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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