
- representation spaces
- quiver representations
- moduli spaces
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- A for aggregates
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- why nag? (2)
- why nag? (3)
- sexing up curves
- the Klein stack
- Alain Connes on everything
- noncommutative topology (1)
- a noncommutative topology 2
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Never thought that I would ever consider Galois descent of semigroup
coalgebras but in preparing for my talks for the master-class it
came about naturally. Let A be a formally smooth algebra
(sometimes called a quasi-free algebra, I prefer the terminology
noncommutative curve) over an arbitrary base-field k. What, if
anything, can be said about the connected components of the affine
k-schemes rep(n,A) of n-dimensional representations
of A? If k is algebraically closed, then one can put a
commutative semigroup structure on the connected components induced by
the sum map
rep(n,A) x rep(m,A) -> rep(n + m,A) : (M,N) -> M + Nas introduced and studied by Kent Morrison a long while ago. So what would be a natural substitute for this if k is arbitrary? Well, define pi(n) to be the maximal unramified sub k-algebra of k(rep(n,A)), the coordinate ring of rep(n,A), then corresponding to the sum-map above is a map
pi(n + m) -> pi(n) \otimes pi(m)and these maps define on the graded space
Pi(A) = pi(0) + pi(1) + pi(2) + ...the structure of a graded commutative k-coalgebra with comultiplication
pi(n) -> sum(a + b=n) pi(a) \otimes pi(b)The relevance of Pi(A) is that if we consider it over the algebraic closure K of k we get the semigroup coalgebra
K G with g -> sum(h.h\' = g) h \otimes h\'where G is Morrison\’s connected component semigroup. That is, Pi(A) is a k-form of this semigroup coalgebra. Perhaps it is a good project for one of the students to work this out in detail (and correct possible mistakes I made) and give some concrete examples for formally smooth algebras A. If you know of a reference on this, please let me know.








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