connected component coalgebra

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 + N

as 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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