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J. L. Taylor's functional calculus theorem (1970) asserts that every commuting
\(n\)-tuple \(T=(T_1,\ldots ,T_n)\) of bounded linear operators on a Banach space \(E\)
admits a holomorphic functional calculus on any neighborhood \(U\) of the joint
spectrum \(\sigma(T)\). This means that there exists a continuous homomorphism
\(\gamma:\mathscr{O}(U)\to\mathscr{B}(E)\) (where \(\mathscr{O}(U)\) is the algebra of holomorphic
functions on \(U\) and \(\mathscr{B}(E)\) is the algebra of bounded linear operators on \(E\))
that takes the coordinates \(z_1,\ldots ,z_n\) to \(T_1,\ldots ,T_n\), respectively.
The original Taylor's proof was quite involved. In 1972, Taylor developed a completely
different and considerably shorter proof based on methods of homological algebra.
Later it was simplified and generalized by M. Putinar (1980) to the case of Fréchet
\(\mathscr{O}(X)\)-modules, where \(X\) is a finite-dimensional Stein space.
The idea of Taylor-Putinar's construction is to establish an isomorphism between
a Fréchet \(\mathscr{O}(X)\)-module \(M\) satisfying \(\sigma(M)\subset U\) and the \(0\)th cohomology
of a certain double complex \(C\) of Fréchet \(\mathscr{O}(U)\)-modules. Unfortunately,
\(C\) depends on the choice of a special cover of \(X\) by Stein open sets, and there
seems to be no canonical way of associating \(C\) to \(M\).
Our goal is to extend Taylor-Putinar's theorem to the setting of derived categories.
We believe that this is exactly the environment in which Taylor-Putinar's theorem is
most naturally formulated and proved.
Given an object \(M\) of the derived category \(\mathsf{D}^-(\mathscr{O}(X)\mbox{-}\underline{\mathop{\mathsf{mod}}})\) of Fréchet
\(\mathscr{O}(X)\)-modules, we define the spectrum \(\sigma(M)\subset X\), and we show that
for every open set \(U\subset X\) containing \(\sigma(M)\) there is an isomorphism
\(
M \cong \mathrm{R}\Gamma(U,\mathscr{O}_X)\mathop{\widehat\otimes}\nolimits_{\mathscr{O}(X)}^{\mathrm{L}}M
\)
in \(\mathsf{D}^-(\mathscr{O}(X)\mbox{-}\underline{\mathop{\mathsf{mod}}})\). In the special case where \(M\) is a Fréchet \(\mathscr{O}(X)\)-module,
this yields Taylor-Putinar's result. Moreover, we have \(C=\mathrm{R}\Gamma(U,\mathscr{O}_X)\mathop{\widehat\otimes}\nolimits_{\mathscr{O}(X)}^{\mathrm{L}}M\),
so \(C\) is natural in \(M\) when viewed as an object of the derived category.
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