e: Hom_Z(R,C) –> Hom_C(A,C) then there is an algebra map from A to R_C, no need of R being a group ring or anything (maybe just flatness to avoid nasty torsion killing on the tensor product). The way of constructing such a map is as follows: Take a character s: R_C –> C, then by composing with the natural inclusion R –> R_C it induces a character s’: R –> C, upon which we can apply our evaluation map e, getting a map e’: Hom_C(R_C,C) –> Hom_C(A,C) defined by e’(s) := e(s’). This is a map between the corresponding maximal spectra, and thus induces an algebra map between the algebras, in the opposite direction.

I feel so stupidly frustrated now… first I was frustrated by not being able to solve the problem, and now because I feel that I should have realized before! Does math keep being like this forever, or it gets any better when you get a permanent position?

]]>Now, any Z-algebra morphism Z[D]–>C determines (and is determined by) a group-morphism D–>C^* (that is, by a character) and as such extends to a C-algebra morphism C[D]–>C.

Max(C[D]) is the set of all characters of D. Lets call Max(A_X) the variety of all 1-dml reps of A_X (exists even when A_X is noncommutative). In fact, Max(A_X) = Max(A_X/[A_X,A_X]) so is the affine variety of a commutative algebra (affine, if A_X is supposed to be affine).

The soule-condition determines for each x in X(D) a map from Max(C[D])–>Max(A_X) and this is a polynomial map because it just sends a finite number of points to points of the variety corresponding to the commutative ring B (A_X mod commutators). So, by duality this gives an algebra map of commutative algebras B —> C[D]. The corresponding C-algebra map of lemma 2.4 is then the composition A_X –>> B –> C[D].

So, this then gives the map of sets X(D)–>Hom_C(A_X,C[D]).

]]>via . According to C-C (Lemma 2.4) that induces a complex algebra map , but I don’t quite see how. Only thing I can think about is assuming that for every , we could define as an element in such that for every character of we have , where , but that leaves the problem on proving existence and uniqueness of such an element of .

Or am I getting something wrong and it follows from some adjunction properties? I find all of it a bit confusing, since we are mixing up complex algebra maps, ring homomorphisms and set maps… any ideas?

]]>