Exercise 2, Appendix 4

Keep the notation of the previous exercise. If V and W are subspaces of E, then we denote by $V\prec W$ the relation “W has finite codimension in V + W.”

a) Let V be a subspace of E. We denote by $\mathfrak{A}$ (resp. $\mathfrak{a},\mathfrak{b}$) the set of endomorphisms $u$ of V such that $u(V)\prec V$ (resp. $u(E)\prec V,u(V)\prec 0$). Prove that $\mathfrak{A}$ is a K-subalgebra of End$_K$(E), that $\mathfrak{a}$ and $\mathfrak{b}$ are two-sided ideals of $\mathfrak{A}$, with sum $\mathfrak{A}$, and that we have $\mathfrak{a}\cap \mathfrak{b}\subset$ End$^{pf}_K(E)$.

b) Let $u, u'$ be permutable elements of $\mathfrak{A}$; we can write $u=a+b$ and $u'=a'+b'$ with $a, a'$ in $\mathfrak{a}$ and $b, b'$ in $\mathfrak{b}$. Prove that the element $[a, a'] =aa'-a'a$ of $\mathfrak{A}$ belongs to $\mathfrak{a}\cap \mathfrak{b}$ and that its trace does not depend on the chosen decompositions of $u$ and $u'$; we denote it by $\langle u, u'\rangle$.

c) Let A be a commutative subalgebra of $\mathfrak{A}$. Prove that there exists a unique K-linear mapping Res$: \Omega_K(A)\rightarrow K$ such that Res($u dv$) $=\langle u, v\rangle$ for all $u, v$ in A (use the identity $[u, vw] + [v, wu] + [w, uv] = 0$).

d) Let $u, v\in A$; suppose $v(V)\subset V$. Let $\pi$ be a projector of E with image V. Prove the equality Res($u dv$) $=$ Tr[$\pi u, v]$. Deduce that if $u(V)\subset V$, then we have Res $u dv= 0$ (observe that the endomorphism $[\pi u, v]$ has square zero).

e) Let $u\in A,n\in \mathbf{N}$. Prove that we have Res $u^ndu= 0$. If $u$ is invertible in A, then we have Res $u^{-n}du= 0$ for $n\geqslant 2$; if moreover $u(V)\subset V$, then we have Res $u^{-1}du=$ dim$_KV/u(V)$.

f ) Take E = K((X)) (IV, §4, No. 9, p. 38), V = K[[X]], and let A = E act on itself by homotheties. Prove that the assumption of c) is satisfied. Let $f(X) =\sum_na_nX^ndX$ be an element of K((X)). Prove that Res $f(X)dX$ is equal to $a_{-1}$. Deduce that if $g$ is an element of K[[X]] such that $g(0) = 0$ and $g'(0)\not= 0$, then the coefficients of $X^{-1}$ in the formal series expansions of $f(X)$ and $f(g(X))g'(X)$ are the same.