Re: question 2b frm hw 5
William Stein
> Dear Professor Stein,
>
> I and Yannick were working on question 2 b from homework 5 and we think that
> we might need more hypothesis on the dimension of V_1 and V_2 (or we might
> be wrong).
You are right. You need that the map is surjective on vector spaces.
I've updated http://wstein.org/edu/2011/581g/hw/5.pdf
>
> For instance if we think of the inclusion from C -> C^2 (i.e. x-> (x,0)) and
> think of the lattices as Z sitting inside C and Z^2 sitting inside C^2, then
> we get an induced map C/Z -> C^2/Z^2. The kernel of this map is {0} and
> hence finite. But we need to show that the kernel is isomorphic to
> (Z^2)/(Zx{0}) which is isomorphic to Z in this example (and hence not
> finite).
>
> Here is problem 2b)
> Let V_i be finite dimensional complex vector spaces
> and let Λ_i ⊂ V_i be lattices (so rank_{Z} (Λ_i ) = 2 dim_{C} V_i and R.Λ_i
> = V_i ).
> Suppose T : V_1 → V_2 is a C-linear map such that T (Λ_1 ) ⊂ Λ_2 . Observe
> that
> T induces a homomorphism φ : V_1 /Λ_1 → V_2 /Λ_2 .
> i. If the kernel of φ is finite, prove that it is isomorphic to Λ_2 /T (Λ_1
> ).
>
>
> Bharathwaj Palvannan
> C8 F Padelford
> Graduate student
> Department of Mathematics
> University of Washington Seattle
>
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org