Does there exists a "local slice" for an action $ \widehat{\mathbb{G}}_a $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?

Question

Every action $ \beta $ of $ \mathbb{G}_a $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_0 \ast x) = \sum_{j=0}^\infty (\delta^j(f)(x)t_0^j)/j! $. In this case, the existence of a local slice is equivalent to the existence of $ g,h \in A $ such that $ \delta(g)=h $ and $ h \in A^{\mathbb{G}_a} $.

Such a pair of polynomials always exists in characteristic zero. However, suppose that we now move to the category of formal schemes and possibly non-algebraic actions $ \beta $ of $ \widehat{\mathbb{G}}_a $ on affine formal schemes $ \operatorname{Spf}(\widehat{A}) $. In this case, the action $ \beta $ is obtained from a derivation $ \delta \in \Omega_{\widehat{A}/k} $ via $ f(t_0 \ast x) = \sum_{j=0}^\infty (\delta^j(f)(x)t_0^j)/j! $. Note that the derivation $ \delta $ may not be locally nilpotent so the co-action $ \beta^{\sharp}: \widehat{A} \to \widehat{A}[[t]] $. Also the action may not be algebraic. Does there exist a pair of $ g,h \in \widehat{A} $ such that $ \delta(g)=h $ and $ h \in \widehat{A}^{\widehat{\mathbb{G}}_a} $?

Answer 1

The answer is no, such a ``local slice'' does not exist in general even for algebraic actions. Let $ \delta \in T_{k[x_{1},x_{2}]/k} $ be the derivation $ x_{1} \frac{\partial}{\partial x_{2}}+x_{2}\frac{\partial}{\partial x_{1}} $. By the infinite series identities \begin{align*} \cos(it)&=\sum_{s=0}^{\infty} t^{2s}/(2s)! \\ -i\sin(it) &= \sum_{s=0}^{\infty} t^{2s+1}/(2s+1)! \end{align*} the action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(k[[x_{1},x_{2}]]) $ obtained from $ \delta $ is the action $ \beta $ which sends $ (t_{0},(a_{1},a_{2})) $ to $ (a_{1}\cos(it_{0})-ia_{2}\sin(it_{0}),a_{2}\cos(it_{0})-ia_{1}\sin(it_{0})) $.

If we replace the basis $ \{x_{1},x_{2}\} $ with the basis $ u_{1}:=x_{1}+x_{2} $ and $ u_{2}:= x_{2}-x_{1} $,then the derivation $ \delta $ becomes $ u_{1} \frac{\partial}{\partial u_{1}}-u_{2}\frac{\partial}{\partial u_{2}} $. The co-action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(k[[u_{1},u_{2}]]) $ sends $ u_{1} $ to $ e^{t} u_{1} $ and $ u_{2} $ to $ e^{-t} u_{2} $.

Suppose that there exists a $ g(u_{1},u_{2}) \in k[[u_{1},u_{2}]] $ such that $ \beta^{\sharp}(g(u_{1},u_{2})) = g(u_{1},u_{2})+h(u_{1},u_{2})t $ for some $ h(u_{1},u_{2}) \in k[[u_{1},u_{2}]]^{\widehat{\mathbb{G}_{a}}} $.

Let $ g(u_{1},u_{2}) $ equal $ \sum_{j=0}^{\infty} g_{j}(u_{1},u_{2}) $ where $ g_{j}(u_{1},u_{2}) $ is a homogeneous polynomial in $ k[u_{1},u_{2}] $ of degree $ j $. Also, let $ g_{j}(u_{1},u_{2}) $ equal $ \sum_{i=0}^{j} a_{i,j} u_{1}^{i} u_{2}^{j-i} $. Under the co-action $ \beta^{\sharp}(g_{j}(u_{1},u_{2})) = \sum_{i=0}^{j} a_{i,j} e^{(2i-j)t} u_{1}^{i}u_{2}^{j-i} $.

If $ \delta(g(u_{1},u_{2})) = \beta^{\sharp}(g(u_{1},u_{2}))-g(u_{1},u_{2}) $, then \begin{align*} h(u_{1},u_{2})t &= \beta^{\sharp}(g(u_{1},u_{2}))-g(u_{1},u_{2}) \\ &= \sum_{j=0}^{\infty} \sum_{i=0}^{j} a_{i,j} (e^{(2i-j)t}-1)u_{1}^{i}u_{2}^{j-i}. \end{align*}

If $ h(u_{1},u_{2}) $ is equal to $ \sum_{j=0}^{\infty} h_{j}(u_{1},u_{2}) $ where $ h_{j}(u_{1},u_{2}) $ is a degree $ j $, homogeneous polynomial in $ k[u_{1},u_{2}] $, then for any $ j $ such that $ h_{j}(u_{1},u_{2}) $ is non-zero, \begin{align*} k[u_{1},u_{2}][t] & \ni h_{j}(u_{1},u_{2})t \\ &= \sum_{i=0}^{j} a_{i,j} e^{(2i-j)t} u_{1}^{i} u_{2}^{j-i} \\ & \notin k[u_{1},u_{2}][t]. \end{align*} As a result, such a $ g(u_{1},u_{2}) $ cannot exist for this action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spf}(k[[x_{1},x_{2}]]) $.