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A special case of Hill's equation is Mathieu's equation,\n\n\[ \n{y}^{\prime \prime } + \left( {\alpha + \beta \cos t}\right) y = 0,\n\]\n\nwhere \( \alpha \) and \( \beta \) are real parameters. We will assume that \( \beta \neq 0 \) . Note that the Floquet multipliers of Mathieu’s equation depend on \( \alpha \) and ... | Let\n\n\[ \n\gamma \left( {\alpha ,\beta }\right) \mathrel{\text{:=}} {\mu }_{1}\left( {\alpha ,\beta }\right) + {\mu }_{2}\left( {\alpha ,\beta }\right)\n\]\n\nThen the Floquet multipliers of Mathieu's equation satisfy the quadratic equation\n\n\[ \n{\mu }^{2} - \gamma \left( {\alpha ,\beta }\right) \mu + 1 = 0.\n\]\n... | Yes |
We consider the ring \( \mathbb{Z}\left\lbrack i\right\rbrack \) with \( {i}^{2} = - 1 \), and try to imagine its prime spectrum \( \operatorname{Spec}\left( {\mathbb{Z}\left\lbrack i\right\rbrack }\right) \), using the inclusion map \( \varphi : \mathbb{Z} \rightarrow \mathbb{Z}\left\lbrack i\right\rbrack \) . This de... | We write \( \omega = \left( 0\right) \in \operatorname{Spec}\mathbb{Z} \) and \( {\omega }^{\prime } = \left( 0\right) \in \operatorname{Spec}\left( {\mathbb{Z}\left\lbrack i\right\rbrack }\right) \) for the points of \( \operatorname{Spec}\mathbb{Z} \) and \( \operatorname{Spec}\left( {\mathbb{Z}\left\lbrack i\right\r... | Yes |
Recall that a subset \( S \subset A \) is a multiplicative set if it contains 1 and is closed under multiplication. For every multiplicative set, we can construct a ring of fractions \( {A}_{S} \) consisting of pairs \( \left( {a, s}\right) \) with \( a \in A \) and \( s \in S \), identified according to the rule\n\n\[... | Algebraic operations are defined by the rules\n\n\[ \left( {a, s}\right) + \left( {{a}^{\prime },{s}^{\prime }}\right) = \left( {a{s}^{\prime } + {a}^{\prime }s, s{s}^{\prime }}\right) ,\]\n\n\[ \left( {a, s}\right) \left( {{a}^{\prime },{s}^{\prime }}\right) = \left( {a{a}^{\prime }, s{s}^{\prime }}\right) . \] | Yes |
If \( A \) is the ring of integers of an algebraic number field \( K \), for example, \( A = \mathbb{Z}, K = \mathbb{Q} \), then \( \operatorname{Spec}A \) consists of maximal ideals together with (0). | For \( x = (0) \), we have \( {\mathcal{O}}_{x} = K \) and hence \( x \) is regular, with 0 -dimensional tangent space. If \( x = \mathfrak{p} \neq \left( 0\right) \) then it is known that \( {\mathcal{O}}_{x} \) is a principal ideal domain. Hence these points are also regular, with 1-dimensional tangent spaces. | No |
To find points that are not regular, consider the ring \( A = \mathbb{Z}\left\lbrack {mi}\right\rbrack = \) \( \mathbb{Z}\left\lbrack y\right\rbrack /\left( {{y}^{2} + {m}^{2}}\right) = \mathbb{Z} + \mathbb{Z}{mi} \), where \( m > 1 \) is an integer and \( {i}^{2} = - 1 \) . The inclusion \( \varphi : A \hookrightarrow... | If we restrict to prime ideals coprime to \( m \) then this is a one-to-one correspondence, and it is easy to check that the local rings of corresponding prime ideals are equal. Hence a point \( x \in \operatorname{Spec}{A}^{\prime } \) is not regular only if the prime ideal divides \( m \) . Prime ideals of \( {A}^{\p... | Yes |
The simplest example of a decomposition of \( \operatorname{Spec}A \) into irreducible components is the case of a ring \( A \) that is a direct sum of a number of rings having no zerodivisors:\n\n\[ A = {A}_{1} \oplus \cdots \oplus {A}_{r} \] | In this case, one checks easily that \( \operatorname{Spec}A \) is a disjoint union of irreducible components \( \operatorname{Spec}\left( {A}_{i}\right) \) . | Yes |
To consider a slightly less trivial example, take the group ring \( \mathbb{Z}\left\lbrack \sigma \right\rbrack \) of the cyclic group of order 2 : | \[ A = \mathbb{Z}\left\lbrack \sigma \right\rbrack = \mathbb{Z} + \mathbb{Z}\sigma ,\;\text{ with }{\sigma }^{2} = 1. \] The nilradical of \( A \) equals (0), but this is not a prime ideal, since \( A \) has zerodivisors: \( \left( {1 + \sigma }\right) \left( {1 - \sigma }\right) = 0 \) . Hence \[ \operatorname{Spec}A ... | Yes |
Example 5.10 To give an example of a ring of bigger dimension, consider the case \( A = \mathbb{Z}\left\lbrack T\right\rbrack \) . Since we expect that the reader has already worked out the structure of Spec \( \mathbb{Z}\left\lbrack T\right\rbrack \) as an exercise in Section 1.1, we assume it known. It is very simple... | \[ \left( {p, f\left( T\right) }\right) \supset \left( {g\left( t\right) }\right) \supset \left( 0\right) \text{ or }\left( {p, f\left( T\right) }\right) \supset \left( p\right) \supset \left( 0\right) . \] Thus \( \dim \mathbb{Z}\left\lbrack T\right\rbrack = 2 \), in agreement with Proposition C. | No |
We show that a homomorphism \( \lambda : A \rightarrow B \) defines a morphism \( \varphi : \operatorname{Spec}B \rightarrow \operatorname{Spec}A \) . | We first set \( \varphi = {}^{a}\lambda \) . For \( U = D\left( f\right) \subset \operatorname{Spec}A \) we have \( {\varphi }^{-1}\left( U\right) = D\left( {\lambda \left( f\right) }\right) \) . Sending \( a/{f}^{n} \mapsto \lambda \left( a\right) /\lambda {\left( f\right) }^{n} \) defines a homomorphism \( {\psi }_{U... | No |
Theorem 5.2 Every local morphism \( \varphi : \operatorname{Spec}B \rightarrow \operatorname{Spec}A \) can be expressed uniquely in the form \( \varphi = {}^{a}\lambda \), where \( \lambda : A \rightarrow B \) is a homomorphism. | Proof There is, of course, only one candidate for \( \lambda \), namely \( {\psi }_{U} \), where \( U = \operatorname{Spec}A \) . We must prove that \( \varphi = {}^{a}\lambda \) . First we need to check this equality on the set Spec \( B \) . This follows at once from the fact that \( \varphi \) is local. Indeed, the ... | Yes |
We explain how the notion of quasiprojective variety fits into the framework of schemes. We start from the case of an affine variety \( X \) over an algebraically closed field \( k \) . The scheme \( \operatorname{Spec}\left( {k\left\lbrack X\right\rbrack }\right) \) defined in Example 5.18 is not equal to \( X \) even... | We now consider an arbitrary quasiprojective variety \( X \) over \( k \), and associate with \( X \) in a similar way a \( k \) -scheme \( \widetilde{X} \) . As the set \( \widetilde{X} \) we take the set of irreducible subvarieties of \( X \) . Let \( U \subset X \) be an open subset and \( \widetilde{U} \) the set o... | No |
Can we recover \( X \) from an open cover \( X = \bigcup {X}_{\alpha } \) where each \( {X}_{\alpha } \) is an affine open set? | We consider this question in somewhat greater generality, without presupposing the open sets \( {U}_{\alpha } \) to be affine.\n\nWe note first that any open set \( U \subset X \) is a scheme. This follows from that fact that each point has an affine neighbourhood \( V \), and the open sets \( D\left( f\right) \subset ... | No |
Let \( X \) be a scheme over an algebraically closed field \( k \) . Any morphism of \( k \) -schemes \( \varphi : \operatorname{Spec}k \rightarrow X \) takes the closed point \( o \in \operatorname{Spec}k \) to a closed point \( x = \varphi \left( o\right) \in X \), with \( k\left( x\right) = k \) . Conversely, any po... | \[ \text{\{morphisms}\operatorname{Spec}k \rightarrow X\} \overset{ \sim }{ \rightarrow }{X}_{\max }\text{,} \] where \( {X}_{\max } \) is the set of closed points of a scheme \( X \) of finite type over \( k \), obviously commutes with morphisms \( X \rightarrow {X}^{\prime } \), that it, is a functor. | Yes |
Proposition If \( X \) is a \( k \) -scheme and \( x \in X \), the set \( {\mathcal{M}}_{x}\left( {\operatorname{Spec}D, X}\right) \) is in one-to-one correspondence with the tangent space \( {\Theta }_{X, x} \) to \( X \) at \( x \) . | It is easy to check that the correspondence \( {\mathcal{M}}_{x}\left( {\operatorname{Spec}D, X}\right) \overset{ \sim }{ \rightarrow }{\Theta }_{X, x} \) commutes with morphisms \( f : X \rightarrow {X}^{\prime } \) and their differentials \( {\mathrm{d}}_{x} : {\Theta }_{X, x} \rightarrow {\Theta }_{{X}^{\prime }, f\... | No |
Proposition 5.1 If the affine scheme \( \operatorname{Spec}A \) is Noetherian then \( A \) is a Noetherian ring. | Proof By assumption there exists a cover (5.17) such that the \( {A}_{i} \) are Noetherian rings. Let \( {\mathfrak{a}}_{1} \subset {\mathfrak{a}}_{2} \subset \cdots \) be a chain of ideals of \( A \) . As we showed in Section 2.2, \( A = \) \( \mathcal{O}\left( X\right) \), where \( \mathcal{O} \) is the structure she... | Yes |
Proposition 5.2 If an affine scheme \( \operatorname{Spec}A \) is of finite type over a ring \( B \) then \( A \) is an algebra of finite type over \( B \) . | Proof By assumption there exist a cover (5.17) such that the algebras \( {A}_{i} \) are of finite type over \( B \) . Since each \( \operatorname{Spec}{A}_{i} \) is compact, it has a finite cover by principal open sets \( D\left( f\right) \) with \( f \in A \) . The corresponding algebras \( {\left( {A}_{i}\right) }_{f... | Yes |
Proposition 5.3 An affine scheme \( X \) over a ring \( B \) is separated, and \( \Delta : X \rightarrow X{ \times }_{B} \) \( X \) is a closed embedding. | Proof Let \( X = \operatorname{Spec}A \), where \( A \) is a \( B \) -algebra. Since by definition \( X{ \times }_{B}X = \) \( \operatorname{Spec}\left( {A{ \otimes }_{B}A}\right) \), the morphism \( \Delta : X \rightarrow X{ \times }_{B}X \) is associated with the homomorphism \( \lambda : A{ \otimes }_{B}A \rightarro... | Yes |
Proposition 5.4 Let \( X = \bigcup {U}_{\alpha } \) be an affine cover that satisfies the conditions: (1) all the sets \( {U}_{\alpha } \cap {U}_{\beta } \) are affine, and (2) the ring \( {\mathcal{O}}_{X}\left( {{U}_{\alpha } \cap {U}_{\beta }}\right) \) is generated by its subrings \( {\rho }_{{U}_{\alpha } \cap {U}... | Proof Let \( u, v : X{ \times }_{B}X \rightarrow X \) be the standard maps of the product. Then\n\n\[{\Delta }^{-1}\left( {{u}^{-1}\left( {U}_{\alpha }\right) \cap {v}^{-1}\left( {U}_{\beta }\right) }\right) = {\Delta }^{-1}\left( {{u}^{-1}\left( {U}_{\alpha }\right) }\right) \cap {\Delta }^{-1}\left( {{v}^{-1}\left( {... | Yes |
Proposition 5.5 In a separated scheme, the intersection of two affine open sets is affine. | Proof Indeed,\n\n\[ U \cap V = {\Delta }^{-1}\left( {U \times V}\right) \]\n\nIf \( U \) and \( V \) are affine then so is \( U \times V \), and if \( X \) is separated then \( \Delta \) is a closed embedding, and hence \( U \cap V \) is a closed subscheme of an affine scheme. This is affine by Proposition of Section 3... | Yes |
Theorem 6.1 If \( X \) is a nonsingular irreducible variety and \( \varphi : X \rightarrow Y \) a rational map to a complete variety \( Y \), the locus of indeterminacy of \( \varphi \) has codimension \( \geq 2 \) . | Proof Let \( V \subset X \) be the set of points at which \( \varphi \) is defined, \( {\Gamma }_{\varphi } \subset V \times Y \) the graph of the morphism \( \varphi : V \rightarrow Y \) and \( Z \) its closure in \( X \times Y \) . The image of \( Z \) under the projection \( p : X \times Y \rightarrow X \) is closed... | Yes |
Let \( V \) and \( W \) be two vector spaces of dimension \( m \) and \( n \). We determine the general form of a morphism \( f : X \times V \rightarrow X \times W \) between two trivial families. | The composite of the isomorphism \( X \rightarrow X \times {e}_{i} \) and the embedding \( X \times {e}_{i} \rightarrow X \times V \) defines a morphism \( {\varphi }_{i} : X \rightarrow X \times V \). Set \( {a}_{ij} = {\varphi }_{i}^{ * }\left( {{f}^{ * }\left( {y}_{j}\right) }\right) \in {\mathcal{O}}_{X}\left( X\ri... | Yes |
Example 6.2 Let \( V \) be an \( \left( {n + 1}\right) \) -dimensional vector space and \( {\mathbb{P}}^{n} \) the vector space of lines \( l \subset V \) through 0 . Write \( {l}_{x} \) for the line corresponding to a point \( x \in {\mathbb{P}}^{n} \) . Consider the subset \( E \subset {\mathbb{P}}^{n} \times V \) of... | In \( V \), we introduce a coordinate system \( \left( {{x}_{0},\ldots ,{x}_{n}}\right) \) . The restriction of \( E \) to the open set \( {U}_{\alpha } \) given by \( {x}_{\alpha } \neq 0 \) consists of points\n\n\[ \xi = \left( {{t}_{1},\ldots ,{t}_{n};{y}_{0},\ldots ,{y}_{n}}\right) \;\text{ such that }\;{y}_{i} = {... | Yes |
Example 6.4 Let \( X = \operatorname{Grass}\left( {r, n}\right) \) be the Grassmannian of \( r \) -dimensional vector subspaces of an \( n \) -dimensional vector space with basis \( {e}_{1},\ldots ,{e}_{n} \) (Example 1.24 of Section 4.1, Chapter 1). Consider in \( X \times V \) the subvariety \( E \) consisting of poi... | Consider the open subset \( {U}_{{k}_{1}\ldots {k}_{r}} \subset \operatorname{Grass}\left( {r, n}\right) \) defined by \( {p}_{{k}_{1}\ldots {k}_{r}} \neq 0 \) ; then for \( x \in {U}_{{k}_{1}\ldots {k}_{r}} \), the vector subspace \( {L}_{x} = {p}^{-1}\left( x\right) \) has a basis\n\n\[ \left\{ {{e}_{i} - \mathop{\su... | Yes |
Consider the vector bundle \( E \) of Example 6.2. Every section \( s : {\mathbb{P}}^{n} \rightarrow E \) determines, in particular, a section \( s : {\mathbb{P}}^{n} \rightarrow {\mathbb{P}}^{n} \times V \), and hence by Corollary 1.2 of Section 5.2, Chapter 1 is of the form \( s\left( x\right) = \left( {x, v}\right) ... | This proves in particular that \( E \) is not a trivial bundle. | Yes |
Theorem 6.2 The correspondence \( E \mapsto {\mathcal{L}}_{E} \) establishes a one-to-one correspondence between vector bundles and locally free sheaves of finite rank (here objects of either type are considered up to isomorphism). | Proof We show how to recover a vector bundle from a locally free sheaf \( \mathcal{F} \) . We can obviously assume that \( X \) is connected. Suppose that \( X = \bigcup {U}_{\alpha } \) is a cover such that \( {\mathcal{F}}_{\mid {U}_{\alpha }} \) is a free sheaf, and let \( {\varphi }_{\alpha } : {\mathcal{F}}_{\mid ... | "No" |
Example 6.6 Let \( E \) and \( F \) be vector bundles, and \( {\mathcal{L}}_{E},{\mathcal{L}}_{F} \) the corresponding locally free sheaves. It is obvious that the sheaves \( {\mathcal{L}}_{E} \oplus {\mathcal{L}}_{F},{\mathcal{L}}_{E} \otimes {\mathcal{L}}_{F},{\mathcal{L}}_{E}^{ * },\mathop{\bigwedge }\limits_{\mathc... | If \( X = \bigcup {U}_{\alpha } \) is a cover in which \( E \) and \( F \) are defined by transition matrixes \( {C}_{\alpha \beta } \) and \( {D}_{\alpha \beta } \) then in the same cover, \( E \oplus F, E \otimes F,{E}^{ * },/{/}^{p}E \) are defined by the transition matrixes \[ \left( \begin{matrix} {C}_{\alpha \bet... | Yes |
Example 6.7 Let \( X \) be a nonsingular variety. Taking an open set \( U \) to the group \( {\Omega }^{p}\left\lbrack U\right\rbrack \) of differential \( p \) -forms regular on \( U \) defines in an obvious way a sheaf of \( {\mathcal{O}}_{X} \) -modules. It is called the sheaf of differential p-forms. | Theorem 3.18 of Section 5.3, Chapter 3 asserts that this sheaf is locally free. Hence by Theorem 6.2 it defines a vector bundle, denoted by \( {\Omega }^{p} \) . In particular, \( {\Omega }^{1} \) is called the cotangent bundle. | Yes |
Example 6.9 (The normal bundle \( {N}_{X/Y} \) ) Let \( X \) be a nonsingular variety and \( Y \subset X \) a nonsingular closed subvariety. We define the normal bundle \( {N}_{X/Y} \) to \( Y \) in \( X \) . The definition used in differential geometry is not applicable in the algebraic situation, since it is based on... | Write \( {\Theta }_{X}^{\prime } \) for the restriction to \( Y \) of the tangent bundle \( {\Theta }_{X} \) . It is defined as the pullback \( {j}^{ * }{\Theta }_{X} \), where \( j : Y \hookrightarrow X \) is the closed embedding. The vector bundle \( {\Theta }_{Y} \) is a subbundle of \( {\Theta }_{X}^{\prime } \) . ... | Yes |
Let \( X = {\mathbb{P}}^{n} \) and let \( D \) be a hyperplane of \( {\mathbb{P}}^{n} \). The line bundle \( {E}_{D} \) corresponding to \( D \) under Theorem 6.3 is denoted by \( \mathcal{O}\left( 1\right) \). If \( D \) is given by \( {x}_{0} = 0 \) then in the open set \( {U}_{\alpha } \) where \( {x}_{\alpha } \neq... | Let us find the sections of \( \mathcal{O}\left( 1\right) \). In \( {U}_{\alpha } \) these are of the form \( {s}_{\alpha } = {P}_{\alpha }/{x}_{\alpha }^{k} \), where \( {P}_{\alpha } \) is a form of degree \( k \) ; and they are related by \( {s}_{\beta } = {c}_{\alpha \beta }{s}_{\alpha } \). It follows that \( k = ... | Yes |
Let \( X \) be a nonsingular variety and \( Y \subset X \) a nonsingular hypersurface. In this case the normal bundle \( {N}_{X/Y} \) is a line bundle. We compute its characteristic class. | Suppose that \( Y \) is given in an affine cover \( X = \bigcup {U}_{\alpha } \) by local equations \( {f}_{\alpha } \) . Then \( {f}_{\beta }/{f}_{\alpha } = {f}_{\alpha \beta } \), where \( {f}_{\alpha \beta } \) and \( {f}_{\alpha \beta }^{-1} \in {\mathcal{O}}_{X}\left( {{U}_{\alpha } \cap {\mathcal{O}}_{\beta }}\r... | Yes |
Theorem 6.4 (Adjunction formula) The genus \( {g}_{Y} \) of a nonsingular curve \( Y \) on a complete nonsingular surface \( X \) is given by the formula\n\n\[ \n{g}_{Y} = \frac{1}{2}Y\left( {Y + K}\right) + 1 \n\]\n\nwhere \( K \) is the canonical class of \( X \) . | Proof Let \( X \) be a nonsingular variety and \( Y \subset X \) an arbitrary nonsingular closed subvariety. By the definition of the normal bundle \( {N}_{X/Y} \) and (6.8), we obtain\n\n\[ \n{\rho }_{Y}\left( {\det {\Theta }_{X}}\right) = \det {\Theta }_{X}^{\prime } = \det {\Theta }_{Y} \otimes \det {N}_{X/Y} \n\]\n... | Yes |
Example 6.13 Let \( X \) be a nonsingular algebraic variety over \( k,{\mathcal{O}}_{X} \) the sheaf of regular functions, and \( {\Omega }^{1} \) the sheaf of regular differential 1-forms. Sending \( f \in \) \( {\mathcal{O}}_{X}\left( U\right) \) to the differential \( \mathrm{d}f \in {\Omega }^{1}\left( U\right) \) ... | \[ \mathrm{d} : {\mathcal{O}}_{X} \rightarrow {\Omega }^{1} \] This is a homomorphism of sheaves of modules over the sheaf of locally constant \( k \) -valued functions, but not over \( {\mathcal{O}}_{X} \) (because, by Leibnitz’ rule, \( \mathrm{d} \) is not \( {\mathcal{O}}_{X} \) -linear). | Yes |
Example 6.14 Let \( X \) be a nonsingular irreducible curve, and \( {\mathcal{K}}^{ * } \) the constant sheaf with \( {\mathcal{K}}^{ * }\left( U\right) = k{\left( X\right) }^{ * } \) the group of nonzero elements of \( k\left( X\right) \) under multiplication; let \( \mathcal{D} \) be the sheaf of local divisors, defi... | Thus it seems natural to define the image of a homomorphism \( f : \mathcal{F} \rightarrow \mathcal{G} \) of sheaves of modules as follows. First define the presheaf \( {\mathcal{I}}^{\prime } \) by setting\n\n\[ \n{\mathcal{I}}^{\prime }\left( U\right) = {f}_{U}\left( {\mathcal{F}\left( U\right) }\right) \;\text{ for ... | Yes |
Proposition If \( S \) is the support of a sheaf \( \mathcal{F} \) and \( U \subset V \) are two open sets such that \( U \cap S = V \cap S \) then the restriction \( {\rho }_{U}^{V} : \mathcal{F}\left( V\right) \rightarrow \mathcal{F}\left( U\right) \) is an isomorphism. | Proof Let \( a \in \mathcal{F}\left( V\right) \) be such that \( {\rho }_{U}^{V}\left( a\right) = 0 \) . By definition of \( S \) every point \( x \in V \) with \( x \notin S \) has a neighbourhood \( {V}_{x} \), which we can assume to be contained in \( V \), for which\n\n\[{\rho }_{{V}_{x}}^{V}\left( a\right) = 0.\]\... | Yes |
For any sheaf \( \mathcal{F} \) on a scheme \( X \), the dual sheaf \( \mathcal{G} = \mathcal{H}{om}\left( {\mathcal{F},{\mathcal{O}}_{X}}\right) \) is the sheafication of the presheaf \( \mathcal{G}\left( U\right) = \operatorname{Hom}\left( {\mathcal{F}\left( U\right) ,{\mathcal{O}}_{X}\left( U\right) }\right) \) . If... | If \( A \) is Noetherian and \( M = A{m}_{1} + \cdots + A{m}_{r} \) is finite then a homomorphism \( M \rightarrow A \) is determined by its values on the generators \( {m}_{i} \), so that \( {\operatorname{Hom}}_{A}\left( {M, A}\right) \subset {A}^{r} \), and is therefore again finite. It follows from this that if \( ... | No |
Under the assumptions of Example 6.20, the sheaf of modules \( {\mathcal{I}}_{Y}/{\mathcal{I}}_{Y}^{2} \) is coherent. We prove that if \( X \) and \( Y \) are nonsingular then it is locally free. | This is a local assertion, and it is enough to check it in the case \( X = \operatorname{Spec}A, Y = \operatorname{Spec}B \) and \( B = A/I \), and we can even assume that \( A \) is the local ring of a point \( x \in \) \( X \) . Since \( X \) and \( Y \) are nonsingular we can assume that \( I = \left( {{u}_{1},\ldot... | Yes |
Proposition 6.1 For any coherent sheaf \( \mathcal{F} \) over a Noetherian reduced irreducible scheme \( X \), there exists a dense open set \( W \) such that \( {\mathcal{F}}_{\mid W} \) is free. | Proof The assertion is local in nature, so that we can restrict to the case \( X = \operatorname{Spec}A \) , where \( A \) is a Noetherian ring without nilpotents and \( \mathcal{F} = \widetilde{M} \) for a finite \( A \) -module \( M \) . In addition, we can obviously assume that \( X \) is irreducible. Then \( X \) r... | Yes |
Proposition 6.3 A coherent sheaf \( \mathcal{F} \) on a Noetherian scheme \( X \) with support \( Y \neq X \) has a chain of subsheaves\n\n\[ \mathcal{F} = {\mathcal{F}}_{0} \supset {\mathcal{F}}_{1} \supset \cdots \supset {\mathcal{F}}_{m} = 0 \]\n\nsuch that each quotient sheaf \( {\overline{\mathcal{F}}}_{i}/{\overl... | Proof In Example 6.16, we gave the example of the sheaf \( {\mathcal{I}}_{Y} \) of ideals of the reduced subscheme \( Y \) . Obviously \( \overline{\mathcal{F}} \) is a coherent sheaf of \( {\mathcal{O}}_{Y} \) -modules if\n\n\[ {\mathcal{I}}_{Y} \cdot \mathcal{F} = 0 \]\n\n(6.42)\n\nIndeed, under this assumption, all ... | Yes |
Proposition 6.4 Let \( X \) be a Noetherian reduced scheme with \( X = \bigcup {X}_{i} \) its decomposition as a union of irreducible components, and suppose that \( \mathcal{F} \) is a coherent sheaf on \( X \) . There exist coherent sheaves \( {\mathcal{F}}_{i} \) on \( X \) and a homomorphism \( \varphi : \mathcal{F... | Proof Set \( {\mathcal{F}}_{i} = \mathcal{F}/\left( {{\mathcal{I}}_{{X}_{i}} \cdot \mathcal{F}}\right) \), and let \( {\varphi }_{i} : \mathcal{F} \rightarrow {\mathcal{F}}_{i} \) be the natural projection and \( \varphi = \bigoplus {\varphi }_{i} \) . We saw in Section 3.2 that the support of \( {\mathcal{F}}_{i} \) i... | Yes |
Let us see that the Grassmannian \( \operatorname{Grass}\left( {r, V}\right) \) really is a universal scheme for \( r \) -dimensional subspaces of a vector space \( V \) . We consider schemes over an algebraically closed field \( k \) . For a \( k \) -scheme \( S \), we define \( \Phi \left( S\right) \) as the set of v... | These maps \( {f}_{S} \) are an exact analogue of writing down the Plücker coordinates (see Example 1.24 of Section 4.1, Chapter 1). Let \( E \rightarrow S \) be a vector bundle of rank \( r \), and \( S = \bigcup {U}_{\alpha } \) a cover such that \( {E}_{\mid {U}_{\alpha }} \cong {U}_{\alpha } \times {\mathbb{A}}^{r}... | Yes |
We now give an example of a situation where the universal scheme does not exist. This is an extremely important case, nonsingular curves of given genus \( g \) . The reason for nonexistence is already present most vividly in the most trivial case, curves of genus 0 . We know that all such curves are isomorphic to \( {\... | However, \( f \) then corresponds to another element of \( \Phi \left( {\mathbb{P}}^{1}\right) \), the direct product \( {\mathbb{P}}^{1} \times {\mathbb{P}}^{1} \) . To nail down the contradiction, it remains to see that the family \( V \rightarrow {\mathbb{P}}^{1} \) is not isomorphic to \( {\mathbb{P}}^{1} \times {\... | Yes |
Example 6.24 (The tangent space to the Grassmannian \( \operatorname{Grass}\left( {r, V}\right) \) ) Suppose that a point \( x \in \operatorname{Grass}\left( {r, V}\right) \) corresponds to a vector subspace \( E \) with basis \( {e}_{1},\ldots ,{e}_{r} \) . By what we said above, \( {\Theta }_{x} \) is isomorphic to t... | If the second basis is \( {e}_{1} + \varepsilon {v}_{1},\ldots ,{e}_{r} + \varepsilon {v}_{r} \) then this will happen if and only if\n\n\[ \n{e}_{i} + \varepsilon {v}_{i} = \mathop{\sum }\limits_{j}\left( {{c}_{ij} + \varepsilon {d}_{ij}}\right) \left( {{e}_{j} + \varepsilon {u}_{j}}\right) \n\]\n\nfor \( i = 1,\ldots... | Yes |
A closed point of this scheme is a multiplication \( E \times E \rightarrow E \) ; if \( E \) has basis \( {e}_{1},\ldots ,{e}_{n} \), the multiplication is given by \( {e}_{i}{e}_{j} = \sum {c}_{ij}^{m}{e}_{m} \) . Tautologically, the scheme is universal for multiplication laws in \( S{ \times }_{k}E \), where now \( ... | The associativity condition can be written out at once by comparing the coefficient of \( \varepsilon \) in \( \left( {{e}_{i}{e}_{j}}\right) {e}_{k} \) and \( {e}_{i}\left( {{e}_{j}{e}_{k}}\right) \) :\n\n\[ \mathop{\sum }\limits_{m}{c}_{ij}^{m}{d}_{mk}^{l} + \mathop{\sum }\limits_{m}{d}_{ij}^{m}{c}_{mk}^{l} = \mathop... | Yes |
Example 6.26 Let \( X \subset {\mathbb{P}}^{N} \) be a 0-dimensional subscheme. Suppose that the underlying set \( {X}_{\text{red }} \) does not intersect the hyperplane \( {\xi }_{0} = 0 \) . Taking a homogeneous polynomial \( F \in {S}^{\left( r\right) } \) to the polynomial \( f = F/{\xi }_{0}^{r} \in k\left\lbrack ... | \[ X = \operatorname{Spec}A,\;A = k\left\lbrack {\mathbb{A}}^{N}\right\rbrack /I\;\text{ and }\;{P}_{X}\left( T\right) = \text{ const. } = {\dim }_{k}A. \] | Yes |
Example 6.28 Let \( X \) be a curve with an ordinary double point \( {x}_{0} \) and let \( {X}^{v} \) be the normalisation of \( X \) . We consider the family \( v : {X}^{v} \rightarrow X \) as a family of 0 - dimensional schemes over the base \( X \) . Then for \( x \neq {x}_{0} \) the fibre \( {v}^{-1}\left( x\right)... | By Example 6.26, we have \( {P}_{{v}^{-1}\left( x\right) }\left( r\right) = \) const. \( = 1 \) for \( x \neq {x}_{0} \) but \( {P}_{{v}^{-1}\left( {x}_{0}\right) }\left( r\right) = \) const. \( = 2. \) | Yes |
Example 6.29 Suppose that \( \operatorname{char}k \neq 2 \) ; let \( g \) be the automorphism of \( X = {\mathbb{A}}^{2} \) of order 2 given by \( g\left( {x, y}\right) = \left( {-x, - y}\right) \) and \( S = X/G \) the quotient of \( X \) by the group \( G = \{ 1, g\} \) (see Example 1.21 of Section 2.3, Chapter 1 and... | Then \( S \subset {\mathbb{A}}^{3} \) is given by \( {uv} = {w}^{2} \), and the morphism \( X \rightarrow S \) by \( u = {x}^{2}, v = {y}^{2} \) and \( w = {xy} \) . We view \( X \rightarrow S \) as a family of 0 -dimensional subschemes of \( {\mathbb{A}}^{2} \) with base \( S \) . For \( s = \left( {a, b, c}\right) \i... | Yes |
Theorem 6.7 Let \( A \) be the local ring of a nonsingular point of a curve over an algebraically closed field, and \( X \subset {\mathbb{P}}_{A}^{N} \) a closed subscheme such that the morphism \( X \rightarrow \operatorname{Spec}A \) is flat. Then the fibres of \( X \) over the closed and generic points of \( \operat... | Proof Let\n\n\[ \n{\mathfrak{a}}_{X} = {\bigoplus }_{r \geq 0}{\mathfrak{a}}_{X}^{\left( r\right) } \subset \Gamma = A\left\lbrack {{T}_{0},\ldots ,{T}_{N}}\right\rbrack \n\] \n\nbe the homogeneous ideal corresponding to the closed subscheme \( X \) . Set \( B = \) \( \Gamma /{\mathfrak{a}}_{X} = {\bigoplus }_{r > 0}{B... | Yes |
Lemma 7.1 If \( X \) is an irreducible algebraic variety and \( Y \subsetneqq X \) a proper subvariety then the set \( X\left( \mathbb{C}\right) \smallsetminus Y\left( \mathbb{C}\right) \) is everywhere dense in \( X\left( \mathbb{C}\right) \) . | Proof Consider first the case that \( X \) is an algebraic curve. Then \( Y \) consists of a finite set of closed points. Let \( v : {X}^{v} \rightarrow X \) be the normalisation morphism, and \( {Y}^{\prime } = {v}^{-1}\left( Y\right) \) . Since \( {X}^{v} \) is nonsingular, every point \( {y}^{\prime } \in {Y}^{\prim... | Yes |
Lemma 7.2 If \( V \subset {\mathbb{A}}^{n} \) is an open subset in the Zariski topology then \( V\left( \mathbb{C}\right) \) is connected. | Proof Write \( {\mathbb{A}}^{n} \smallsetminus V = Y \), and let \( {x}_{1},{x}_{2} \in V\left( \mathbb{C}\right) \) . Pass a line \( L \) through \( {x}_{1} \) and \( {x}_{2} \) . Then \( L \) is not contained in any irreducible component of \( Y \), so that \( L \cap Y \) is a finite set \( \left\{ {{y}_{1},\ldots ,{... | Yes |
Lemma 7.3 \( X\left( \mathbb{C}\right) \) is connected if \( X \) is an irreducible curve. | Proof Because the connectedness of \( X\left( \mathbb{C}\right) \) is not affected by adding or deleting a finite number of points, we can restrict ourselves at once to the case that \( X \) is a nonsingular projective curve.\n\nSuppose that \( X\left( \mathbb{C}\right) = {M}_{1} \sqcup {M}_{2} \) is a decomposition of... | Yes |
Lemma 7.4 For an irreducible nonsingular n-dimensional variety \( X \), there exists an open set \( U \), an irreducible \( \left( {n - 1}\right) \)-dimensional variety \( V \), and a surjective morphism \( f : U \rightarrow V \) with the following properties:\n\n(1) Every fibre of \( f \) is 1-dimensional.\n\n(2) Ever... | Proof We will choose \( V \) as a model of a subfield \( K \subset \mathbb{C}\left( X\right) \) of transcendence degree \( n - 1 \) to which we can apply the Bertini theorems, Theorem 2.26 of Section 6.1, Chapter 2 and Theorem 2.27 of Section 6.2, Chapter 2. The only difficulty arises in connection with the first of th... | Yes |
Theorem 7.1 If \( X \) is an irreducible algebraic variety over \( \mathbb{C} \), then \( X\left( \mathbb{C}\right) \) is connected. | Proof By induction on \( n = \dim X \) . Consider the map \( f : U \rightarrow V \) whose existence was proved in Lemma 7.4. Suppose that \( X\left( \mathbb{C}\right) = {M}_{1} \sqcup {M}_{2} \) is a decomposition into disjoint closed sets. Because \( f \) has connected fibres, every fibre is contained entirely in \( {... | Yes |
Lemma 7.6 Suppose that \( f\left( {{z}_{1},\ldots ,{z}_{n}}\right) \) is an analytic function on the whole of \( {\mathbb{C}}^{n} \) , and that there exists a constant \( C \) such that\n\n\[ \left| {f\left( z\right) }\right| < C{\left| z\right| }^{k}\;\text{ for }z = \left( {{z}_{1},\ldots ,{z}_{n}}\right) ,\text{ whe... | Proof Suppose that the homogeneous component \( {F}_{l} \) of the Taylor series expansion\n\n\[ f = {F}_{0} + {F}_{1} + \cdots \]\n\nof \( f \) about 0 is not identically 0 for some \( l > k \) . There exist \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) with \( {F}_{l}\left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) \ne... | Yes |
Theorem 7.2 Let \( X \) and \( Y \) be nonsingular irreducible varieties and \( f : X \rightarrow Y \) a proper morphism such that \( f\left( X\right) \subset Y \) is dense and \( X \) remains irreducible in the algebraic closure of \( \mathbb{C}\left( Y\right) \) (compare Section 6.1, Chapter 2). Then every fibre of \... | Proof By the Bertini theorems, Theorem 2.26 of Section 6.1, Chapter 2 and Theorem 2.27 of Section 6.2, Chapter 2, there exists a subvariety \( S \subsetneqq Y \) such that for every point \( y \notin S \) the fibre \( {f}^{-1}\left( y\right) \) is nonsingular and irreducible. We need to consider the fibres \( {f}^{-1}\... | Yes |
Theorem 7.3 For every morphism \( f : X \rightarrow Y \) of nonsingular curves with \( f\left( X\right) \) dense in \( Y \), and every \( x \in X \), there exist neighbourhoods \( U \ni x \) and \( V \ni f\left( x\right) \), and homeomorphisms \( u : U \rightarrow \mathbb{C} \) and \( v : V \rightarrow \mathbb{C} \) on... | If we interpret \( u \) and \( v \) as coordinates in open sets \( U \) and \( V \), then Theorem 1 asserts that \( f \), restricted to these neighbourhoods and expressed in terms of these coordinates has the very simple form\n\n\[ v = {u}^{k} \]\n\n(7.12)\n\n(see Figure 29). The open sets \( U \) and \( V \) can obvio... | Yes |
Theorem 7.5 If \( X \) is a nonsingular projective curve then the space \( X\left( \mathbb{C}\right) \) can be triangulated, and is a combinatorial surface. | Proof We prove first that \( {\mathbb{P}}_{\mathbb{C}}^{1} \) has a triangulation. We indicate a triangulation which, although not the most economic, is useful for subsequent applications.\n\nThe decomposition of the surface of an octahedron into its faces, edges and vertexes gives a triangulation. Suppose that the oct... | Yes |
Theorem 7.6 If \( X \) is a nonsingular projective curve then \( X\left( \mathbb{C}\right) \) is an orientable surface. | Proof This is of course a particular case of Proposition of Section 1.2, but we give another, much more elementary, proof. We use an arbitrary morphism \( f : X \rightarrow {\mathbb{P}}^{1} \) , and consider compatible triangulations \( \Phi \) and \( \Psi \) of \( X\left( \mathbb{C}\right) \) and \( {\mathbb{P}}_{\mat... | Yes |
Theorem 7.7 The Euler characteristic of \( X\left( \mathbb{C}\right) \) is \( 2 - {2g} \), where \( g \) is the genus of \( X \) . | Proof We again use a regular map \( f : X \rightarrow {\mathbb{P}}^{1} \) and compatible triangulations \( \Phi \) and \( \Psi \) of \( X\left( \mathbb{C}\right) \) and \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . We write \( {c}_{0},{c}_{1},{c}_{2} \) for the numbers appearing in the definition of the Euler charac... | Yes |
Proposition 7.2 Let \( E \) be a 1-simplex of the triangulation \( \Phi \) of \( X\left( \mathbb{C}\right) \) contained in \( X\left( \mathbb{R}\right) \), and \( {E}^{\prime } \) and \( {E}^{\prime \prime } \) the two 2-simplexes meeting along \( E \) . Then \( \tau \left( {E}^{\prime }\right) = {E}^{\prime \prime } \... | Proof Since \( {E}^{\prime } \) and \( {E}^{\prime \prime } \) are the only two simplexes of the triangulation \( \Phi \) with \( E \) as boundary, \( \Phi \) is \( \tau \) -invariant and \( \tau \left( E\right) = E \), it follows that either \( \tau \left( {E}^{\prime }\right) = {E}^{\prime } \) and \( \tau \left( {E}... | Yes |
Example 8.1 (Quotients of \( {\mathbb{C}}^{n} \) by a lattice) We view the \( n \) -dimensional complex vector space \( {\mathbb{C}}^{n} \) as a \( {2n} \) -dimensional real vector space, and choose \( m \) linearly independent vectors \( {a}_{1},\ldots ,{a}_{m} \) . Write \( \Omega \) for the set of vectors of the for... | Suppose that \( m = {2n} \) . In this case the manifold \( {\mathbb{C}}^{n}/G \) has a very simple topological structure. Since\n\n\[ {\mathbb{C}}^{n} = \mathbb{R}{a}_{1} + \cdots + \mathbb{R}{a}_{2n}\;\text{ and }\;\Omega = \mathbb{Z}{a}_{1} + \cdots + \mathbb{Z}{a}_{2n}, \]\n\n\( {\mathbb{C}}^{n}/G \) is homeomorphic... | Yes |
Example 8.2 (Hopf manifolds) Write \( X = {\mathbb{C}}^{n} \smallsetminus 0 \) and let \( c \) be a real number with \( c > 1 \) . Let \( G \) be the group of transformations\n\n\[ \n\left( {{z}_{1},\ldots ,{z}_{n}}\right) \mapsto \left( {{c}^{k}{z}_{1},\ldots ,{c}^{k}{z}_{n}}\right) \;\text{ with }k \in \mathbb{Z}.\n\... | Write any point \( z \in X \) in the form\n\n\[ \nz = {ru}\n\]\n\nwhere \( r \) is a positive number and \( u = \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) a vector such that \( {\left| {u}_{1}\right| }^{2} + \cdots + \) \( {\left| {u}_{n}\right| }^{2} = 1 \) . This representation is obviously unique and defines a homeo... | Yes |
In this example we start with the argument where we left off in Example 8.3. If \( A \) were algebraic, we would be able to find an algebraic curve \( C \subset A \) . If \( v : {C}^{v} \rightarrow C \) is the normalisation map, then triangulating \( {C}^{v} \) using Theorem 7.5 makes \( {C}^{v} \) into a singular cycl... | We now prove that \( C \) is not homologous to 0, so that not all the \( {a}_{i, j} \) are equal to 0 . For this note that the differential form\n\n\[ \frac{1}{2i}\left( {\mathrm{\;d}{z}_{1} \land \mathrm{d}{\bar{z}}_{1} + \mathrm{d}{z}_{2} \land \mathrm{d}{\bar{z}}_{2}}\right) \]\n\non \( {\mathbb{C}}^{2} \) is invari... | Yes |
Theorem 8.1 Every divisor is a difference of two effective divisors with no common components. | Proof of Theorem 8.1 We can assume that \( D \) is given by a cover \( \bigcup {U}_{\alpha } \) and a collection of meromorphic fractions \( {\varphi }_{\alpha } \) with\n\n\[{\varphi }_{\alpha } = {f}_{\alpha }/{g}_{\alpha }\;\text{ in }{U}_{\alpha }\n\]\n\nwhere \( {f}_{\alpha } \) and \( {g}_{\alpha } \) are holomor... | Yes |
Theorem 8.2 A function \( \varphi \) that is holomorphic at every point of a compact connected manifold \( X \) is constant. | Proof The modulus \( \left| \varphi \right| \) is obviously a continuous function on \( X \), and it therefore takes its maximum at some point \( {x}_{0} \) . Consider a neighbourhood \( U \) of \( {x}_{0} \) isomorphic to an open set \( V \subset {\mathbb{C}}^{n} \) ; we can assume that \( V \) consists of points \( \... | Yes |
Theorem 8.5 If \( X \) and \( Y \) are complete algebraic varieties then any holomorphic map \( f : {X}_{\mathrm{{an}}} \rightarrow {Y}_{\mathrm{{an}}} \) is of the form \( f = {g}_{\mathrm{{an}}} \) where \( g : X \rightarrow Y \) is a morphism. | Proof Choose a point \( x \in X \) and set \( y = f\left( x\right) \) ; let \( U \) be an affine neighbourhood of \( y \) . Suppose that \( U \subset {\mathbb{A}}^{N} \), and write \( {t}_{1},\ldots ,{t}_{N} \) for the coordinates in \( {\mathbb{A}}^{N} \) . By Theorem 8.4, the holomorphic functions \( {f}^{ * }\left( ... | Yes |
Let \( X \) be a complex Kähler manifold with Kähler metric \( \varphi \) and \( G \) a group of analytic automorphisms of \( X \) acting freely and discretely (see Section 1.2). If each automorphism \( g \in G \) preserves the Kähler metric \( \varphi \) then it induces a metric \( {\varphi }^{ * } \) on the quotient ... | To define \( {\varphi }^{ * } \), we must take an open set \( U \subset X/G \) whose inverse image \( {\pi }^{-1}\left( U\right) \) under the natural projection \( \pi : X \rightarrow X/G \) breaks up as a disjoint union of open sets \( {U}_{\alpha } \), each of which maps isomorphically to \( U \) under \( \pi \) . We... | Yes |
Let \( {\zeta }_{0},\ldots ,{\zeta }_{n} \) be homogeneous coordinates on \( {\mathbb{P}}^{n} \) and \( \zeta \) an arbitrary linear form. Then \( {\zeta }_{\alpha }/\zeta = {z}_{\alpha } \) are rational functions on \( {\mathbb{P}}^{n} \) . Set\n\n\[ H = \log \mathop{\sum }\limits_{{\alpha = 0}}^{n}{\left| {z}_{\alpha... | Note first that the 2 -form \( \omega \) is independent of the choice of the linear form \( \zeta \) . For this it is enough to check that if \( \eta \) is another linear form then \( {\mathrm{d}}^{\prime }{\mathrm{d}}^{\prime \prime }\log {\left| \zeta /\eta \right| }^{2} = 0 \) .\n\nIndeed, wherever \( \eta \neq 0 \)... | Yes |
Example 8.8 (The induced Kähler metric on a projective manifold \( X \subset {\mathbb{P}}^{n} \) ) Let \( X \) be a Kähler manifold and \( Y \subset X \) a complex submanifold. The restriction of differential forms from \( X \) to \( Y \) takes a closed form to a closed form. The restriction of Hermitian forms takes a ... | In particular we see that any projective variety \( X \) has a Kähler metric. This metric is defined not by intrinsic properties of \( X \) but by its embedding into \( {\mathbb{P}}^{n} \) . | Yes |
Proposition 8.1 The 2-form \( \omega \) associated with a Kähler form can be written in the form\n\n\[ \omega = {\mathrm{d}}^{\prime }{\mathrm{d}}^{\prime \prime }H \]\n\nin a neighbourhood of any point, where \( H \) is a \( {C}^{\infty } \) function of the real coordinates. A form of type \( {\mathrm{d}}^{\prime }{\m... | We observe first that the condition for a 2 -form \( \omega \) to be closed is \( \mathrm{d}\omega = {\mathrm{d}}^{\prime }\omega + {\mathrm{d}}^{\prime \prime }\omega = 0 \) . But \( \omega \) is of type \( \left( {1,1}\right) \) (that it, is of degree 1 in both the \( \mathrm{d}{z}_{\alpha } \) and the \( \mathrm{d}{... | Yes |
Proposition 8.2 For a Kähler metric, there exists a complex analytic coordinate system at each point \( P \) such that the matrix entries \( {c}_{\alpha \beta } \) of the metric satisfy the conditions\n\n\[ \n{c}_{\alpha \beta }\left( P\right) = {\delta }_{\alpha \beta },\;\text{ and }\;\frac{\partial {c}_{\alpha \beta... | Proof We start from the associated form \( \omega \) of the Kähler metric, and its representation \( \omega = {\mathrm{d}}^{\prime }{\mathrm{d}}^{\prime \prime }H \) established in Proposition 8.1, where \( H \) is an analytic function in the coordinates \( {z}_{\alpha } \) and \( {\bar{z}}_{\alpha } \) . In the Taylor... | Yes |
Theorem 9.2 \( \dim {\mathcal{L}}_{k} = k \) . | Proof Let \( f\left( z\right) \) be any of the functions \( {f}_{i} \) . The first of the conditions (9.9) shows that \( f\left( z\right) = \varphi \left( t\right) \), where \( \varphi \) is a holomorphic function on \( {\mathbb{C}}^{1} \smallsetminus 0 \), and \( t = {e}^{2\pi iz} \) . Indeed, \( \varphi \left( t\righ... | Yes |
Theorem 1.3. If \( g \in {L}_{loc}^{1}\left( \Omega \right) \) and \( {\int }_{\Omega }{g\varphi }\mathrm{d}x = 0 \) for each \( \varphi \in {C}_{0}^{\infty }\left( \Omega \right) \) then \( g = 0 \) almost everywhere on \( \Omega \) . | Proof. Consider a function \( \phi \) satisfying (see (14.3.3) for a concrete example)\n\n\[ \phi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) ,\;\phi \geq 0,\;\operatorname{supp}\phi \subseteq \overline{B\left( {0,1}\right) }, \]\n\n\[ \text{and}{\int }_{{\mathbb{R}}^{n}}\phi \left( x\right) \mathrm{d}x = 1\tex... | Yes |
Consider the function\n\n\[ f : \mathbb{R} \rightarrow \mathbb{R},\;f\left( x\right) \mathrel{\text{:=}} \left\{ {\begin{array}{ll} x, & x > 0, \\ 0, & x \leq 0, \end{array}\;\forall x \in \mathbb{R}.}\right. \]\n\nNote that \( f \) is continuous on \( \mathbb{R} \) but not differentiable at 0 . Nonetheless, \( f \) ha... | Indeed, if \( \varphi \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \), then integration by parts yields\n\n\[ - {\int }_{-\infty }^{\infty }f\left( x\right) {\varphi }^{\prime }\left( x\right) \mathrm{d}x = - {\int }_{0}^{\infty }x{\varphi }^{\prime }\left( x\right) \mathrm{d}x = - {\left. x\varphi \left( x\right) \ri... | Yes |
Does there exist a function \( g \in {L}_{loc}^{1}\left( \mathbb{R}\right) \) such that \( g \) is a weak derivative (of order one) of the Heaviside function? | To answer this question first observe that for each \( \varphi \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) we have\n\n\[ - {\int }_{-\infty }^{\infty }H\left( x\right) {\varphi }^{\prime }\left( x\right) \mathrm{d}x = - {\int }_{0}^{\infty }{\varphi }^{\prime }\left( x\right) \mathrm{d}x = \varphi \left( 0\right) ... | Yes |
Example 1.11. Let \( f \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) be given. Then a sequence of functions from \( {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) that converges to \( f \) in \( \mathcal{E}\left( {\mathbb{R}}^{n}\right) \) may be constructed as follows. Recall \( \phi \) from (1.2.3) and define \[... | Clearly, for each \( j \in \mathbb{N} \) we have \[ {\phi }_{j} \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) ,\;\operatorname{supp}{\phi }_{j} \subseteq \overline{B\left( {0,1/j}\right) },\;\text{ and }{\int }_{{\mathbb{R}}^{n}}{\phi }_{j}\mathrm{\;d}x = 1. \] Now if we further set for each \( j \in \mathbb{N} \... | Yes |
Consider the function\n\n\[ \varphi \left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{lll} \frac{1}{{\left| x - \frac{1}{2}\right| }^{2} - \frac{1}{4}} & \text{ if }0 < x < 1, & \text{ for each }x \in \mathbb{R}. \\ 0 & \text{ if }x \leq 0\text{ or }x > 1, & \end{array}\right. \]\n\nNote that \( \varphi \in {C}... | Clearly this limit function does not have compact support. | Yes |
Consider next the issue whether \( {\left\{ {\varphi }_{j}\right\} }_{j \in \mathbb{N}} \) converge in \( \mathcal{D}\left( \mathbb{R}\right) \) . | If this were to be the case, there would exist \( r \in \left( {0,\infty }\right) \) such that \( \operatorname{supp}{\varphi }_{j} \subseteq \left\lbrack {-r, r}\right\rbrack \) for every \( j \) . However, \( \mathop{\bigcup }\limits_{{j = 1}}^{\infty }\operatorname{supp}{\varphi }_{j} = \lbrack 1,\infty ) \) which l... | Yes |
Lemma 1.24. Suppose \( A \in {\mathcal{M}}_{n \times n}\left( \mathbb{R}\right) \) is such that \( \det A \neq 0 \) . Then the composition mapping\n\n\[ \mathcal{D}\left( {\mathbb{R}}^{n}\right) \ni \varphi \mapsto \varphi \circ A \in \mathcal{D}\left( {\mathbb{R}}^{n}\right) \]\n\n(1.3.18)\n\nis well defined, linear a... | Proof. Let \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . By the Chain Rule we have \( \varphi \circ A \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . We claim that\n\n\[ \operatorname{supp}\left( {\varphi \circ A}\right) = \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \in \operatorname{supp}\varphi ... | Yes |
For each \( f \in {L}_{loc}^{1}\left( \Omega \right) \) define the functional \( {u}_{f} : \mathcal{D}\left( \Omega \right) \rightarrow \mathbb{C} \) by\n\n\[ \n{u}_{f}\left( \varphi \right) \mathrel{\text{:=}} {\int }_{\Omega }f\left( x\right) \varphi \left( x\right) \mathrm{d}x,\;\forall \varphi \in {C}_{0}^{\infty }... | Hence, by Proposition 2.4, \( {u}_{f} \) is a distribution in \( \Omega \) . Moreover,(2.1.7) also shows that \( {u}_{f} \) is a distribution of order 0 . | Yes |
We have that \( \ln \left| x\right| \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \), thus \( \ln \left| x\right| \) is a distribution in \( {\mathbb{R}}^{n} \) . | To see that indeed \( \ln \left| x\right| \) is locally integrable in \( {\mathbb{R}}^{n} \), observe first that\n\n\[ \left| {\ln t}\right| \leq \frac{1}{\varepsilon }\max \left\{ {{t}^{\varepsilon },{t}^{-\varepsilon }}\right\} \;\text{ for all }t > 0\text{ and }\varepsilon > 0. \]\n\n(2.1.9)\n\nThis is justified by ... | Yes |
For a given \( f \in {L}_{loc}^{1}\left( \Omega \right) \) and multi-index \( \alpha \in {\mathbb{N}}_{0}^{n} \), consider the functional \( {g}_{\alpha } : \mathcal{D}\left( \Omega \right) \rightarrow \mathbb{C} \) defined by\n\n\[ \n{g}_{\alpha }\left( \varphi \right) \mathrel{\text{:=}} {\left( -1\right) }^{\left| \... | Clearly this is a linear mapping. Moreover, if \( {\varphi }_{j}\xrightarrow[{j \rightarrow \infty }]{\mathcal{D}\left( \Omega \right) }0 \), then there exists \( K \) compact subset of \( \Omega \) such that supp \( {\varphi }_{j} \subseteq K \) for every \( j \in \mathbb{N} \), hence\n\n\[ \n\left| {{g}_{\alpha }\lef... | Yes |
A natural question to ask is whether \( \delta \) is a distribution of function type. | To answer this question, suppose there exists \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) such that\n\n\[ \varphi \left( 0\right) = \langle \delta ,\varphi \rangle = {\int }_{{\mathbb{R}}^{n}}{f\varphi }\mathrm{d}x\text{ for all }\varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) . \]\n\n(2.1.20)\n... | Yes |
Example 2.15. Let \( \mu \) be either a complex Borel measure on \( \Omega \), or a Borel positive measure on \( \Omega \) that is locally finite (i.e., satisfies \( \mu \left( K\right) < \infty \) for every compact \( K \subset \) \( \Omega ) \) . Consider\n\n\[ \mu : \mathcal{D}\left( \Omega \right) \rightarrow \math... | The mapping in (2.1.22) is well defined, linear, and if \( K \) is an arbitrary compact set in \( \Omega \), then\n\n\[ \left| {\mu \left( \varphi \right) }\right| \leq \left| \mu \right| \left( K\right) \mathop{\sup }\limits_{{x \in K}}\left| {\varphi \left( x\right) }\right| ,\;\forall \varphi \in {C}^{\infty }\left(... | Yes |
Proposition 2.16. Let \( u \) be a distribution in \( \Omega \) of order zero. Then the distribution \( u \) extends uniquely to a linear map \( {\Lambda }_{u} : {C}_{0}^{0}\left( \Omega \right) \rightarrow \mathbb{C} \) that is locally bounded, in the following sense: for each compact set \( K \subset \Omega \) there ... | \[ \left| {{\Lambda }_{u}\left( \varphi \right) }\right| \leq {C}_{K}\mathop{\sup }\limits_{{x \in K}}\left| {\varphi \left( x\right) }\right| ,\;\forall \varphi \in {C}_{0}^{0}\left( \Omega \right) \text{ with }\operatorname{supp}\varphi \subseteq K. \] (2.1.24) In addition, the functional \( {\Lambda }_{u} \) satisfi... | Yes |
Proposition 2.19. Let \( u \) be a distribution in \( \Omega \) of order \( k \in {\mathbb{N}}_{0} \). Then there exists a linear map \( {\Lambda }_{u} : {C}_{0}^{k}\left( \Omega \right) \rightarrow \mathbb{C} \) satisfying \( {\left. {\Lambda }_{u}\right| }_{{C}_{0}^{\infty }\left( \Omega \right) } = u \) and with the... | Proof. Let \( K \) be an arbitrary compact subset of \( \Omega \) and let \( \varphi \in {C}_{0}^{k}\left( \Omega \right) \) be such that \( \operatorname{supp}\varphi \subseteq K \). Apply Lemma 2.17 to obtain a sequence \( {\left\{ {\varphi }_{j}\right\} }_{j \in \mathbb{N}} \subset {C}_{0}^{\infty }\left( \Omega \ri... | Yes |
For each multi-index \( \alpha \in {\mathbb{N}}_{0}^{n} \) the distribution \( {\partial }^{\alpha }\delta \) is a distribution of order \( \left| \alpha \right| \) in \( {\mathbb{R}}^{n} \). | By Proposition 2.19 (and formula (2.1.36)) this distribution extends uniquely to the linear map\n\n\[ \n{\Lambda }_{{\partial }^{\alpha }\delta } : {C}_{0}^{\left| \alpha \right| }\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{C}\;\text{defined by} \n\]\n\n(2.1.46)\n\n\[ \n{\Lambda }_{{\partial }^{\alpha }\delta }\... | Yes |
Example 2.24. Let \( \phi \) be as in (1.2.3) and recall the sequence of functions \( {\left\{ {\phi }_{j}\right\} }_{j \in \mathbb{N}} \) from (1.3.7). Interpreting each \( {\phi }_{j} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) as distribution in \( {\mathbb{R}}^{n} \), for each function \( \varphi \in {C}_{0}... | \[ \n\left\langle {{\phi }_{j},\varphi }\right\rangle = {\int }_{{\mathbb{R}}^{n}}\phi \left( y\right) \varphi \left( {y/j}\right) \mathrm{d}y\underset{j \rightarrow \infty }{ \rightarrow }\varphi \left( 0\right) {\int }_{{\mathbb{R}}^{n}}\phi \left( y\right) \mathrm{d}y = \varphi \left( 0\right) = \langle \delta ,\var... | Yes |
Proposition 2.27. Let \( A \in {\mathcal{M}}_{n \times n}\left( \mathbb{R}\right) \) be such that \( \det A \neq 0 \) . For each distribution \( u \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), define the mapping \( u \circ A : \mathcal{D}\left( {\mathbb{R}}^{n}\right) \rightarrow \mathbb{C} \) by settin... | Proof. This is an immediate consequence of (1.3.18). | No |
Proposition 2.29. Let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) and \( a \in {C}^{\infty }\left( \Omega \right) \) . Then the mapping \[ \text{ au } : \mathcal{D}\left( \Omega \right) \rightarrow \mathbb{C}\text{ defined by }\;\left( {au}\right) \left( \varphi \right) \mathrel{\text{:=}} \langle u,{a\var... | Proof. The fact that \( {au} \) is linear is immediate. To show that \( {au} \) is also continuous we make use of Remark 2.3. To this end, consider a sequence \( {\varphi }_{j}\xrightarrow[{j \rightarrow \infty }]{\mathcal{D}\left( \Omega \right) }0 \) . By (2) in Exercise 1.20 we have \( a{\varphi }_{j}\xrightarrow[{j... | Yes |
Recall the Dirac distribution defined in (2.1.19) and assume that some function \( a \in {C}^{\infty }\left( \Omega \right) \) has been given. Then, for every \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) we may write\n\n\[ \langle {a\delta },\varphi \rangle = \langle \delta ,{a\varphi }\rangle = \l... | \[ \langle {a\delta },\varphi \rangle = \langle \delta ,{a\varphi }\rangle = \left( {a\varphi }\right) \left( 0\right) = a\left( 0\right) \varphi \left( 0\right) = \langle a\left( 0\right) \delta ,\varphi \rangle . \] | Yes |
The goal is to solve the equation\n\n\\[ \n{xu} = 1\\;\\text{ in }\\;{\\mathcal{D}}^{\\prime }\\left( \\mathbb{R}\\right) .\n\\]\n\n(2.3.5) | Clearly, this equation does not have a solution \\( u \\) of function type. Recall the distribution defined in Example 2.11. Then for every \\( \\varphi \\in {C}_{0}^{\\infty }\\left( \\mathbb{R}\\right) \\) we may write\n\n\\[ \n\\left\\langle {x\\left( {\\text{ P.V. }\\frac{1}{x}}\\right) ,\\varphi }\\right\\rangle =... | Yes |
Proposition 2.39. For each \( \alpha \in {\mathbb{N}}_{0}^{n} \) and each \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) we have \( {\partial }^{\alpha }u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) . | Proof. Fix \( \alpha \in {\mathbb{N}}_{0}^{n} \) and \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) . That \( {\partial }^{\alpha }u : \mathcal{D}\left( \Omega \right) \rightarrow \mathbb{C} \) is a linear map is easy to see. To prove that it is also continuous, let \( {\varphi }_{j}\xrightarrow[{j \rightarro... | No |
Proposition 2.41. Let \( m \in {\mathbb{N}}_{0} \) and assume that\n\n\[ P\left( {x,\partial }\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\left| \alpha \right| \leq m}}{a}_{\alpha }\left( x\right) {\partial }^{\alpha },\;{a}_{\alpha } \in {C}^{\infty }\left( \Omega \right) ,\;\alpha \in {\mathbb{N}}_{0}^{n},\;\... | Proof. This follows from (2.4.3), Exercise 2.40, and Exercise 2.30. | No |
(1) Any distribution is infinitely differentiable (i.e., \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) is stable under the action of \( {\partial }^{\alpha } \) for any \( \alpha \in {\mathbb{N}}_{0} \) ). | Proof. The first property follows immediately from the definition of distributional derivatives. | No |
Proposition 2.48. Let \( I \) be an open interval in \( \mathbb{R}, g \in {C}^{\infty }\left( I\right) \), and \( f \in {C}^{k}\left( I\right) \) for some \( k \in {\mathbb{N}}_{0} \) . If \( u \in {\mathcal{D}}^{\prime }\left( I\right) \) satisfies \( {u}^{\prime } + {gu} = f \) in \( {\mathcal{D}}^{\prime }\left( I\r... | Proof. Fix \( a \in I \) and define \( F\left( x\right) \mathrel{\text{:=}} {\mathrm{e}}^{{\int }_{a}^{x}{g}_{0}\left( t\right) \mathrm{d}t} \) for \( x \in I \) . Then \( F \in {C}^{\infty }\left( I\right) \) and we may use (4) in Proposition 2.43 and the equation satisfied by \( u \) to write (keeping in mind that \(... | Yes |
Proposition 2.49 (Generalized Leibniz Formula). Suppose \( f \in {C}^{\infty }\left( \Omega \right) \) and let \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) . Then for every \( \alpha \in {\mathbb{N}}_{0}^{n} \) one has\n\n\[ \n{\partial }^{\alpha }\left( {fu}\right) = \mathop{\sum }\limits_{{\beta \leq \alp... | Proof. The first step is to observe that for each \( j \in \{ 1,\ldots, n\} \) and each \( k \in {\mathbb{N}}_{0} \) we\n\nhave\n\[ \n{\partial }_{j}^{k}\left( {fu}\right) = \mathop{\sum }\limits_{{0 \leq \ell \leq k}}\frac{k!}{\ell !\left( {k - \ell }\right) !}\left( {{\partial }_{j}^{\ell }f}\right) \left( {{\partial... | Yes |
Proposition 2.50. Let \( \Omega \) be a non-empty open subset of \( {\mathbb{R}}^{n} \) and suppose \( \omega \) is a nonempty open subset of \( \Omega \). Also, recall the map \( \iota \) from (1.3.16). Then for every \( u \in \) \( {\mathcal{D}}^{\prime }\left( \Omega \right) \), the mapping arising as the restrictio... | Proof. It is immediate that the map in (2.5.2) is well defined and linear. To see that it is also continuous we use Proposition 2.4. Let \( K \) be a compact set contained in \( \omega \). Then \( K \subset \Omega \) and, since \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \), Proposition 2.4 applies and gives ... | Yes |
Proposition 2.52. If \( {u}_{1},{u}_{2} \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) are such that for each \( {x}_{0} \in \Omega \) there exists an open subset \( \omega \) of \( \Omega \) with \( {x}_{0} \in \omega \) and satisfying \( {\left. {u}_{1}\right| }_{\omega } = {\left. {u}_{2}\right| }_{\omega } \) ... | Proof. Observe that this proposition may be viewed as a reconstruction problem, thus it is meaningful to try to use a partition of unity. Let \( \varphi \in {C}_{0}^{\infty }\left( \Omega \right) \) be arbitrary, fixed and set \( K \mathrel{\text{:=}} \operatorname{supp}\varphi \) . The goal is to prove that \( \left\l... | Yes |
Recall the Dirac distribution \( \delta \) from (2.1.19). We claim that supp \( \delta = \) \( \{ 0\} \) . | Indeed, if \( \varphi \in {C}_{0}^{\infty }\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \) it follows that \( \langle \delta ,\varphi \rangle = \varphi \left( 0\right) = 0 \) . By Proposition 2.52, \( {\left. \delta \right| }_{{\mathbb{R}}^{n}\smallsetminus \{ 0\} } = 0 \), thus supp \( \delta \subseteq \{ 0\... | Yes |
If \( f \in {C}^{0}\left( \Omega \right) \) then \( \operatorname{supp}{u}_{f} = \operatorname{supp}f \), where \( {u}_{f} \) is the distribution from (2.1.6). | Indeed, since \( f = 0 \) in \( \Omega \smallsetminus \operatorname{supp}f \), we have \( {\int }_{\Omega }f\left( x\right) \varphi \left( x\right) \mathrm{d}x = 0 \) for every \( \varphi \in {C}_{0}^{\infty }\left( {\Omega \smallsetminus \operatorname{supp}f}\right) \), hence \( \operatorname{supp}{u}_{f} \subseteq \o... | Yes |
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