Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, endowed with some natural topology, say, the topology of uniform convergence on compact sets in $A$. For each $s\in{\tt Spec}[A]$ let us call a tangent space in $s$, ${\tt T}_s[A]$, the set of all tangent vectors in $s$, i.e. continuous linear functionals $\tau:A\to{\mathbb C}$ such that $$ \tau(f\cdot g)=s(f)\cdot\tau(g)+\tau(f)\cdot s(g),\qquad f,g\in A $$ (again, we endow ${\tt T}_s[A]$ with some natural topology, say, the topology of uniform convergence on compact sets in $A$).

Let $M$ be a Stein manifold and $A={\mathcal O}(M)$, the algebra of holomorphic functions on $M$ (with the usual topology of uniform convergence on compact sets in $M$).

I believe that $$ {\tt Spec}[{\mathcal O}(M)]\cong M, $$ (a homeomorphism of topological spaces), and for each $s\in M$ $$ {\tt T}_s[{\mathcal O}(M)]\cong{\tt T}_s(M) $$ (an isomorphism with the usual tangent space of $M$ in the point $s\in M$), but I can't find a reference. Can anybody help me?

EDIT. Ben McKay gave a reference for the first equality, so it remains to prove the second one. As far as I understand, here everything follows from the following variant of

Hadamard's lemma: Let $s$ be a point on a Stein manifold $M$, and $f_1,...,f_n\in {\mathcal O}(M)$ a sequence of functions which form local coordinates in a neighbourhood of $s$ and $$ f_1(s)=...=f_n(s)=0 $$ (such $f_k$ always exist). Then for every function $f\in{\mathcal O}(M)$ there exist functions $g_1,...,g_m,h_1,...,h_m\in {\mathcal O}(M)$ such that $$ g_1(s)=...=g_m(s)=h_1(s)=...=h_m(s)=0 $$ and $$ f=f(s)+\sum_{k=1}^n\frac{\partial f}{\partial f_k}(s)\cdot f_k+ \sum_{k=1}^m g_k\cdot h_k $$ where $\frac{\partial f}{\partial f_k}(s)$ are partial derivatives in $s$ along the coordinates $f_k$.

That is not true?


1 Answer 1


The statement that $\operatorname{Spec} \mathcal{O}(M)=M$ on a complex manifold $M$ whose holomorphic functions separate points precisely when $M$ is a Stein manifold is in H. Rossi, On envelops of holomorphy, Communications in Pure and Applied Mathematics, XVI, 1963, 9-17, theorem 2.6 page 10.

  • 2
    $\begingroup$ Rossi persistently misspells the noun "envelope" as the verb "envelop". $\endgroup$
    – Ben McKay
    May 1, 2016 at 8:02
  • $\begingroup$ Ah, OK! And what about the tangent space? $\endgroup$ May 1, 2016 at 8:04
  • $\begingroup$ There is no such description of the tangent space in Rossi's paper. I don't know a reference. $\endgroup$
    – Ben McKay
    May 1, 2016 at 8:17
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    $\begingroup$ The aptly-titled book "Theory of Stein Spaces" discusses full faithfulness for the functor $X\rightsquigarrow O(X)$ from general Stein spaces (could be singular, or non-reduced) into a suitable category of Frechet $\mathbf{C}$-algebras. The desired definition of the tangent space consists of the continuous $\mathbf{C}$-algebra maps $O(M)\rightarrow \mathbf{C}[\varepsilon]/(\varepsilon^2)$ lifting evaluation at $s$, so by full faithfulness (!) these correspond to maps of analytic spaces ${\rm{Spec}}(\mathbf{C}[\epsilon]/(\varepsilon^2))\rightarrow M$ lifting $s$. Working locally at $s$, we win. $\endgroup$
    – nfdc23
    May 1, 2016 at 8:25
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    $\begingroup$ There is no such manifold: it is a 1-point Stein space in the wider sense of complex-analytic spaces (which may be singular or even have non-reduced structure sheaf) and has a 1-dimensional tangent space with preferred basis (dual to $\varepsilon$). Complex-analytic spaces are the analytic counterpart to schemes, and provide a more robust framework for developing both coherent sheaf theory and analytic geometry (much as singular varieties are useful in algebraic geometry, when making intersections or forming zero loci); they are developed from scratch in the book "Coherent Analytic Sheaves". $\endgroup$
    – nfdc23
    May 1, 2016 at 18:40

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