Exact sequences and Resultants

Question

$F$ is an arbitrary field and $V_k$ denotes subspace of $F[x]$ spanned by ordered basis $B_k = \{x^{k-1}, x^{k-2},\dotsc,1\}$. $f, g$ are elements of $F[x]$. $m = \deg(f)$, $n = \deg(g)$. $T: V_n \oplus V_m \to V_{m+n}$ with $T(u,v) = uf + vg$. How would you give explicit descriptions of $\mathrm{im}(T)$ and $\ker(T)$? In addition how would you show the sequence $0 \to V_a \xrightarrow{i} V_m \oplus V_m \xrightarrow{T} V_{m+n} \xrightarrow{p} V_b \to 0$ is exact? where $i, T, p$ are mappings.

Answer 1

For the kernel:

I assume that $f,g \neq 0$. If $(u,v) \in \ker(T)$, i.e. $uf=-vg$, then $f$ divides $vg$. It follows that $f/\mathrm{gcd}(f,g)$ divides $v g/\mathrm{gcd}(f,g)$. Since $f/\mathrm{gcd}(f,g)$ and $g/\mathrm{gcd}(f,g)$ are coprime, it follows that $f/\mathrm{gcd}(f,g)$ divides $v$. Hence, we can write $v = p \cdot f/\mathrm{gcd}(f,g)$ for some $p$. It also follows that $u = -p \cdot g/\mathrm{gcd}(f,g)$. Let $d=\deg(\mathrm{gcd}(f,g))$. Since $\deg(u) < n$, we $\deg(p) < d$. It follows that $0 \to T_d \xrightarrow{i} T_n \oplus T_m \xrightarrow{T} T_{n+m}$ is exact, where $i(x^k) = (-g/\mathrm{gcd}(f,g),f/\mathrm{gcd}(f,g)) \cdot x^k$.