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Theorem 3.3. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( f \) complex differentiable in \( B \) . Then \( f \) is holomorphic in \( B \) . | Proof. Let \( {\mathfrak{z}}_{0} \in B \) . For the sake of simplicity we assume \( {\mathfrak{z}}_{0} = 0 \) . Then there exists a polycylinder \( P \) about \( {\mathfrak{z}}_{0} \) such that \( \bar{P} \subset B \) . Let \( T \) be the distinguished boundary of \( P \) . From Theorem \( {3.1f}\left| {P = \operatorna... | No |
Theorem 3.4. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( f \) holomorphic in \( B \) and \( {\mathfrak{z}}_{0} \) a point in \( B \) . If \( P \subset B \) is a polycylinder about \( {\mathfrak{z}}_{0} \) with \( \bar{P} \subset B \), then there is a power series \( \mathfrak{P}\left( \mathfrak{z}\right) = \ma... | Proof. If \( f \) is holomorphic in \( B \), then \( f \mid P = \operatorname{ch}\left( {f \mid T}\right) \), where the distinguished boundary of \( P \) is denoted by \( T \) . From Theorem \( {3.2f} \mid P \) can be expanded as a power series in all of \( P \) . | No |
Theorem 3.5. Let the sequence of functions \( \left( {f}_{v}\right) \) converge uniformly to \( f \) on the region \( B \) with all \( {f}_{v} \) holomorphic in \( B \) . Then \( f \) is holomorphic in \( B \) . | Proof. Let \( {\mathfrak{z}}_{0} \in B \) . Again, we assume that \( {\mathfrak{z}}_{0} = 0 \) . Let \( P \) be a polycylinder about \( {\mathfrak{z}}_{0} \) with \( \bar{P} \subset \bar{B} \) . Let \( \mathfrak{z} = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in P.N\left( \xi \right) \mathrel{\text{:=}} \left( {{\xi }_{1... | Yes |
Theorem 3.6. Let \( \mathfrak{P}\left( \mathfrak{z}\right) = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{\mathfrak{z}}^{v} \) be a formal power series and \( G \) the domain of convergence for \( \mathfrak{P}\left( \mathfrak{z}\right) \) . Then \( f \) with \( f\left( \mathfrak{z}\right) \mathrel{\text{:=}} \mathf... | Proof. Let \( \mathfrak{I} \) be the set of all multi-indices \( v = \left( {{v}_{1},\ldots ,{v}_{n}}\right) ,{I}_{0} \subset \mathfrak{I} \) a finite subset. Clearly the polynomial \( \mathop{\sum }\limits_{{v \in {I}_{0}}}{a}_{v}{\mathfrak{z}}^{v} \) is holomorphic on all of \( {\mathbb{C}}^{n} \) . Let \( {\mathfrak... | Yes |
Theorem 4.1 (Identity theorem for holomorphic functions). Let \( G \subset {\mathbb{C}}^{n} \) be a domain and \( {f}_{1},{f}_{2} \) be holomorphic in \( G \) . Let \( B \subset G \) be a nonempty region with \( {f}_{1}\left| {B = {f}_{2}}\right| B \) . Then \( {f}_{1}\left| {G = {f}_{2}}\right| G \) . | Proof. Let \( {B}_{0} \) be the interior of the set \( \left\{ {\mathfrak{z} \in G : {f}_{1}\left( \mathfrak{z}\right) = {f}_{2}\left( \mathfrak{z}\right) }\right\} \) and \( {W}_{0} \mathrel{\text{:=}} \) \( G - {B}_{0} \) . Because \( B \subset {B}_{0},{B}_{0} \neq \varnothing \) . Since \( G \) is connected it suffi... | Yes |
Theorem 4.2 (Identity theorem for power series). Let \( G \subset {\mathbb{C}}^{n} \) be a domain with \( 0 \in G \), and \( \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{\mathfrak{z}}^{v},\mathop{\sum }\limits_{{v = 0}}^{\infty }{b}_{v}{\mathfrak{z}}^{v} \) two power series convergent in \( G \) . If there is an \(... | Proof. Let \( f\left( 3\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{3}^{v}, g\left( 3\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{v = 0}}^{\infty }{b}_{v}{3}^{v} \) for \( 3 \in G \) . By Theorem 3.6 \( f \) and \( g \) are holomorphic in \( G \), and differentiation gives:\n\n\[ ... | Yes |
Theorem 5.2. Let \( G \) be a domain in \( {\mathbb{C}}^{n},{E}_{0} \mathrel{\text{:=}} \left\{ {\mathfrak{z} = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in {\mathbb{C}}^{n} : {z}_{v} = 0}\right. \) for at least one \( v\} \) . Then \( {G}_{0} \mathrel{\text{:=}} G - {E}_{0} \) is also a domain. | Proof. For each \( \mu \) with \( 1 \leq \mu \leq n,{G}_{\mu } : G - {E}_{\mu } \) is connected, where \( {E}_{\mu } \mathrel{\text{:=}} \left\{ {\mathfrak{z} = \left( {{z}_{1},\ldots ,{z}_{n}}\right) \in {\mathbb{C}}^{n} : {z}_{\mu } = 0}\right\} \) . This follows from Theorem 5.1 by a simple permutation of the coordi... | Yes |
Theorem 5.3. Let \( G \subset {\mathbb{C}}^{n} \) be a proper Reinhardt domain, \( f \) holomorphic on \( G,{\mathfrak{z}}_{0} \in G \cap {\mathfrak{C}}^{n} \) . Then \( \operatorname{ch}\left( {f \mid {T}_{\mathfrak{d}0}}\right) \) coincides with \( f \) in a neighborhood of the origin. | Proof. We have \( {G}_{0} \mathrel{\text{:=}} \tau \left( {G \cap {\mathring{\mathbb{C}}}^{n}}\right) \subset \left\{ {\mathrm{r} \in V : {r}_{j} \neq 0}\right. \) for \( \left. {j = 1,\ldots, n}\right\} \) .\n\n1. \( {G}_{0} \) is a domain:\n\na. \( G \cap {\mathring{\mathbb{C}}}^{n} \) is a Reinhardt domain; therefor... | Yes |
Theorem 5.4. Let \( G \subset {\mathbb{C}}^{n} \) be a proper Reinhardt domain, \( f \) holomorphic in \( G \) . Then there is a power series \( \mathfrak{P}\left( \mathfrak{z}\right) = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{\mathfrak{z}}^{v} \) which converges in \( G \) with \( f\left( 3\right) = \mathfrak{... | Proof. If \( {\mathfrak{z}}_{0} \in G \) then there is a \( {\mathfrak{z}}_{1} \in G \) with \( \left| {z}_{j}^{\left( 0\right) }\right| < \left| {z}_{j}^{\left( 1\right) }\right| \) for \( j = 1,\ldots, n \) ; therefore \( {\mathfrak{z}}_{0} \in {P}_{{\mathfrak{z}}_{1}} \) . Let \( \operatorname{ch}\left( {f \mid {T}_... | No |
Theorem 5.5. Let \( G \) be a proper Reinhardt domain, \( f \) holomorphic in \( G \) . Then there is exactly one holomorphic function \( F \) in \( \widehat{G} \) with \( F \mid G = f \) . | Proof. By Theorem 5.4 we can write in \( G \) \[ f\left( 3\right) = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v3}{}^{v} \] The series is still convergent on \( \widehat{G} \), and actually converges to a holomorphic function \( F \) . Clearly \( F \mid G = f \) . The uniqueness of the continuation follows from the ... | No |
Theorem 6.1. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( {\mathfrak{z}}_{0} \in B \) a point, \( f \) a complex function on \( B \) . \( f \) is real differentiable at \( {\mathfrak{z}}_{0} \) if and only if there are functions \( {\Delta }_{v}^{\prime },{\Delta }_{v}^{\prime \prime } \) on \( B \) which are c... | Proof\n\n1. Let \( f \) be real differentiable at \( {}_{30} \) . We use the equations\n\n\[ {x}_{v} - {x}_{v}^{\left( 0\right) } = \frac{1}{2}\left\lbrack {\left( {{z}_{v} - {z}_{v}^{\left( 0\right) }}\right) + \left( {{\bar{z}}_{v} - {\bar{z}}_{v}^{\left( 0\right) }}\right) }\right\rbrack \]\n\nand\n\n\[ {y}_{v} - {y... | Yes |
Theorem 6.2. Let \( B \subset {\mathbb{C}}^{n} \) be a region \( {\mathfrak{z}}_{0} \in B, f \) a complex function on \( B.f \) is complex differentiable at \( {}_{30} \) if and only if \( f \) is real differentiable at \( {}_{30} \) and \( {f}_{{z}_{v}}\left( {\mathfrak{z}}_{0}\right) = 0 \) for \( 1 \leq v \leq n \) ... | Proof.\n\n1. Let \( f\left( 3\right) = f\left( {3}_{0}\right) + \mathop{\sum }\limits_{{v = 1}}^{n}\left( {{z}_{v} - {z}_{v}^{\left( 0\right) }}\right) {\Delta }_{v}\left( 3\right) ,{\Delta }_{v}\left( 3\right) \) continuous at \( {3}_{0} \) . Then\n\n\[ \nf\left( z\right) = f\left( {z}_{0}\right) + \mathop{\sum }\limi... | Yes |
Theorem 6.3 (Chain rule). Let \( {B}_{1},{B}_{2} \) be regions in \( {\mathbb{C}}^{n} \), respectively \( {\mathbb{C}}^{m} \). \( g = \left( {{g}_{1},\ldots ,{g}_{m}}\right) : {B}_{1} \rightarrow {\mathbb{C}}^{m} \) be a mapping with \( g\left( {B}_{1}\right) \subset {B}_{2} \). Let \( {\mathfrak{z}}_{0} \in {B}_{1} \)... | Proof. As in the real case, the proof follows from the definitions. | No |
Theorem 7.1. Let \( {B}_{1} \subset {\mathbb{C}}^{n},{B}_{2} \subset {\mathbb{C}}^{m} \) be regions, \( g = \left( {{g}_{1},\ldots ,{g}_{m}}\right) : {B}_{1} \rightarrow {B}_{2} \) be a mapping. \( g \) is holomorphic if and only if for each holomorphic function \( f \) on \( {B}_{2}f \circ g \) is a holomorphic functi... | Proof. Let \( g \) be a holomorphic mapping. Then all the component functions \( {g}_{\mu } \) are holomorphic, that is, \( {\left( {g}_{\mu }\right) }_{{z}_{v}} = 0 \) for all \( v \) and \( \mu \) . If \( f \) is holomorphic, then \( {f}_{{\bar{w}}_{\mu }} = 0 \) for all \( \mu, f \circ g \) is real differentiable, a... | Yes |
Theorem 7.2. Let \( {\mathfrak{z}}_{0} \in B,{\mathfrak{w}}_{0} = g\left( {\mathfrak{z}}_{0}\right), f \) and \( g \) as above. Then\n\n\[{\mathfrak{M}}_{f \circ g}\left( {\mathfrak{z}}_{0}\right) = {\mathfrak{M}}_{f}\left( {\mathfrak{w}}_{0}\right) \circ {\mathfrak{M}}_{g}\left( {\mathfrak{z}}_{0}\right)\] | Proof. \( {\left( {\mathfrak{M}}_{f \circ g}\right) }_{v\mu } = {\left( {f}_{v} \circ g\right) }_{{z}_{\mu }} = \mathop{\sum }\limits_{{\lambda = 1}}^{m}{f}_{v,{w}_{\lambda }} \cdot {g}_{\lambda ,{z}_{\mu }} = {\left( {\mathfrak{M}}_{f} \circ {\mathfrak{M}}_{g}\right) }_{v\mu } \) | Yes |
Theorem 7.3. Let the notation be as above and let \( m = n = 1 \) . Then \( {M}_{f \circ g} = \) \( {M}_{f} \cdot {M}_{g} \) . | \[ \n{\Delta }_{g} = \det \left( {\frac{\left( {g}_{v,{z}_{\mu }}\right) }{\left( {\bar{g}}_{v,{z}_{\mu }}\right) } - \frac{\left( {g}_{v,{\bar{z}}_{\mu }}\right) }{\left( {\bar{g}}_{v,{\bar{z}}_{\mu }}\right) }}\right) = \det \left( {\frac{\left( {g}_{v,{z}_{\mu }}\right) }{0} - \left| {\; - \frac{0}{\left( {g}_{v,{z}... | No |
Theorem 7.4. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( g : B \rightarrow {\mathbb{C}}^{n} \) a holomorphic mapping. Let \( {\mathfrak{z}}_{0} \in B \) and \( {\mathfrak{w}}_{0} = g\left( {\mathfrak{z}}_{0}\right) \) . There are open neighborhoods \( U = U\left( {\mathfrak{z}}_{0}\right) \subset B \) and \( V... | Proof\n\n1. There are open neighborhoods \( U, V \) such that \( g : U \rightarrow V \) is biholo-morphic. Then \( 1 = {M}_{{\mathrm{{id}}}_{U}}\left( {\mathfrak{z}}_{0}\right) = {M}_{{g}^{-1}}\left( {\mathfrak{w}}_{0}\right) \cdot {M}_{g}\left( {\mathfrak{z}}_{0}\right) \), hence \( {M}_{g}\left( {\mathfrak{z}}_{0}\ri... | Yes |
Theorem 7.6. Let \( {B}_{1} \subset {\mathbb{C}}^{n} \) be a region, \( g : {B}_{1} \rightarrow {\mathbb{C}}^{n} \) one-to-one and holomorphic. Then \( {B}_{2} \mathrel{\text{:=}} g\left( {B}_{1}\right) \) is also an open set and \( {g}^{-1} : {B}_{2} \rightarrow {B}_{1} \) is holomorphic. | ## Proof\n\n1. Let \( {\mathfrak{w}}_{0} \in {B}_{2} \) . Then there exists a \( {\mathfrak{z}}_{0} \in {B}_{1} \) with \( g\left( {\mathfrak{z}}_{0}\right) = {\mathfrak{w}}_{0} \) . From Theorem \( {7.5}{M}_{g} \neq 0 \) on \( {B}_{1} \), and therefore there are open neighborhoods \( U\left( {\mathfrak{z}}_{0}\right) ... | Yes |
Theorem 1.1. Let \( \left( {\widetilde{P},\widetilde{H}}\right) \) be a general Hartogs figure in \( {\mathbb{C}}^{n} \) , \( f \) holomorphic in \( \widetilde{H} \) . Then there is exactly one holomorphic function \( F \) on \( \widetilde{P} \) with \( F \mid \widetilde{H} = f \) . | Proof. Let \( \left( {\widetilde{P},\widetilde{H}}\right) = \left( {g\left( P\right), g\left( H\right) }\right), g : P \rightarrow {\mathbb{C}}^{n} \) be biholomorphic. Then \( f \circ g \) is holomorphic in \( H \) and by Theorem 5.5 of Chapter I there is exactly one holomorphic function \( {F}^{ \star } \) on \( P \)... | Yes |
Theorem 1.2 (Continuity theorem). Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( \left( {\widetilde{P},\widetilde{H}}\right) \) a general Hartogs figure with \( \widetilde{H} \subset B, f \) a holomorphic function in \( B \) . If \( \widetilde{P} \cap B \) is connected, then \( f \) can be continued uniquely to \... | Proof. \( {f}_{1} \mathrel{\text{:=}} f \mid \widetilde{H} \) is holomorphic in \( \widetilde{H} \) . Therefore there exists exactly one holomorphic function \( {f}_{2} \) in \( \widetilde{P} \) with \( {f}_{2} \mid \widetilde{H} = {f}_{1} \) .\n\nLet\n\[ F\left( 3\right) \mathrel{\text{:=}} \left\{ \begin{matrix} f\le... | Yes |
Theorem 1.4. Let \( n \geq 2, B \subset {\mathbb{C}}^{n} \) be a region, and \( {\mathfrak{z}}_{0} \in B \) . Let \( f \) be holomorphic in \( {B}^{\prime } \mathrel{\text{:=}} B - \left\{ {\mathfrak{z}}_{0}\right\} \) . Then \( f \) has a unique holomorphic extension on \( B \) . (For \( n \geq 2 \) there are no isola... | Proof. Without loss of generality we assume that \( {\mathfrak{z}}_{0} = 0 \) . Let \( P \) be a poly-cylinder about \( {\mathfrak{z}}_{0} \) with \( P \subset B,{P}^{\prime } \mathrel{\text{:=}} P - \left\{ {\mathfrak{z}}_{0}\right\} \) . This is the situation of Theorem 1.3; so there is a holomorphic function \( {F}^... | Yes |
Theorem 2.1. Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( G \subset {\mathbb{C}}^{n} \) a domain with \( B \cap G \neq \varnothing \) and \( \left( {{\mathbb{C}}^{n} - B}\right) \cap G \neq \varnothing \) . Then for each connected component \( Q \) of \( B \cap G \)\n\n\[ G \cap \partial Q \cap \partial B \neq ... | Proof. We have \( G = Q \cup \left( {G - Q}\right) .Q \) is open and not empty, and because \( \left( {{\mathbb{C}}^{n} - B}\right) \cap G \neq \varnothing, G - Q \) is also non-empty. Since \( G \) is a domain it does not split into two non-empty open subsets. Hence \( G - Q \) is not open. Let \( {z}_{1} \in G - Q \)... | Yes |
Theorem 2.2. Let \( G \) be a domain of holomorphy. Then \( G \) is pseudoconvex. | Proof. Assume that \( G \) is not pseudoconvex. Then there is a Hartogs figure \( \left( {P, H}\right) \) with \( H \subset G \) but \( P \cap G \neq P \) . We choose an arbitrary \( {\mathfrak{z}}_{0} \) in \( H \) and set \( Q \mathrel{\text{:=}} {C}_{P \cap G}\left( {\mathfrak{z}}_{0}\right) \) . Since \( H \) lies ... | Yes |
Theorem 2.3. Let \( G \subset {\mathbb{C}}^{n} \) be a domain with \( {C}^{2} \) boundary, \( U \) an open set with \( U \cap \partial G \neq \varnothing \) . Let \( \varphi ,\psi \) be two functions on \( U \) which satisfy the conditions of Def. 2.4. Then there is a uniquely determined positive differentiable functio... | Proof. We only need to show that for each \( {\mathfrak{z}}_{0} \in U \cap \partial G \) there is a neighborhood \( V\left( {\mathfrak{z}}_{0}\right) \subset U \) and in \( V \) exactly one differentiable function \( h \) with \( \varphi \mid V = \) \( h \cdot \left( {\psi \mid V}\right) \) . Therefore let \( {\mathfra... | Yes |
Theorem 2.4. Let \( G \subset {\mathbb{C}}^{n} \) be a domain with \( {C}^{2} \) boundary, \( {\mathfrak{z}}_{0} \in \partial G \) and \( U = \) \( U\left( {\mathfrak{z}}_{0}\right) \) an open neighborhood. Let \( \varphi ,\psi \) be two functions on \( U \) which satisfy the conditions of Def. 2.4. If \( \varphi \) sa... | Proof. We can find a twice continuously differentiable positive function \( h \) on \( U \) with \( \psi = h \cdot \varphi \) . Now let \( \mathfrak{w} \in {\mathbb{C}}^{n} \) and \( \mathop{\sum }\limits_{{i = 1}}^{n}{\psi }_{{z}_{i}}\left( {\mathfrak{z}}_{0}\right) {w}_{i} = 0 \) . Then at \( {\mathfrak{z}}_{0} \)\n\... | Yes |
Theorem 2.5. Let \( G \subset {\mathbb{C}}^{n} \) be a domain with \( {C}^{2} \) boundary. Then \( G \) is pseudo-convex if and only if the Levi condition is satisfied for every boundary point of \( G \) . | This theorem will not be proved here. | No |
Theorem 3.1 (The properties of the geometrically convex hull). Let \( M \subset {\mathbb{R}}^{2} \) be an arbitrary subset. Then:\n\n1. \( M \subset {\widehat{M}}_{e} \) .\n\n2. \( {\widehat{M}}_{e} \) is closed and geometrically convex.\n\n3. \( {\widehat{M}}_{e} = {\widehat{M}}_{e} \) .\n\n4. Let \( {M}_{1} \subset {... | ## Proof\n\n1. Let \( \mathfrak{x} \in M \) . Then for each \( \ell \in L,\ell \left( \mathfrak{x}\right) \leq \sup \ell \left( M\right) \) . Therefore \( \mathfrak{x} \) lies in \( {\widehat{M}}_{e} \) .\n\n2. Let \( {\mathfrak{x}}_{0} \notin {\widehat{M}}_{e} \) . Then there exists an \( \ell \in L \) with \( \ell \l... | Yes |
Theorem 3.3 (The properties of the holomorphically convex hull). Let \( B \subset {\mathbb{C}}^{n} \) be a region, \( K \subset B \) a subset. Then:\n\n1. \( K \subset \widehat{K} \)\n\n2. \( \widehat{K} \) is closed in \( B \) .\n\n3. \( \widehat{\widehat{K}} = \widehat{K} \)\n\n4. Let \( {K}_{1} \subset {K}_{2} \subs... | Proof\n\n1. For \( \mathfrak{z} \in K,\left| {f\left( \mathfrak{z}\right) }\right| \leq \sup \left| {f\left( K\right) }\right| \).\n\n2. Let \( \mathfrak{z} \in B - \widehat{K} \) . Then there exists a holomorphic function \( f \) on \( B \) with \( \left| {f\left( 3\right) }\right| > \sup \left| {f\left( K\right) }\ri... | Yes |
Theorem 4.1. \( {B}_{\varepsilon } \) is closed. | Proof. Let \( {\mathfrak{z}}_{0} \in {\mathbb{C}}^{n} - {B}_{\varepsilon } \) . We define \( \delta \mathrel{\text{:=}} {\operatorname{dist}}^{\prime }\left( {{\mathfrak{z}}_{0},{\mathbb{C}}^{n} - B}\right) .\varepsilon > \delta \geq 0 \), so \( \varepsilon - \delta > 0 \) . Let \( U \mathrel{\text{:=}} {U}_{\varepsilo... | Yes |
Theorem 4.2. Let \( B \) be a region, \( f \) holomorphic in \( \bar{B},\left| {f\left( \bar{B}\right) }\right| \leq M,\varepsilon > 0 \), and \( {\mathfrak{z}}_{0} \in {B}_{\varepsilon } \) a point. In a neighborhood \( U = U\left( {\mathfrak{z}}_{0}\right) \subset B \), let \( f \) have the power series expansion \( ... | Proof. Let \( P \mathrel{\text{:=}} \left\{ {3 \in {\mathbb{C}}^{n} : {\operatorname{dist}}^{\prime }\left( {3,{3}_{0}}\right) < \varepsilon }\right\} \) . Then for \( 3 \in P,{\operatorname{dist}}^{\prime }\left( {3,{\mathbb{C}}^{n} - B}\right) \geq \) \( {\operatorname{dist}}^{\prime }\left( {{\mathfrak{z}}_{0},{\mat... | Yes |
Theorem 5.1. Let \( B \subset {\mathbb{C}}^{n} \) be a region. Then there exists a sequence of subsets \( {K}_{v} \subset B \) with the following properties.\n\n1. \( {K}_{v} \) is compact for all \( v \in \mathbb{N} \) .\n\n2. \( \mathop{\bigcup }\limits_{{v = 1}}^{\infty }{K}_{v} = B \) .\n\n3. \( {K}_{v} \subset {\m... | Proof. It is clear how the \( {K}_{v} \) should be chosen: If \( {\bar{P}}_{v} \mathrel{\text{:=}} \left\{ {{}_{3} : \left| {z}_{\lambda }\right| \leq v}\right. \) for all \( \lambda \} \), we define \( {K}_{v} \mathrel{\text{:=}} {\bar{P}}_{v} \cap {B}_{1/v} \) . Obviously \( {K}_{v} \) is compact and lies in \( B \) ... | Yes |
Theorem 5.4. Let \( B \subset {\mathbb{C}}^{n} \) be a region. If \( B \) is holomorphically convex then there exists a normal exhausting \( \left( {K}_{v}\right) \) of \( B \) with the property that \( {K}_{v} = {\widehat{K}}_{v} \) for every \( v \in \mathbb{N} \) . | Proof. Let \( \left( {K}_{v}\right) \) be any normal exhaustion of \( B \) . Then for all \( v,{K}_{v} \subset \subset B \) and as \( B \) is holomorphically convex, it follows that \( {\widehat{K}}_{v} \subset \subset B.{\widehat{K}}_{v} \) is therefore a compact subset of \( B \) . We now construct a subsequence of t... | Yes |
Theorem 5.5. Let \( B \subset {\mathbb{C}}^{n} \) be a region. If \( B \) is holomorphically convex, then \( B \) is a region of holomorphy. | Proof. By the preceding theorem there is a normal exhausting \( \left( {K}_{v}\right) \) of \( B \) with \( {K}_{v} = {\widehat{K}}_{v} \) for all \( v \) and hence sequences \( \left( {\lambda }_{\mu }\right) \) and \( \left( {3}_{\mu }\right) \) in the sense of Theorem 5.2 and a function \( f \) holomorphic in \( B \... | Yes |
Theorem 6.1. Let \( B \subset \mathbb{C} \) be a region. Then \( B \) is a region of holomorphy. (Hence for every open set \( B \) in \( \mathbb{C} \) there exists a holomorphic function which cannot be extended to any proper open superset of \( B \).) | Proof. It was shown in Section 3, that every region in \( \mathbb{C} \) is holomorphically convex. From Theorem 5.4 it follows that \( B \) is a region of holomorphy. | Yes |
Theorem 6.2. Let \( B \subset {\mathbb{C}}^{n} \) be a region. Then the following statements are are equivalent:\n\n1. \( B \) is pseudo-convex.\n\n2. \( B \) is a region of holomorphy.\n\n3. \( B \) is holomorphically convex.\n\n4. For every infinite set \( D \) discrete in \( B \) there exists a function fholomorphic... | Proof. The statements have all been proved in the preceding paragraphs (apart from the solution of the Levi problem: if \( B \) is pseudoconvex, then \( B \) is a region of holomorphy). | No |
Theorem 6.4. Let \( {V}_{1},\ldots ,{V}_{n} \subset \mathbb{C} \) be regions. Then \( V \mathrel{\text{:=}} {V}_{1} \times \cdots \times {V}_{n} \subset \) \( {\mathbb{C}}^{n} \) is a region of holomorphy. | Proof. Let \( D \subset V \) be a discrete infinite set and \( \left( {\mathfrak{z}}_{\mu }\right) \) a sequence of distinct points of \( D \) with \( {\mathfrak{z}}_{\mu } = \left( {{z}_{1}^{\left( \mu \right) },\ldots ,{z}_{n}^{\left( \mu \right) }}\right) \) . If the sequence \( \left( {z}_{1}^{\left( \mu \right) }\... | Yes |
Theorem 6.5. Let \( B \subset {\mathbb{C}}^{n} \) be a region. Then every analytic polyhedron in \( B \) is a region of holomorphy. | Proof. Let \( U,{V}_{1},\ldots ,{V}_{k},{f}_{1},\ldots ,{f}_{k} \) and \( P \) be given as in Def. 6.1. Then \( F : = \left( {{f}_{1}\left| {U,\ldots ,{f}_{k}}\right| U}\right) : U \rightarrow {\mathbb{C}}^{k} \) is a holomorphic mapping and \( P = \) \( {F}^{-1}\left( {{V}_{1} \times \cdots \times {V}_{k}}\right) \) .... | Yes |
Theorem 6.6. Every region of holomorphy \( B \subset {\mathbb{C}}^{n} \) can be exhausted by analytic polyhedra in the sense that there exists a sequence \( \left( {P}_{j}\right) \) of special analytic polyhedra in \( B \) with \( {P}_{j} \subset \subset {P}_{j + 1} \) and \( \mathop{\bigcup }\limits_{{j + 1}}^{\infty ... | ## Proof\n\n1. Let \( \left( {K}_{j}\right) \) be a normal exhaustion of \( B \) with \( {K}_{j} = {\widehat{K}}_{j} \) . If \( \mathfrak{z} \in \partial {K}_{j + 1} \) is an arbitrary point, then \( \mathfrak{z} \) does not lie in \( {K}_{j} \subset {\widehat{K}}_{j + 1} \), and therefore not in \( {\widehat{K}}_{j} \... | No |
Theorem 7.1 (Uniqueness of lifting). Let \( \mathfrak{G} = \left( {G,\pi ,{x}_{0}}\right) \) be a domain over \( {\mathbb{C}}^{n} \) with base point, \( Y \) a connected topological space and \( {y}_{0} \in Y \) a point. If \( {\psi }_{1},{\psi }_{2} : Y \rightarrow G \) are continuous mappings with \( {\psi }_{1}\left... | Proof. Let \( M \mathrel{\text{:=}} \left\{ {y \in Y : {\psi }_{1}\left( y\right) = {\psi }_{2}\left( y\right) }\right\} \) . By assumption \( {y}_{0} \in M \), so \( M \neq \varnothing \) . Since \( G \) is a Hausdorff space it follows immediately that \( M \) is closed. Now let \( {y}_{1} \in M \) be chosen arbitrari... | Yes |
Theorem 7.2. Let \( {\mathfrak{G}}_{j} = \left( {{G}_{j},{\pi }_{j},{x}_{j}}\right) \) be domains with base point over \( {\mathbb{C}}^{n} \) for \( j = 1,2 \) . Then there exists at most one continuous fiber-preserving mapping \( \varphi : {G}_{1} \rightarrow {G}_{2} \) with \( \varphi \left( {x}_{1}\right) = {x}_{2}.... | Proof. If there are two continuous mappings \( \varphi ,\psi : {G}_{1} \rightarrow {G}_{2} \) with \( {\pi }_{2} \circ \varphi = \) \( {\pi }_{1} = {\pi }_{2} \circ \psi \) and \( \varphi \left( {x}_{1}\right) = \psi \left( {x}_{1}\right) = {x}_{2} \), then it follows from Theorem 7.1 that \( \varphi = \psi \) . | Yes |
Theorem 7.4. Two domains \( {\mathfrak{G}}_{j} = \left( {{G}_{j},{\pi }_{j},{x}_{j}}\right), j = 1,2 \), are isomorphic if and and if there exists a topological fiber preserving mapping \( \varphi : {G}_{1} \rightarrow {G}_{2} \) with \( \varphi \left( {x}_{1}\right) = {x}_{2} \) . | Proof. \( {\mathfrak{G}}_{1} \simeq {\mathfrak{G}}_{2} \) means that there exist continuous fiber-preserving mappings \( {\varphi }_{1} : {G}_{1} \rightarrow {G}_{2} \) with \( {\varphi }_{1}\left( {x}_{1}\right) = {x}_{2} \) and \( {\varphi }_{2} : {G}_{2} \rightarrow {G}_{1} \) with \( {\varphi }_{2}\left( {x}_{2}\ri... | Yes |
Theorem 7.5. \( {\mathfrak{G}}_{1} < {\mathfrak{G}}_{2} \Leftrightarrow {G}_{1} \subset {G}_{2} \) if \( {\mathfrak{G}}_{1},{\mathfrak{G}}_{2} \) are schlicht domains. | The proof is trivial. | No |
Lemma 1. The equivalence relation \( \sim \) defined on \( X \) by \( {\mathfrak{X}}^{ \star } \) has property \( \left( P\right) \) . Furthermore, the equivalence classes \( {X}_{v} \) in each case contain only points over the same fundamental point \( {\mathfrak{Z}}_{v} \in {\mathbb{C}}^{n} \) . | Proof. The equivalence relation \( \kappa \in K \) will also be denoted by \ | No |
Lemma 2. Let \( \left( {{y}_{1},{\iota }_{1}}\right) ,\left( {{y}_{2},{\iota }_{2}}\right) \in X \) be equivalent, \( {\mathfrak{z}}_{1} \in {\mathbb{C}}^{n} \) the common fundamental point, \( V = V\left( {\mathfrak{z}}_{1}\right) \subset {\mathbb{C}}^{n} \) a connected open neighborhood and \( {U}_{i} = \) \( {U}_{i}... | Proof. Let \( \varphi \) be a path in \( V \) which joins \( {\mathfrak{z}}_{1} \) with \( \mathfrak{z} \) . Then \( {\psi }_{1} \mathrel{\text{:=}} {\left( {\pi }_{{\iota }_{1}} \mid {U}_{{\iota }_{1}}\right) }^{-1} \circ \varphi \) and \( {\psi }_{2} \mathrel{\text{:=}} {\left( {\pi }_{{\iota }_{2}} \mid {U}_{{\iota ... | Yes |
Lemma 3. For all \( {\iota }_{1},{\iota }_{2} \in I \) it is true that: If \( M \subset {G}_{{\iota }_{1}} \) is open, then \( {\varphi }_{{\imath }_{2}}^{-1}\left( {{\varphi }_{{\imath }_{1}}\left( M\right) }\right) \subset {G}_{{\imath }_{2}} \) is open. | Proof. \( {\varphi }_{{\iota }_{2}}^{-1}\left( {{\varphi }_{{\iota }_{1}}\left( M\right) }\right) = \left\{ {x \in G}\right. \) : There is a \( \left. {y \in M\text{with}{\varphi }_{{\iota }_{1}}\left( y\right) = {\varphi }_{{\iota }_{2}}\left( x\right) }\right\} = \) \( \left\{ {x \in {G}_{{\iota }_{2}}\text{: There i... | Yes |
Lemma 4. Let \( {M}_{{\iota }_{1}} \subset {G}_{{\iota }_{1}},{M}_{{\iota }_{2}} \subset {G}_{{\iota }_{2}} \) be arbitrary subsets. Then \( {\varphi }_{{\iota }_{1}}\left( {M}_{{\iota }_{1}}\right) \cap {\varphi }_{{\iota }_{2}}\left( {M}_{{\iota }_{2}}\right) = {\varphi }_{{\iota }_{2}}\left( {{M}_{{\iota }_{2}} \cap... | Proof\n\n1. Let \( y \in {\varphi }_{{\iota }_{1}}\left( {M}_{{\iota }_{1}}\right) \cap {\varphi }_{{\iota }_{2}}\left( {M}_{{\iota }_{2}}\right) \) . Then there are points \( {y}_{1} \in {M}_{{\iota }_{1}},{y}_{2} \in {M}_{{\iota }_{2}} \) with \( {\varphi }_{{\iota }_{1}}\left( {y}_{1}\right) = {\varphi }_{{\iota }_{... | Yes |
1. \( \widetilde{\mathfrak{G}} = \left( {\widetilde{G},\widetilde{\pi },{\widetilde{x}}_{0}}\right) \) is a domain over \( {\mathbb{C}}^{n} \) with base point. | 1a. \( \widetilde{G} \) is a topological space and \( \widetilde{\pi }\left( {\widetilde{x}}_{0}\right) = {\mathfrak{z}}_{0} = {\pi }_{\imath }\left( {x}_{\imath }\right) \).\nb. \( \widetilde{G} \) is connected: If \( y \in \widetilde{G} \), then there is an \( \iota \in I \) and a \( {y}_{\iota } \in {G}_{\iota } \) ... | Yes |
Lemma 1. Let \( \left( {{G}_{1},{\pi }_{1},{y}_{1}}\right) ,\ldots ,\left( {{G}_{t},{\pi }_{t},{y}_{t}}\right) ,\left( {G,\pi, y}\right) \) be domains with base point over \( {\mathbb{C}}^{n} \) and let \( \mathfrak{z} = \pi \left( y\right) \) . If \( {\varphi }_{i} : G \rightarrow {G}_{i} \) are fiber-preserving mappi... | Proof. We can find open neighborhoods \( \widehat{U}\left( y\right) ,\widehat{V}\left( 3\right) \), and \( {\widehat{U}}_{i}\left( {y}_{i}\right) \) such that the mappings \( \pi \mid \widehat{U} : \widehat{U} \rightarrow \widehat{V} \) and \( {\pi }_{i} \mid {\widehat{U}}_{i} : {\widehat{U}}_{i} \rightarrow V \) are t... | Yes |
Theorem 8.1. Under the conditions of Def. 8.2, \( f \mid {G}_{1} \) is holomorphic on \( {G}_{1} \) whenever \( f \) is holomorphic on \( {G}_{2} \) . | Proof. Let \( {y}_{1} \in {G}_{1} \) be arbitrary, \( {y}_{2} \mathrel{\text{:=}} \varphi \left( {y}_{1}\right) \in {G}_{2} \) and \( {\mathfrak{z}}_{1} \mathrel{\text{:=}} {\pi }_{1}\left( {y}_{1}\right) = {\pi }_{2}\left( {y}_{2}\right) \) . By Lemma 1 we obtain a commutative diagram of topological mappings:\n\n![cd0... | Yes |
Theorem 8.3. Let \( {\mathfrak{G}}_{\lambda } = \left( {{G}_{\lambda },{\pi }_{\lambda },{x}_{\lambda }}\right) \) be domains over \( {\mathbb{C}}^{n} \) with \( {\pi }_{\lambda }\left( {x}_{\lambda }\right) = {\mathfrak{z}}_{0} \) , \( \lambda = 1,2 \), and with \( {\mathfrak{G}}_{1} < {\mathfrak{G}}_{2} \) . Let \( f... | Proof. Let \( {F}_{1},{F}_{2} \) be holomorphic extensions of \( f \) to \( {G}_{2} \) . By Lemma 1 there exist neighborhoods \( {U}_{\lambda }\left( {x}_{\lambda }\right) \) such that the restriction of the canonical mapping \( \varphi : {G}_{1} \rightarrow {G}_{2} \) to \( {U}_{1} \) maps the set \( {U}_{1} \) topolo... | Yes |
Theorem 8.4. Let \( \mathfrak{G} = \left( {G,\pi ,{x}_{0}}\right) \) be a domain over \( {\mathbb{C}}^{n},\mathcal{F} \) a non-empty set of functions holomorphic on \( G \) and \( {H}_{\mathcal{F}}\left( \mathfrak{G}\right) = \left( {\widehat{G},\widehat{\pi },\widehat{x}}\right) \) the holomorphic hull of G relative o... | ## Proof\n\n1. Let \ | No |
Theorem 8.5. Let \( {\mathfrak{G}}_{\lambda } = \left( {{G}_{\lambda },{\pi }_{\lambda }, x}\right) ,\lambda = 1,2 \) be domains over \( {\mathbb{C}}^{n} \) with \( {\mathfrak{G}}_{1} \cup \) \( {\mathfrak{G}}_{2} = \left( {\widetilde{G},\widetilde{\pi },\widetilde{x}}\right) \), and \( {f}_{1} : {G}_{1} \rightarrow \m... | Proof. Let \( f \mathrel{\text{:=}} {f}_{1}\left| {G = {f}_{2}}\right| G,\mathcal{F} \mathrel{\text{:=}} \{ f\} \) . Then \( {f}_{1} \) is a holomorphic extension of \( f \) to \( {G}_{1} \) and \( {f}_{2} \) is a holomorphic extension of \( f \) to \( {G}_{2} \) . Therefore by Theorem 8.4: \( {\mathfrak{G}}_{1} < {H}_... | Yes |
Theorem 8.6. Let \( \left( {G,\pi }\right) \) be a domain over \( {\mathbb{C}}^{n},\left( {\mathfrak{P},\mathfrak{H}}\right) \) a generalized Hartogs figure, and \( {x}_{0} \in G \) a point for which \( \mathfrak{H} < \mathfrak{G} \mathrel{\text{:=}} \left( {G,\pi ,{x}_{0}}\right) \) . Then every function \( f \in A\le... | Proof. \( f \mid H \) has a holomorphic extension \( F \in A\left( G\right) \) . Let \( {\mathfrak{G}}_{1} \mathrel{\text{:=}} \mathfrak{G},{\mathfrak{G}}_{2} \mathrel{\text{:=}} \) \( \mathfrak{P},{f}_{1} : = f,{f}_{2} : = F \) . Because \( \mathfrak{H} < {\mathfrak{G}}_{1},\mathfrak{H} < {\mathfrak{G}}_{2} \) and \( ... | Yes |
1. If \( \mathfrak{G} = \left( {G,\pi ,{x}_{0}}\right) \) is a domain over \( {\mathbb{C}}^{n} \) and \( F \) a non-empty set of functions holomorphic on \( G \), then \( {H}_{\mathfrak{F}}\left( \mathfrak{G}\right) \) is a pseudoconvex domain. | The proof is trivial. | No |
Theorem 8.8. (Oka, 1953). If \( \mathfrak{G} \) is pseudoconvex then \( \mathfrak{G} \) is holomorphically convex and is a domain of holomorphy. | The proof is tedious. | No |
Theorem 1.1. \( f \in \mathbb{C}\{ 3\} \) is convergent if and only if there is a \( \mathfrak{t} \in {\mathbb{R}}_{ + }^{n} \) with \( \parallel f{\parallel }_{\mathrm{t}} < \infty \) . | Proof\n\n1. Let \( f\left( 3\right) = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{3}^{v} \) be convergent in the polycylinder \( P \) . Then there exists a \( \mathrm{t} \in {\mathbb{R}}_{ + }^{n} \) with \( {P}_{\mathrm{t}} \subset P \), and therefore \( \parallel f{\parallel }_{\mathrm{t}} < \infty \) .\n\n2. If... | Yes |
Theorem 2.2. If \( B \) is a Banach algebra, \( f \in B \) and \( \parallel 1 - f\parallel < 1 \), then \( f \) is a unit in \( B \) and \( \begin{Vmatrix}{f}^{-1}\end{Vmatrix} \leq 1/\left( {1 - \parallel 1 - f\parallel }\right) \) . | Proof. Let \( g \mathrel{\text{:=}} \mathop{\sum }\limits_{{\lambda = 0}}^{\infty }{\left( 1 - f\right) }^{\lambda },\varepsilon \mathrel{\text{:=}} \parallel 1 - f\parallel \) . Then \( 0 \leq \varepsilon < 1 \) and \( \mathop{\sum }\limits_{{\lambda = 0}}^{\infty }{\varepsilon }^{\lambda } \) dominates \( g \) . Ther... | Yes |
Theorem 2.3 (Weierstrass preparation theorem). If \( g \in B \) satisfies the \( W \) - condition at \( s \), then there exists exactly one normalized polynomial \( \omega \in {B}^{\prime }\left\lbrack {z}_{1}\right\rbrack \) with \( \deg \left( \omega \right) = s \) and one unit \( e \in B \) such that \( g = e \cdot ... | Proof. We apply the Weierstrass formula to \( f = {z}_{1}^{s} \) : There are uniquely determined elements \( q \in B \) and \( r \in {B}^{\prime }\left\lbrack {z}_{1}\right\rbrack \) with \( {z}_{1}^{s} = q \cdot g + r \) and \( \deg \left( r\right) < s \) (we take an \( \varepsilon < \frac{1}{2} \) which satisfies the... | Yes |
Theorem 3.1. For \( {g}_{1},{g}_{2} \in \mathbb{C}\{ 3\} \)\n\n1. \( {g}_{1} \cdot {g}_{2} \) is regular in \( {z}_{1} \) if and only if \( {g}_{1} \) and \( {g}_{2} \) are regular in \( {z}_{1} \) ,\n\n2. \( \operatorname{ord}\left( {{g}_{1} \cdot {g}_{2}}\right) = \operatorname{ord}\left( {g}_{1}\right) + \operatorna... | Proof. \( \left( {{g}_{s} \cdot {g}_{2}}\right) \left( {{z}_{1},0}\right) = {g}_{1}\left( {{z}_{1},0}\right) \cdot {g}_{2}\left( {{z}_{1},0}\right) \) . Since \( \mathbb{C}\left\{ {z}_{1}\right\} \) is an integral domain, (1) holds; (2) is obtained by multiplying out. | No |
Theorem 3.3. \( \sum \) is an abelian group of biholomorphic mappings of \( {\mathbb{C}}^{n} \) onto itself. | Proof. Linear shearings are, of course, holomorphic. It follows from the equalities\n\n\[ \n{\sigma }_{{\mathbb{C}}_{1} + {\mathbb{C}}_{2}} = {\sigma }_{{\mathbb{C}}_{1}} \circ {\sigma }_{{\mathbb{C}}_{2}} \n\]\n\nand\n\n\[ \n{\sigma }_{\mathbb{C}} \circ {\sigma }_{-\mathbb{C}} = {\sigma }_{0} = {\mathrm{{id}}}_{{\math... | Yes |
Theorem 3.4. Let \( g \in {H}_{n}, g \neq 0 \) . Then there exists a shearing \( \sigma \in \sum \) such that \( g \circ \sigma \) is regular in \( {z}_{1} \) . | 1. Let \( g = \mathop{\sum }\limits_{{v = 0}}^{\infty }{a}_{v}{3}^{v} = \mathop{\sum }\limits_{{\lambda = 0}}^{\infty }{p}_{\lambda }\left( 3\right) \) with \( {p}_{\lambda }\left( 3\right) = \mathop{\sum }\limits_{{\left| v\right| = \lambda }}{a}_{v}{3}^{v} \) be the expansion of \( g \) into a series of homogeneous p... | Yes |
Theorem 3.5 (Weierstrass formula for convergent power series). Let \( g \in {H}_{n} \) be regular of order \( s \) in \( {z}_{1} \) . Then for every \( f \in {H}_{n} \) there is exactly one \( q \in {H}_{n} \) and one \( r \in {H}_{n - 1}\left\lbrack {z}_{1}\right\rbrack \) with \( \deg \left( r\right) < s \) such that... | Proof\n\n1. There is a \( \mathfrak{t} \in {\mathbb{R}}_{ + }^{n} \) such that \( f \) and \( g \) lie in \( {B}_{\mathrm{t}} \) and \( {g}_{s} \) is a unit in \( {B}_{\mathrm{t}} \) and \( {\begin{Vmatrix}{z}_{1}^{s} - g{g}_{s}^{-1}\end{Vmatrix}}_{\mathrm{t}} \leq \varepsilon \cdot {t}_{1}^{s} \) for an \( \varepsilon... | Yes |
Theorem 3.6 (Weierstrass preparation theorem for power series). Let \( g \in {H}_{n} \) be regular of order \( s \) in \( {t}_{1} \) . Then there exists a unit \( e \in {H}_{n} \) and a normalized polynomial \( \omega \in {H}_{n - 1}\left\lbrack {z}_{1}\right\rbrack \) of degree \( s \) with\n\n\[ g = e \cdot \omega \] | Proof\n\n1. There exists a \( \mathfrak{t} \in {\mathbb{R}}_{ + }^{n} \) such that \( g \) satisfies the W-condition in \( {B}_{\mathfrak{t}} \) . The existence of the decomposition \ | No |
Theorem 3.7. \( f \in {H}_{n} \) is a unit if and only if \( f\left( 0\right) \neq 0 \) . | Proof\n\n1. If \( f \in {H}_{n} \) is a unit, then there exists a \( g \in {H}_{n} \) with \( f \cdot g = 1 \) . In particular \( f\left( 0\right) \cdot g\left( 0\right) = 1 \), so \( f\left( 0\right) \neq 0 \) .\n\n2. If \( f \in {H}_{n} \) and \( f\left( 0\right) \neq 0 \), then \( g \mathrel{\text{:=}} \left\lbrack ... | Yes |
Theorem 4.1. If \( k \) is a field, then \( k\left\lbrack X\right\rbrack \) is a unique factorization domain. | Proof. The euclidean algorithm is valid in \( k\left\lbrack X\right\rbrack \), hence \( k\left\lbrack X\right\rbrack \) is a principal ideal domain. But every principal ideal domain is a unique factorization domain. (Details are found in van der Waerden.) | Yes |
Theorem 4.2. Let \( I \) be a unique factorization domain, \( Q \mathrel{\text{:=}} Q\left( I\right) \) the quotient field. Furthermore, let \( {\omega }_{1},{\omega }_{2} \in {Q}^{0}\left\lbrack X\right\rbrack ,\omega \in {I}^{0}\left\lbrack X\right\rbrack \), and \( \omega = {\omega }_{1} \cdot {\omega }_{2} \) . The... | Proof. For \( \lambda = 1,2,{\omega }_{\lambda } = {X}^{{s}_{\lambda }} + {A}_{\lambda ,1}{X}^{{s}_{\lambda } - 1} + \cdots + {A}_{\lambda, s} \) with \( {A}_{\lambda ,\mu } \in Q \) . Therefore there exists a \( {d}_{\lambda } \in I \) such that \( {d}_{\lambda } \cdot {\omega }_{\lambda } \in I\left\lbrack X\right\rb... | Yes |
Theorem 4.3 (Gauss' lemma). If \( I \) is a unique factorization domain, then so is \( {I}^{0}\left\lbrack X\right\rbrack \) ; that is, every element of \( {I}^{0}\left\lbrack X\right\rbrack \) is a product of finitely many prime elements of \( {I}^{0}\left\lbrack X\right\rbrack \) . | 1. Let \( \omega \in {I}^{0}\left\lbrack X\right\rbrack \subset Q\left\lbrack X\right\rbrack \) . Then \( \omega = {\omega }_{1} \cdot {\omega }_{2}\cdots {\omega }_{t} \) with \( {\omega }_{\lambda } \in Q\left\lbrack X\right\rbrack \) prime (Theorem 4.1). In each case let \( {a}_{\lambda } \) be the coefficient of th... | Yes |
Theorem 4.4. \( {H}_{n} \) is a unique factorization domain. | Proof. We proceed by induction on \( n \) . For \( n = 0,{H}_{n} = \mathbb{C} \) is a field, and the statement is trivial. Suppose the proposition has been proved for \( n - 1 \) .\n\n1. If \( f \in {H}_{n} \) is not a unit, and \( f = {f}_{1} \cdot {f}_{2} \) a proper decomposition, then \( \operatorname{ord}\left( f\... | Yes |
Theorem 5.1. \( {H}_{n} \) is a local \( \mathbb{C} \) -algebra. | Proof\n\n1. \( \mathfrak{m} = \left\{ {f \in {H}_{n} : f\left( 0\right) = 0}\right\} \) is clearly an ideal in \( {H}_{n} \).\n\n2. For \( f \in {H}_{n}, f = \imath \left( {f\left( 0\right) }\right) + \left( {f - \imath \left( {f\left( 0\right) }\right) }\right) \) with \( f - \imath \left( {f\left( 0\right) }\right) \... | Yes |
Theorem 5.2. \( {H}_{n} \) is a henselian ring. | This theorem follows directly from Hensel's lemma: | No |
Theorem 5.3 (Hensel’s lemma). Let \( \omega \left( {u,\mathfrak{z}}\right) \in {H}_{n}^{0}\left\lbrack u\right\rbrack \) have the decomposition\n\n\( \omega \left( {u,0}\right) = \mathop{\prod }\limits_{{\lambda = 1}}^{\ell }{\left( u - {c}_{\lambda }\right) }^{{s}_{\lambda }} \) into linear factors (with \( {c}_{v} \n... | Proof. We proceed by induction on \( \ell \) . The case \( \ell = 1 \) is trivial; we assume that the theorem has been proved for \( \ell - 1 \) .\n\n1. First assume that \( \omega \left( {0,0}\right) = 0 \) . Without loss of generality we can assume that \( {c}_{1} = 0 \) ; thus \( \omega \left( {u,0}\right) = {u}^{{s... | Yes |
Theorem 5.4. If \( R \) is a noetherian ring and \( q \in \mathbb{N} \), then \( {R}^{q} \) is a noetherian \( R \) -module. | Proof. We proceed by induction on \( q \) .\n\nThe case \( q = 1 \) is trivial. Assume the theorem is proved for \( q - 1 \) . Let \( M \subset {R}^{q} \) be an \( R \) -submodule. Then \( \mathcal{I} \mathrel{\text{:=}} \left\{ {{r}_{1} \in R : }\right. \) There exist \( {r}_{2},\ldots ,{r}_{q} \in R \) with \( \left.... | Yes |
Theorem 6.1. If \( B \subset {\mathbb{C}}^{n} \) is a region and \( M \subset B \) is an analytic set in \( B \), then \( M \) is closed in \( B \) . | Proof. We will show that \( B - M \) is open. If \( {\mathfrak{z}}_{0} \in B - M \), then there exists an open neighborhood \( U = U\left( {\mathfrak{z}}_{0}\right) \subset B \) and functions \( {f}_{1},\ldots ,{f}_{\ell } \in A\left( U\right) \) with \( N\left( {{f}_{1},\ldots ,{f}_{t}}\right) = U \cap M \) such that,... | Yes |
Theorem 6.2. Let \( G \subset {\mathbb{C}}^{n} \) be a domain. Then the ring \( A\left( G\right) \) of functions holomorphic on \( G \) is an integral domain. | Proof. We need only to show that \( A\left( G\right) \) has no zero divisors: Suppose \( {f}_{1},{f}_{2} \) are two elements of \( A\left( G\right) \) with \( {f}_{1} \neq 0 \) and \( {f}_{1} \cdot {f}_{2} = 0 \) . Then there is a \( {}_{30} \in G \) with \( {f}_{1}\left( {\mathfrak{z}}_{0}\right) \neq 0 \), and hence ... | Yes |
Theorem 6.3. Let \( I \) be an integral domain, \( Q = Q\left( I\right) \) the quotient field of \( I \) . \( {I}^{0}\left\lbrack X\right\rbrack \) is a unique factorization domain if \( I \) satisfies the condition:\n\n\[ \n{\omega }_{1},{\omega }_{2} \in {Q}^{0}\left\lbrack X\right\rbrack \text{ and }{\omega }_{1} \c... | Proof. Although Gauss’ lemma assumed that \( I \) was a unique factorization domain, the proof only used the above property of \( I \), which is satisfied for every unique factorization domain. | Yes |
Theorem 6.4. If \( {\omega }_{1},{\omega }_{2} \) are elements of \( {Q}^{0}\left\lbrack u\right\rbrack \) with \( {\omega }_{1} \cdot {\omega }_{2} \in {A}^{0}\left\lbrack u\right\rbrack \) then \( {\omega }_{1},{\omega }_{2} \in {A}^{0}\left\lbrack u\right\rbrack \) . | Proof\n\n1. If \( \omega \in {Q}^{0}\left\lbrack u\right\rbrack \), then \( \omega \) has the form \( \omega = {u}^{s} + {A}_{1}{u}^{s - 1} + \cdots + {A}_{s} \) with \( {A}_{i} \in Q \) for \( i = 1,\ldots, s \) . Let \( {\left( \omega \right) }_{3} \mathrel{\text{:=}} {u}^{s} + {\left( {A}_{1}\right) }_{3}{u}^{s - 1}... | Yes |
Theorem 6.5. Let \( G \subset {\mathbb{C}}^{n} \) be a domain, \( A = A\left( G\right) \) . Then \( {A}^{0}\left\lbrack u\right\rbrack \) is a unique factorization domain. | The proof follows directly from Theorems 6.3 and 6.4. | No |
Theorem 6.7. Let \( \omega \in {A}^{0}\left\lbrack u\right\rbrack \) be a pseudopolynomial without multiple factors. Then there are elements \( {q}_{1},{q}_{2} \in A\left\lbrack u\right\rbrack \) such that \( h \mathrel{\text{:=}} {q}_{1} \cdot \omega + {q}_{2} \cdot D\left( \omega \right) \) lies in \( A \) and does n... | Proof. We have shown above that \( \gcd \left( {\omega, D\left( \omega \right) }\right) = 1 \), so there exist elements \( {p}_{1},{p}_{2} \in Q\left\lbrack u\right\rbrack \) with \( {p}_{1}\omega + {p}_{2} \cdot D\left( \omega \right) = 1 \) . If we multiply the equation by an appropriate factor \( h \in A \) (with \(... | Yes |
Theorem 6.10. Let \( f\left( X\right) = \mathop{\prod }\limits_{{\rho = 1}}^{s}\left( {X - {X}_{\rho }}\right) \in \mathbb{C}\left\lbrack X\right\rbrack \) . \( f \) has a multiple root if and only if \( \Delta \left( f\right) = 0 \) . | Proof\n\n\[ f\left( X\right) = \left( {X - {X}_{1}}\right) \left( {X - {X}_{2}}\right) \cdots \left( {X - {X}_{s}}\right) \]\n\n\[ = {X}^{s} - \left( {{X}_{1} + \cdots + {X}_{s}}\right) {X}^{s - 1} + \left( {{X}_{1}{X}_{2} + \cdots }\right) {X}^{s - 2} + \cdots \]\n\n\[ + {\left( -1\right) }^{s}{X}_{1} \cdot {X}_{2}\cd... | Yes |
Theorem 6.11. Let \( G \subset {\mathbb{C}}^{n} \) be a domain, \( \omega \left( {u,\mathfrak{z}}\right) \in {A}^{0}\left\lbrack u\right\rbrack \) a pseudopolynomial. \( {\Delta }_{\omega } \) does not vanish identically if and only if \( \omega \) has no multiple factors. | 1. Let \( \omega = {\omega }_{1}^{2} \cdot \widetilde{\omega } \) with \( \deg \left( {\omega }_{1}\right) > 0 \) . If \( \mathfrak{z} \in G \), then we can decompose \( {\omega }_{1}\left( {u,3}\right) \) into linear factors,\n\n\[{\omega }_{1}\left( {u,3}\right) = \left( {u - {c}_{1}}\right) \cdots \left( {u - {c}_{t... | Yes |
Theorem 6.14. Let \( G \subset {\mathbb{C}}^{n} \) be a domain. Then:\n\n1. \( \varnothing \) and \( G \) are analytic subsets of \( G \) .\n\n2. If \( {M}_{1},\ldots ,{M}_{\ell } \) are analytic in \( G \), so is \( \mathop{\bigcup }\limits_{{i = 1}}^{\ell }{M}_{i} \) .\n\n3. If \( {M}_{1},\ldots ,{M}_{\ell } \) are a... | Proof\n\n1. \( \varnothing = \{ 3 \in G : 1 = 0\}, G = \{ 3 \in G : 0 = 0\} \) .\n\n2. Let \( {\mathfrak{z}}_{0} \in M \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{i = 1}}^{l}{M}_{i} \) . Then there exists an open neighborhood\n\n\( U\left( {\mathfrak{z}}_{0}\right) \subset G \) and holomorphic functions \( {f}_{i, j... | No |
Theorem 6.17. Let \( G \subset {\mathbb{C}}^{n} \) be a domain, \( M \) analytic. Then there is a countable system \( \left( {M}_{i}\right) \) of irreducible analytic subsets of \( G \) such that\n\n1. \( \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{M}_{i} = M \) .\n\n2. The system \( {\left( {M}_{i}\right) }_{i \in \... | The proof is lengthy and requires the help of sheaf theory. | No |
Theorem 6.19. Let \( G \subset {\mathbb{C}}^{n} \) be a domain, let \( {f}_{1},\ldots ,{f}_{n - k} \) be holomorphic functions in \( G, M \mathrel{\text{:=}} \left\{ {3 \in G : {f}_{1}\left( 3\right) = \cdots = {f}_{n - k}\left( 3\right) = 0}\right\} ,{M}^{\prime } \subset M \) an irreducible component. Then \( {\dim }... | Proof. \( G \) itself is an irreducible analytic set. Then, by Theorem 6.18, \( {\dim }_{\mathbb{C}}\left( \left\{ {3 \in G : {f}_{1}\left( 3\right) = 0}\right\} \right) \geq n - 1 \), and the set \( {M}_{1} = \left\{ {3 \in G : {f}_{1}\left( 3\right) = 0}\right\} \) is pure dimensional. Let \( {M}_{1} = \mathop{\bigcu... | Yes |
Theorem 1.1. Let \( \left( {\mathcal{S},\pi }\right) \) be a sheaf over \( B, W \subset B \) open and \( s \in \Gamma \left( {W,\mathcal{S}}\right) \) . Then \( \pi : s\left( W\right) \rightarrow W \) is topological and \( s = {\left( \pi \mid s\left( W\right) \right) }^{-1} \) . | Proof. By definition \( \pi \circ s = {\operatorname{id}}_{W} \) . For \( \mathfrak{z} \in W \)\n\n\[ s \circ \left( {\pi \mid s\left( W\right) }\right) \left( {s\left( 3\right) }\right) = s \circ \pi \circ s\left( 3\right) = s\left( 3\right) . \]\n\nTherefore \( s \circ \left( {\pi \mid s\left( W\right) }\right) = {\o... | Yes |
Theorem 1.2. Let \( \left( {\mathcal{S},\pi }\right) \) be a sheaf over \( B, W \subset B \) open and \( s : W \rightarrow \mathcal{S} \) a mapping with \( \pi \circ s = {\operatorname{id}}_{W} \) . Then \( s \in \Gamma \left( {W,\mathcal{S}}\right) \) if and only if \( s\left( W\right) \) is open in \( \mathcal{S} \) ... | ## Proof\n\n1. Let \( s \) be continuous, \( {\sigma }_{0} \in s\left( W\right) \), and \( {\mathfrak{z}}_{0} \mathrel{\text{:=}} \pi \left( {\sigma }_{0}\right) \) . Then \( s\left( {\mathfrak{z}}_{0}\right) = {\sigma }_{0} \) and there are open neighborhoods \( V\left( {\mathfrak{z}}_{0}\right) \subset W \) and \( U\... | Yes |
Theorem 1.3. Let \( \left( {\mathcal{S},\pi }\right) \) be a sheaf over \( B,\sigma \in \mathcal{S} \). Then there exists an open set \( V \subset B \) and a section \( s \in \Gamma \left( {V,\mathcal{S}}\right) \) with \( \sigma \in s\left( V\right) \). | Proof. Let \( \mathfrak{z} \mathrel{\text{:=}} \pi \left( \sigma \right) \). Let open neighborhoods \( U\left( \sigma \right) \subset \mathcal{S} \) and \( V\left( \mathfrak{z}\right) \subset B \) be chosen so that \( \pi \mid U : U \rightarrow V \) is topological. Then \( V \) and \( s \mathrel{\text{:=}} {\left( \pi ... | Yes |
Theorem 1.4. Let \( \left( {\mathcal{S},\pi }\right) \) be a sheaf over \( B, W \subset B \) open. If for two sections \( {s}_{1},{s}_{2} \in \Gamma \left( {W,\mathcal{S}}\right) \) there is a point \( \mathfrak{z} \in W \) with \( {s}_{1}\left( \mathfrak{z}\right) = {s}_{2}\left( \mathfrak{z}\right) \), then there is ... | Proof. Let \( \sigma \mathrel{\text{:=}} {s}_{1}\left( 3\right) = {s}_{2}\left( 3\right) \) . Then \( U \mathrel{\text{:=}} {s}_{1}\left( W\right) \cap {s}_{2}\left( W\right) \) is an open neighborhood of \( \sigma \) and \( \pi \mid U : U \rightarrow V \mathrel{\text{:=}} \pi \left( U\right) \subset W \) is a topologi... | Yes |
Theorem 1.5. Let \( \left( {{\mathcal{S}}_{1},{\pi }_{1}}\right) ,\left( {{\mathcal{S}}_{2},{\pi }_{2}}\right) \) be sheaves over \( B,\varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) a stalk preserving mapping. Then the following statements are equivalent:\n\n1. \( \varphi \) is a sheaf morphism.\n\n2. For... | ## Proof\n\na. If \( \varphi \) is continuous, \( W \subset B \) open and \( s \in \Gamma \left( {W,{\mathcal{S}}_{1}}\right) \) then \( \varphi \circ s \) is also continuous. Moreover: \( {\pi }_{2} \circ \left( {\varphi \circ s}\right) = \left( {{\pi }_{2} \circ \varphi }\right) \circ s = {\pi }_{1} \circ s = {\mathr... | Yes |
Theorem 1.7. If \( \mathcal{S} \) is a sheaf over \( B \), then the sheaf defined by the canonical pre-sheaf \( \left\{ {\Gamma \left( {W,\mathcal{S}}\right) ,{r}_{V}^{W}}\right\} \) is canonically isomorphic to \( \mathcal{S} \) . | Proof. Let \( \left( {\widehat{\mathcal{S}},\widehat{\pi }}\right) \) be the sheaf defined by the canonical pre-sheaf.\n\na. If \( \left( {{W}_{1},{s}_{1}}\right) \sim \left( {{W}_{2},{s}_{2}}\right) \) then \( {s}_{1}\left( 3\right) = {s}_{2}\left( 3\right) \) and the converse also holds. Therefore \( \varphi : {\left... | Yes |
Theorem 1.8. Every sheaf morphism is an open mapping. | Proof. Let \( \varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) be a sheaf morphism. Since \( {\mathcal{S}}_{1} \) is canonically isomorphic to the sheaf \( {\mathcal{P}}_{1} \) defined by the canonical pre-sheaf \( \left\{ {\Gamma \left( {W,{\mathcal{S}}_{1}}\right) ,{r}_{V}^{W}}\right\} \) , the sets \( s\... | Yes |
Theorem 1.9. Let \( \left( {{\mathcal{S}}_{1},{\pi }_{1}}\right) ,\ldots ,\left( {{\mathcal{S}}_{t},{\pi }_{t}}\right) \) be sheaves over \( B \), and let \( \mathcal{S} = \) \( {\mathcal{S}}_{1} \oplus \cdots \oplus {\mathcal{S}}_{t} \) be their Whitney sum. Then for every \( \mathfrak{z} \in B \) there is a bijection... | ## Proof\n\na. Let \( {s}_{\lambda } = \left( {{s}_{1}^{\left( \lambda \right) },\ldots ,{s}_{\ell }^{\left( \lambda \right) }}\right) \in \Gamma \left( {{W}_{\lambda },\mathcal{S}}\right) \) for \( \lambda = 1,2,\mathfrak{z} \in {W}_{1} \cap {W}_{2} \) .\n\n\( \left( {{W}_{1},{s}_{1}}\right) \sim \left( {{W}_{2},{s}_{... | Yes |
Theorem 1.10. Let \( \left( {{\mathcal{S}}_{1},{\pi }_{1}}\right) ,\ldots ,\left( {{\mathcal{S}}_{\ell },{\pi }_{\ell }}\right) \) be sheaves over B. Then the canonical projections \( {p}_{i} : {\mathcal{S}}_{1} \oplus \cdots \oplus {\mathcal{S}}_{\ell } \mapsto {\mathcal{S}}_{i} \) (with \( \left. {{p}_{i}\left( {{\si... | Proof. The mappings \( {p}_{i} \) are stalk preserving, by definition. If \( \sigma \in \left( {{\mathcal{S}}_{1} \oplus \cdots }\right. \) \( {\left( \oplus {\mathcal{S}}_{\ell }\right) }_{\mathfrak{z}} = {\left( {\mathcal{S}}_{1}\right) }_{\mathfrak{z}} \times \cdots \times {\left( {\mathcal{S}}_{\ell }\right) }_{\ma... | No |
Theorem 2.1. Let \( {\mathcal{S}}_{1},\ldots ,{\mathcal{S}}_{\ell } \) , \( \mathcal{S} \) be sheaves over \( B \) given by pre-sheaves \( \left\{ {{M}_{W}^{\left( i\right) },{r}_{iV}^{W}}\right\}, i = 1,\ldots ,\ell \) and \( \left\{ {{M}_{W},{r}_{V}^{W}}\right\} \) . Suppose that for every open set \( W \subset B \) ... | Proof\n\n1. Let \( W,\widetilde{W} \) be open in \( B,\mathfrak{z} \in W \cap \widetilde{W} \) and \( \left( {W,{s}_{i}}\right) \sim \left( {\widetilde{W},{\widetilde{s}}_{i}}\right) \) for \( i = 1,\ldots ,\ell \) . Then there exists a neighborhood \( V\left( \mathfrak{z}\right) \subset W \cap \widetilde{W} \) with \(... | Yes |
Theorem 2.2. Let \( \\left\\{ {{M}_{W},{r}_{V}^{W}}\\right\\} \) be a pre-sheaf of \( \\mathbb{C} \) -algebras, \( \\mathcal{S} \) the corresponding sheaf. Then \( \\mathcal{S} \) is a sheaf of \( \\mathbb{C} \) -algebras and for every open set \( W \\subset B \) \( r : {M}_{W} \\rightarrow \\Gamma \\left( {W,\\mathcal... | Proof. For \( W \\subset B \) let \( {\\varphi }_{W} : {M}_{W} \\times {M}_{W} \\rightarrow {M}_{W} \) be defined by \( {\\varphi }_{W}\\left( {{s}_{1},{s}_{2}}\\right) \\mathrel{\\text{:=}} \) \( {s}_{1} + {s}_{2} \) . Then\n\n\\[ \n{r}_{V}^{W}{\\varphi }_{W}\\left( {{s}_{1},{s}_{2}}\\right) = {r}_{V}^{W}\\left( {{s}_... | Yes |
Theorem 2.4. Let \( \mathcal{I} \subset \mathcal{O} \) be an ideal sheaf over \( B \) . Then \( N\left( \mathcal{I}\right) = \left\{ {\mathfrak{z} \in B : }\right. \) For all \( \left. {{f}_{3} \in {\mathcal{I}}_{3}, f\left( 3\right) = 0}\right\} \) . | Proof\n\n1. Let \( \mathfrak{z} \in N\left( \mathcal{I}\right) ,{f}_{\mathfrak{z}} \in {\mathcal{I}}_{\mathfrak{z}} \), but \( f\left( \mathfrak{z}\right) \neq 0 \) . Then on a neighborhood of \( W\left( \mathfrak{z}\right) \) , \( 1/f \) is holomorphic and \( {\mathbf{l}}_{3} = {r1}\left( 3\right) = r\left( {1/f}\righ... | Yes |
Theorem 2.5. Let \( {\mathcal{S}}_{1},\ldots ,{\mathcal{S}}_{\ell } \) be analytic sheaves over \( B \) . Then \( \mathcal{S} \mathrel{\text{:=}} \) \( {\mathcal{S}}_{1} \oplus \cdots \oplus {\mathcal{S}}_{t} \) is analytic. | Proof. Clearly \( {\mathcal{S}}_{3} = {\left( {\mathcal{S}}_{1}\right) }_{3} \times \cdots \times {\left( {\mathcal{S}}_{\ell }\right) }_{3} \) is always an \( {\mathcal{O}}_{3} \) -module. It remains to show that the operations are continuous. We only carry out the proof for addition:\n\nLet\n\n\[ \left( {s,\widetilde... | Yes |
Theorem 2.6. Let \( \mathcal{S} \) be an analytic sheaf over \( B,{\mathcal{S}}^{ \star } \subset \mathcal{S} \) an analytic sub-sheaf, \( \mathcal{Q} = \mathcal{S}/{\mathcal{S}}^{ \star } \) the quotient sheaf. Then for every \( \mathfrak{z} \in B \) there is an isomorphism \( \psi : {\mathcal{Q}}_{3} \rightarrow {\ma... | Proof\n\n1. \( \left( {{W}_{1},\left\langle {s}_{1}\right\rangle }\right) \sim \left( {{W}_{2},\left\langle {s}_{2}\right\rangle }\right) \) if and only if there is a neighborhood \( V\left( 3\right) \subset \) \( {W}_{1} \cap {W}_{2} \) such that\n\n\[ \left\langle {{s}_{1} \mid V}\right\rangle = {r}_{V}^{{W}_{1}}\lef... | Yes |
Theorem 3.1. If \( \varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) is an analytic sheaf morphism, then \( \operatorname{Im}\varphi \) and \( \operatorname{Ker}\varphi \) are analytic sheaves. | ## Proof\n\n1. Since every sheaf morphism is an open mapping, \( \operatorname{Im}\varphi = \varphi \left( {\mathcal{S}}_{1}\right) \subset {\mathcal{S}}_{2} \) is open in \( {\mathcal{S}}_{2} \), and is therefore a subsheaf. Since \( {\left( \operatorname{Im}\varphi \right) }_{3} = \varphi \left( {\left( {\mathcal{S}}... | Yes |
Theorem 3.2. If \( \varphi : {\mathcal{S}}_{1} \rightarrow {\mathcal{S}}_{2} \) is an analytic sheaf morphism, then \( {\mathcal{S}}_{1}/\operatorname{Ker}\varphi \simeq \) \( \operatorname{Im}\varphi \) . | Proof. \( \bar{\varphi }\left( \bar{\sigma }\right) \mathrel{\text{:=}} \varphi \left( \sigma \right) \) defines a stalk preserving bijective mapping \( \bar{\varphi } : {\mathcal{S}}_{1}/ \) Ker \( \varphi \rightarrow \operatorname{Im}\varphi \) which induces a \( {\mathcal{O}}_{3} \) -module isomorphism in every stal... | Yes |
Theorem 4.1. If \( \mathcal{S} \) is finitely generated, then \( \operatorname{Supp}\left( \mathcal{S}\right) \) is closed in \( B \) . | Proof. We show that \( B - \operatorname{Supp}\left( \mathcal{S}\right) \) is open in \( {\mathbb{C}}^{n} \) . Let \( {\mathfrak{z}}_{0} \in B - \operatorname{Supp}\left( \mathcal{S}\right) \) be chosen arbitrarily, and let \( W\left( {\mathfrak{z}}_{0}\right) \subset B \) be an open neighborhood over which a sheaf epi... | Yes |
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