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The formal power series ring \( R \mathrel{\text{:=}} K\left\lbrack \left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \right\rbrack \) in \( n \) indeterminates over a field is a regular local ring. | This can be seen by doing Exercise 12.6(a) (the result is \( \operatorname{gr}\left( R\right) \cong K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) ) and applying Theorem 13.4, or by observing that the maximal ideal is generated by \( {x}_{1},\ldots ,{x}_{n} \) and using Exercise 7.10 to conclude that \( \dim \... | No |
Let \( S \) be a Noetherian ring, and suppose that \( P \in \operatorname{Spec}\left( S\right) \) is contained in more than one irreducible component of \( \operatorname{Spec}\left( S\right) \). This means that \( P \) contains more than one minimal prime ideal of \( S \). | By Theorem 6.5, it follows that the localization \( {S}_{P} \) has more than one minimal prime ideal. But since an integral domain has \( \{ 0\} \) as the only minimal prime ideal, \( {S}_{P} \) is not an integral domain, and by Corollary 13.6(a) we conclude that \( {S}_{P} \) is not regular. So \( P \) is a singular p... | Yes |
Proposition 13.8 (Facts about separable field extensions). (a) Every finitely generated field extension of a perfect field is separable. | Proof. (a) The proof follows Mac Lane [35]. Let \( K \) be a perfect field, which we may assume to have positive characteristic \( p \) . We will prove the following by induction on \( n \) : If \( L \) is a finitely generated extension of \( K \) with a transcendence basis \( T \) such that \( L \) has degree \( n \) ... | Yes |
Lemma 13.9. Let \( L \) be an extension of a field \( K \) of characteristic \( p > 0 \) . Let \( T \) be a finite transcendence basis, and write \( {T}^{p} \) for the set of all \( p \) th powers of elements of \( T \) . If the minimal polynomial \( g \mathrel{\text{:=}} \operatorname{irr}\left( {\alpha, K\left( T\rig... | Proof. Since \( K\left\lbrack T\right\rbrack \) is factorial, there exists \( 0 \neq h \in K\left\lbrack T\right\rbrack \) such that \( f \mathrel{\text{:=}} {hg} \in \) \( K\left\lbrack T\right\rbrack \left\lbrack x\right\rbrack \) is a primitive polynomial, so by the Gauss lemma, \( f \) is irreducible (see Lang [33,... | Yes |
Lemma 13.12. Let \( I = \left( {{f}_{1},\ldots ,{f}_{m}}\right) \subseteq K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be an ideal in a polynomial ring over a field, and let \( P \subset K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a prime ideal containing \( I \) . If \( L \mathrel{\text{:=}} ... | Proof. We will construct linear maps \( \varphi : {L}^{m} \rightarrow {P}_{P}/{P}_{P}^{2} \) and \( \psi : {P}_{P}/{P}_{P}^{2} \rightarrow {L}^{n} \) . First\n\n\[ \varphi : {L}^{m} \rightarrow {P}_{P}/{P}_{P}^{2},\left( {{g}_{1} + {P}_{P},\ldots ,{g}_{m} + {P}_{P}}\right) \mapsto \mathop{\sum }\limits_{{j = 1}}^{m}{g}... | Yes |
(a) If \( A \) is an affine algebra over a perfect field, then the singular locus \( {X}_{\text{sing }} \) in \( X \mathrel{\text{:=}} \operatorname{Spec}\left( A\right) \) is closed. | Proof. (a) Write \( A = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack /I \) with \( I = \left( {{f}_{1},\ldots ,{f}_{m}}\right) \), and let \( {Q}_{1},\ldots ,{Q}_{k} \in \operatorname{Spec}\left( {K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack }\right) \) be the prime ideals that are minimal over \( I \) ... | Yes |
Consider the algebra \( A = K\left\lbrack x\right\rbrack /\left( {x}^{2}\right) \) with \( K \) a field. Then \( P = \left( x\right) \) is the only prime ideal in \( K\left\lbrack x\right\rbrack \) containing \( I \mathrel{\text{:=}} \left( {x}^{2}\right) \) . The Jacobian matrix reduced modulo \( P \) is zero. So by t... | This is also clear since \( R \) is zero-dimensional but not a field. | Yes |
Theorem 14.1. A Noetherian local ring of dimension one is regular if and only if it is normal. | Proof. Regularity implies normality by Corollary 13.6(b).\n\nFor the converse, assume that \( R \) is a one-dimensional normal Noetherian local domain with maximal ideal \( \mathfrak{m} \) . By Corollary 7.9 there exists \( a \in \mathfrak{m} \) with \( \sqrt{\left( a\right) } = \mathfrak{m} \) . By the Noether propert... | Yes |
We wish to desingularize the plane complex curve \( X \subseteq {\mathbb{C}}^{2} \) given by the equation \( {x}_{1}^{4} + {x}_{2}^{4} - {x}_{1}^{2} = 0 \), which is irreducible by the Eisenstein criterion (see Lang [33, Chapter V, Theorem 7.1]). The curve \( X \) is shown in Fig. 14.1. The idea is to desingularize \( ... | The Jacobian criterion (Theorem 13.10) yields \( \left( {0,0}\right) \) as the only singular point. By Theorem 14.1, the localization of the coordinate ring \( A = \mathbb{C}\left\lbrack X\right\rbrack \) is normal at all points except \( \left( {0,0}\right) \) . So the normalization \( \widetilde{A} \) is contained in... | Yes |
In the ring \( R \mathrel{\text{:=}} \mathbb{Z}\left\lbrack \sqrt{-5}\right\rbrack \subseteq \mathbb{C} \), consider the ideal \( I = \) \( {\left( 2,1 + \sqrt{-5}\right) }_{R} \subseteq R \). If \( J \mathrel{\text{:=}} {\left( 1,\frac{1 - \sqrt{-5}}{2}\right) }_{R} \subseteq \operatorname{Quot}\left( R\right) \), the... | Indeed, from the assumption \( I = {\left( z\right) }_{R} \) with \( z = a + b\sqrt{-5}, a, b \in \mathbb{Z} \), we deduce that \( {a}^{2} + 5{b}^{2} \) (the norm of \( z \) , which by definition is the product of \( z \) and its complex conjugate) divides 4 and 6, the norms of 2 and of \( 1 + \sqrt{-5} \) . This impli... | Yes |
Proposition 14.6 (Invertible ideals are locally principal). Let \( R \) be an integral domain and \( I \subseteq K \mathrel{\text{:=}} \operatorname{Quot}\left( R\right) \) a fractional ideal. Then the following statements are equivalent:\n\n(a) \( I \) is invertible.\n\n(b) If \( {I}^{\prime } \mathrel{\text{:=}} \{ a... | Proof. We start by showing that (a) implies (c). So we assume that there exists a fractional ideal \( J \subseteq K \) with \( I \cdot J = R \) . In particular, we have \( 1 = \) \( \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{b}_{i} \) with \( {a}_{i} \in I \) and \( {b}_{i} \in J \) . So every \( x \in I \) satisfies \... | Yes |
Corollary 14.7 (Properties of invertible ideals). Let \( I \in C\left( R\right) \) be an invertible fractional ideal of a Noetherian domain \( R \). (a) There exist invertible ideals \( {J}_{1},{J}_{2} \subseteq R \) with \( I = {J}_{1} \cdot {J}_{2}^{-1} \). | Proof. (a) By Proposition 14.6, \( I \) is finitely generated. If \( a \in R \smallsetminus \{ 0\} \) is a common denominator of all elements in a generating set, then \( {J}_{1} \mathrel{\text{:=}} \) \( I \cdot \left( a\right) \subseteq R \) and \( I = {J}_{1} \cdot {\left( a\right) }^{-1} \) . Since \( {J}_{2} \math... | Yes |
Theorem 14.8 (Invertible ideals in a locally factorial ring). Let \( R \) be a Noetherian domain.\n\n(a) If \( R \) is locally factorial, then every height-one prime ideal of \( R \) is invertible.\n\n(b) If every height-one prime ideal of \( R \) is invertible, then an ideal \( I \subseteq R \) is invertible if and on... | Proof of Theorem 14.8. (a) Let \( Q \subset R \) be a prime ideal of height 1 . We use Proposition 14.6. Clearly \( Q \) is finitely generated and nonzero, so we need to show only that \( {Q}_{P} \subseteq {R}_{P} \) is a principal ideal for every \( P \in \) \( \operatorname{Spec}\left( R\right) \) . If \( Q \nsubsete... | Yes |
Lemma 14.9. Let \( R \) be a Noetherian domain and let \( I \subseteq R \) be a nonzero ideal that is contained in an invertible prime ideal \( P \) . Then \( I \subsetneqq I \cdot {P}^{-1} \subseteq R \) . | Proof. From \( I \subseteq P \) it follows that \( J \mathrel{\text{:=}} I \cdot {P}^{-1} \subseteq P \cdot {P}^{-1} = R \) . Moreover, \( I = \) \( J \cdot P \subseteq J \) . Assume that \( I = J \) . Then \( I = P \cdot I \) . This localizes to \( {I}_{P} = {P}_{P} \cdot {I}_{P} \) , which by Nakayama’s lemma (Theore... | Yes |
Proposition 14.10 (Unique factorization of invertible ideals). Let \( R \) be an integral domain and let \( I \subseteq R \) be an invertible ideal that has a factorization\n\n\[ I = {P}_{1}\cdots {P}_{n} \]\n\nwith \( {P}_{i} \) prime ideals (where \( n = 0 \) occurs if \( I = R \) ). Then this factorization is unique... | Proof. We use induction on \( n \) . Let \( I = {Q}_{1}\cdots {Q}_{m} \) be another factorization with \( {Q}_{i} \in \operatorname{Spec}\left( R\right) \) . If \( n = 0 \) then \( m = 0 \), since otherwise \( I \subseteq {Q}_{1} \subsetneqq R = I \) . Consider the case \( n > 0 \) . By renumbering, we may assume that ... | Yes |
How do the corresponding principal ideals \( {\left( 2\right) }_{R},{\left( 3\right) }_{R} \), etc. factorize? | In Exercise 14.9 it is shown that every ideal of a Dedekind domain is generated by two elements. With this in mind, it is not too hard to find the following factorizations, which are easy to verify:\n\n\[{\left( 2\right) }_{R} = {\left( 2,1 + \sqrt{-5}\right) }_{R}^{2}\]\n\n\[{\left( 3\right) }_{R} = {\left( 3,1 + \sqr... | Yes |
Theorem 14.13 (Factorial Dedekind domains). For a Dedekind domain \( R \) , the following statements are equivalent:\n\n(a) \( R \) is factorial;\n\n(b) \( R \) is a principal ideal domain. | Proof. First assume that \( R \) is factorial. By Lemma 5.14, it follows that every prime ideal of height 1 is principal. Since every nonzero ideal is a product of height-one prime ideals, this implies (b).\n\nThe fact that every principal ideal domain is factorial is usually part of an abstract algebra course (see Lan... | Yes |
Lemma 2.1. A module \( M \) over a noetherian ring \( A \) is flat if and only if for every prime ideal \( \mathfrak{p} \subseteq A,{\operatorname{Tor}}_{1}^{A}\left( {M, A/\mathfrak{p}}\right) = 0 \) . | Proof. The exactness of the functor \( N \mapsto N{ \otimes }_{A}M \) is equivalent to \( {\operatorname{Tor}}_{1}\left( {M, N}\right) = 0 \) for all \( A \) -modules \( N \) . Since Tor commutes with direct limits, it is sufficient to require \( {\operatorname{Tor}}_{1}\left( {M, N}\right) = 0 \) for all finitely gene... | Yes |
Proposition 2.2. Let \( {A}^{\prime } \rightarrow A \) be a surjective homomorphism of noetherian rings whose kernel \( J \) has square zero. Then an \( {A}^{\prime } \) -module \( {M}^{\prime } \) is flat over \( {A}^{\prime } \) if and only if\n\n(1) \( M = {M}^{\prime }{ \otimes }_{{A}^{\prime }}A \) is flat over \(... | Proof. Note that since \( J \) has square zero, it is an \( A \) -module and we can identify \( {M}^{\prime }{ \otimes }_{{A}^{\prime }}J \) with \( M{ \otimes }_{A}J \) .\n\nIf \( {M}^{\prime } \) is flat over \( {A}^{\prime } \), then (1) follows by base extension, and (2) follows by tensoring \( {M}^{\prime } \) wit... | Yes |
Proposition 2.3. In the situation above, to give \( {I}^{\prime } \subseteq {B}^{\prime } \) such that \( {B}^{\prime }/{I}^{\prime } \) is flat over \( D \) and the image of \( {I}^{\prime } \) in \( B \) is \( I \) is equivalent to giving an element \( \varphi \in {\operatorname{Hom}}_{B}\left( {I, B/I}\right) \) . I... | Proof. We will make use of the splitting \( {B}^{\prime } = B \oplus {tB} \) as \( B \) -modules, or, equivalently, of the section \( \sigma : B \rightarrow {B}^{\prime } \) given by \( \sigma \left( b\right) = b + 0 \cdot t \), which makes \( {B}^{\prime } \) into a \( B \) -module.\n\nTake any element \( x \in I \) .... | Yes |
Corollary 2.5. If \( Y \) is a closed subscheme of the projective space \( X = {\mathbb{P}}_{k}^{n} \), then the Zariski tangent space of the Hilbert scheme \( H \) at the point \( y \) corresponding to \( Y \) is isomorphic to \( {H}^{0}\left( {Y,{\mathcal{N}}_{Y/X}}\right) \) . | Proof. The Zariski tangent space to \( H \) at \( y \) can be interpreted as the set of morphisms from the dual numbers \( D \) to \( H \) sending the closed point to \( y\lbrack {57} \) , II, Ex. 2.8]. Because of the universal property of the Hilbert scheme \( \left( {{1.1}\left( a\right) }\right) \) , this set is in ... | Yes |
Proposition 2.6. Let \( X \) be a scheme over \( k \), and \( \mathcal{L} \) an invertible sheaf on \( X \). The set of isomorphism classes of invertible sheaves \( {\mathcal{L}}^{\prime } \) on \( X \times D \) such that \( {\mathcal{L}}^{\prime } \otimes {\mathcal{O}}_{X} \cong \mathcal{L} \) is in natural one-to-one... | Proof. We use the fact that on any ringed space \( X \), the isomorphism classes of invertible sheaves are classified by \( {H}^{1}\left( {X,{\mathcal{O}}_{X}^{ * }}\right) \), where \( {\mathcal{O}}_{X}^{ * } \) is the sheaf of multiplicative groups of units in \( {\mathcal{O}}_{X} \) [57, III, Ex. 4.5]. The exact seq... | Yes |
Theorem 2.7. Let \( X \) be a scheme over \( k \), and let \( \mathcal{F} \) be a coherent sheaf on \( X \) . The (equivalence classes of) deformations of \( \mathcal{F} \) over \( D \) are in natural one-to-one correspondence with the elements of the group \( {\operatorname{Ext}}_{X}^{1}\left( {\mathcal{F},\mathcal{F}... | Proof. By (2.2), the flatness of \( {\mathcal{F}}^{\prime } \) over \( D \) is equivalent to the exactness of the sequence\n\n\[ 0 \rightarrow \mathcal{F}\overset{t}{ \rightarrow }{\mathcal{F}}^{\prime } \rightarrow \mathcal{F} \rightarrow 0 \]\n\nobtained by tensoring \( {\mathcal{F}}^{\prime } \) with \( 0 \rightarro... | Yes |
If \( \mathcal{E} \) is a vector bundle over \( X \), then the deformations of \( \mathcal{E} \) over \( D \) are in natural one-to-one correspondence with the elements of \( {H}^{1}\left( {X,\mathcal{E}{nd}\mathcal{E}}\right) \), where \( \mathcal{E}{nd}\mathcal{E} = \mathcal{H}{om}\left( {\mathcal{E},\mathcal{E}}\rig... | In this case, since \( \mathcal{E} \) is locally free, \( {\operatorname{Ext}}^{1}\left( {\mathcal{E},\mathcal{E}}\right) = {\operatorname{Ext}}^{1}\left( {{\mathcal{O}}_{X},\mathcal{E}{nd}\mathcal{E}}\right) = \) \( {H}^{1}\left( {X,\mathcal{E}{nd}\mathcal{E}}\right) \). | Yes |
Lemma 3.2. The modules \( {T}^{i}\left( {B/A, M}\right) \) constructed above are independent of the choice of \( F \) (keeping \( R \) fixed). | Proof. If \( F \) and \( {F}^{\prime } \) are two choices of free \( R \) -modules mapping onto \( I \), then \( F \oplus {F}^{\prime } \) is a third choice, so by symmetry it is sufficient to compare \( F \) with \( F \oplus {F}^{\prime } \) . Since \( {F}^{\prime } \) is free, the map \( {j}^{\prime } : {F}^{\prime }... | Yes |
Theorem 3.4. Let \( A \rightarrow B \) be a homomorphism of rings. Then for \( i = 0,1,2 \) , \( {T}^{i}\left( {B/A, \cdot }\right) \) is a covariant, additive functor from the category of \( B \) -modules to itself. If \[ 0 \rightarrow {M}^{\prime } \rightarrow M \rightarrow {M}^{\prime \prime } \rightarrow 0 \] is a ... | Proof. We have seen that the \( {T}^{i}\left( {B/A, M}\right) \) are well-defined. By construction they are covariant additive functors. Given a short exact sequence of modules as above, since the terms \( {L}_{1} \) and \( {L}_{0} \) of the complex \( {L}_{ \bullet } \) are free, we get a sequence of complexes \[ 0 \r... | Yes |
Proposition 3.6. For any \( A \rightarrow B \) and any \( M,{T}^{0}\left( {B/A, M}\right) = \) \( {\operatorname{Hom}}_{B}\left( {{\Omega }_{B/A}, M}\right) \; = \;{\operatorname{Der}}_{A}\left( {B, M}\right) .\; \) In \( \; \) particular, \( \;{T}^{0}\left( {B/A, B}\right) \; = \) \( {\operatorname{Hom}}_{B}\left( {{\... | Proof. Write \( B \) as a quotient of a polynomial ring \( R \), with kernel \( I \) . Then there is an exact sequence \( \left\lbrack {{57},\mathrm{{II}},{8.4}\mathrm{\;A}}\right\rbrack \)\n\n\[ \nI/{I}^{2}\overset{d}{ \rightarrow }{\Omega }_{R/A}{ \otimes }_{R}B \rightarrow {\Omega }_{B/A} \rightarrow 0.\n\]\n\nSince... | Yes |
Proposition 3.7. If \( B \) is a polynomial ring over \( A \), then \( {T}^{i}\left( {B/A, M}\right) = 0 \) for \( i = 1,2 \) and for all \( M \) . | Proof. In this case we can take \( R = B \) in the construction. Then \( I = 0 \) , \( F = 0 \), so \( {L}_{2} = {L}_{1} = 0 \), and the complex \( {L}_{ \bullet } \) is reduced to the \( {L}_{0} \) term. Therefore \( {T}^{i} = 0 \) for \( i = 1,2 \) and any \( M \) . | Yes |
Proposition 3.8. If \( A \rightarrow B \) is a surjective ring homomorphism with kernel \( I \), then \( {T}^{0}\left( {B/A, M}\right) = 0 \) for all \( M \), and \( {T}^{1}\left( {B/A, M}\right) = {\operatorname{Hom}}_{B}\left( {I/{I}^{2}, M}\right) \) . In particular, \( {T}^{1}\left( {B/A, B}\right) = {\operatorname... | Proof. In this case we can take \( R = A \), so that \( {L}_{0} = 0 \) . Thus \( {T}^{0} = 0 \) for any \( M \) . Furthermore, the exact sequence\n\n\[ 0 \rightarrow Q \rightarrow F \rightarrow I \rightarrow 0 \]\n\n\ntensored with \( B \), gives an exact sequence\n\n\[ Q{ \otimes }_{A}B \rightarrow F{ \otimes }_{A}B \... | Yes |
Corollary 3.9. If \( A \) is a local ring and \( B \) is a quotient \( A/I \), where \( I \) is generated by a regular sequence \( {a}_{1},\ldots ,{a}_{r} \), then \( {T}^{2}\left( {B/A, M}\right) = 0 \) for all \( M \) . | Proof. Indeed, in this case, since the Koszul complex of a regular sequence is exact \( \left\lbrack {{104},{16.5}}\right\rbrack \), we find \( Q = {F}_{0} \) in the construction of the \( {T}^{i} \) -functors. Thus \( {L}_{2} = 0 \) and \( {T}^{2}\left( {B/A, M}\right) = 0 \) for all \( M \) . | Yes |
Proposition 3.10. Suppose \( A = k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) and \( B = A/I \) . Then for any \( M \) there is an exact sequence\n\n\[ 0 \rightarrow {T}^{0}\left( {B/k, M}\right) \rightarrow \operatorname{Hom}\left( {{\Omega }_{A/k}, M}\right) \rightarrow \operatorname{Hom}\left( {I/{I}^{2},... | Proof. Write the long exact sequence of \( {T}^{i} \) -functors for the composition \( k \rightarrow \) \( A \rightarrow B \) and use (3.6),(3.7), and (3.8). The same works for any base ring \( k \) , not necessarily a field. | No |
Lemma 4.5. Let \( {B}^{\prime } \rightarrow B \) be a surjective homomorphism of \( k \) -algebras with kernel \( I \) of square zero. Let \( R \rightarrow B \) be a homomorphism of \( k \) -algebras.\n\n(a) If \( f, g : R \rightarrow {B}^{\prime } \) are two liftings of the map \( R \rightarrow B \) to \( {B}^{\prime ... | Proof. (a) Let \( f, g : R \rightarrow {B}^{\prime } \) and let \( \theta = g - f \) . As a \( k \) -linear map, \( \theta \) followed by the projection \( {B}^{\prime } \rightarrow B \) is zero, so \( \theta \) sends \( R \) to \( I \) . Let \( x, y \in R \) . Then\n\n\[ \theta \left( {xy}\right) = g\left( {xy}\right)... | Yes |
Proposition 4.6. Let \( X \) be a scheme of finite type over \( k \) algebraically closed. Suppose that for every morphism \( f : Y \rightarrow X \) of a punctual scheme \( Y \) (meaning \( Y \) is the Spec of a local Artin ring), finite over \( k \), and for every infinitesimal thickening \( Y \subseteq {Y}^{\prime } ... | Proof. It is sufficient (4.1) to show that the local ring \( {\mathcal{O}}_{P, X} \) is a regular local ring for every closed point \( P \in X \) . So again we reduce to an algebraic question, namely, let \( A,\mathfrak{m} \) be a local \( k \) -algebra, essentially of finite type over \( k \), and with residue field \... | Yes |
Corollary 4.7. Let \( A \) be a local ring, essentially of finite type over an algebraically closed field \( k \), with residue field \( k \) . Then \( A \) is a regular local ring if and only if it has the infinitesimal lifting property for local Artin rings \( {B}^{\prime } \rightarrow B \) finite over \( k \) . | Proof. Just localize (4.4) and (4.6). | No |
Corollary 4.8. Let \( X \) be a nonsingular affine scheme over \( k \) . Let \( A \) be a local Artin ring over \( k \), and let \( {X}^{\prime } \) be a scheme, flat over \( \operatorname{Spec}A \), such that \( {X}^{\prime }{ \times }_{A}k \) (where by abuse of notation we mean \( {X}^{\prime }{ \times }_{\operatorna... | Proof. We apply (4.4) to the identity map of \( X \) to \( X \) and the infinitesimal thickening \( i : X \hookrightarrow {X}^{\prime } \) defined by the isomorphism \( {X}^{\prime }{ \times }_{A}k \cong X \) . Therefore there is a lifting \( p : {X}^{\prime } \rightarrow X \) such that \( p \circ i = {\operatorname{id... | Yes |
Theorem 4.9. Let \( X = \operatorname{Spec}B \) be an affine scheme over \( k \) algebraically closed. Then \( X \) is nonsingular if and only if \( {T}^{1}\left( {B/k, M}\right) = 0 \) for all \( B \) -modules \( M \) . Furthermore, if \( X \) is nonsingular, then also \( {T}^{2}\left( {B/k, M}\right) = 0 \) for all \... | Proof. Write \( B \) as a quotient of a polynomial ring \( A = k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) over \( k \) . Then \( \operatorname{Spec}A \) is nonsingular, and we can use the criterion of (4.3), which shows that \( X \) is nonsingular if and only if the conormal sequence\n\n\[ 0 \rightarrow I/... | Yes |
Corollary 4.10. Let \( B \) be a local \( k \) -algebra with residue field \( k \) algebraically closed. Then \( B \) is a regular local ring if and only if \( {T}^{1}\left( {B/k, M}\right) = 0 \) for all \( B \) -modules \( M \), and in this case \( {T}^{2}\left( {B/k, M}\right) = 0 \) for all \( M \) . | Proof. By localization, using (4.1) and (4.9). | No |
Lemma 4.12 (Dévissage). Let \( B \) be a noetherian ring, and let \( F \) be a semi-exact additive functor from finitely generated B-modules to finitely generated \( B \) -modules. Assume that \( F\left( {B/\mathfrak{m}}\right) = 0 \) for every maximal ideal \( \mathfrak{m} \) of \( B \) . Then \( F\left( M\right) = 0 ... | Proof. Any finitely generated \( B \) -module \( M \) has a composition series whose quotients are \( B/{\mathfrak{p}}_{i} \) for various prime ideals \( {\mathfrak{p}}_{i} \) . By semi-exactness, it is sufficient to show that \( F \) vanishes on each of these. Thus we may assume \( M = B/\mathfrak{p} \) .\n\nWe procee... | Yes |
Theorem 4.13. Let \( A \) be a regular local \( k \) -algebra with residue field \( k \) algebraically closed, and let \( B = A/I \) be a quotient of \( A \) . Then \( B \) is a local complete intersection in \( A \) if and only if \( {T}^{2}\left( {B/k, M}\right) = 0 \) for all \( B \) -modules \( M \) . | Proof. Since \( A \) is regular, we have \( {T}^{1}\left( {A/k, M}\right) = 0 \) for \( i = 1,2 \) and all \( M \) by (4.10). Then from the exact sequence (3.5) we obtain \( {T}^{2}\left( {B/k, M}\right) = \) \( {T}^{2}\left( {B/A, M}\right) \) for all \( M \) . If \( B \) is a local complete intersection in \( A \), t... | Yes |
Corollary 5.2. Let \( k \) be a field and let \( B \) be a \( k \) -algebra. Then the set of deformations of \( B \) over the dual numbers is in natural one-to-one correspondence with the group \( {T}^{1}\left( {B/k, B}\right) \) . | Proof. This follows from the theorem and the discussion at the beginning of this section, which showed that such deformations are in one-to-one correspondence with the \( k \) -algebra extensions of \( B \) by \( B \) . | No |
Theorem 5.3. Let \( X \) be a nonsingular variety over \( k \) . Then the deformations of \( X \) over the dual numbers are in natural one-to-one correspondence with the elements of the group \( {H}^{1}\left( {X,{\mathcal{T}}_{X}}\right) \), where \( {\mathcal{T}}_{X} = {\operatorname{Hom}}_{X}\left( {{\Omega }_{X/k},{... | Proof (cf. [57, III,9.13.2]). Let \( {X}^{\prime } \) be a deformation of \( X \), and let \( \mathcal{U} = \left( {U}_{i}\right) \) be an open affine covering of \( X \) . Over each \( {U}_{i} \) the induced deformation \( {U}_{i}^{\prime } \) is trivial by (4.8), or by (4.9) combined with (5.2), so we can choose an i... | Yes |
If \( X = {\mathbb{P}}_{k}^{n} \) for \( n \geq 1 \), then \( {H}^{1}\left( {\mathcal{T}}_{X}\right) = 0 \), so every deformation of \( X \) over the dual numbers is trivial. Thus \( X \) is an example of a rigid scheme, by which we mean a scheme all of whose deformations over the dual numbers are trivial. | We have already seen that any affine nonsingular scheme is rigid (4.8). This result also follows from (5.3), since an affine scheme has no higher cohomology. | No |
Let \( C \) be a nonsingular projective curve of genus \( g \) . Then by Serre duality \( {H}^{1}\left( {\mathcal{T}}_{C}\right) \) is dual to \( {H}^{0}\left( {\Omega }_{C}^{\otimes 2}\right) \), which has degree \( {4g} - 4 \) . | For \( g \geq 2 \) this is nonspecial, so by Riemann-Roch, \( {H}^{1}\left( {\mathcal{T}}_{C}\right) \) has dimension \( {3g} - 3 \) . | No |
Theorem 5.4. In the situation above, if \( {\operatorname{depth}}_{x}B \geq 2 \), then there is an exact sequence\n\n\[ 0 \rightarrow {T}_{B/k}^{1} \rightarrow {H}^{1}\left( {U,{\mathcal{T}}_{U}}\right) \rightarrow {H}^{1}\left( {U,{T}_{R} \mid {}_{U}}\right) \] | Proof. Since \( U \) is nonsingular, we have an exact conormal sequence of sheaves\n\n\[ {\left. 0 \rightarrow {\mathcal{T}}_{U} \rightarrow {T}_{R}\right| }_{U} \rightarrow {\mathcal{N}}_{U/R} \rightarrow 0 \]\n\nWe consider the following diagram, where the second row is the cohomology sequence of this exact sequence ... | Yes |
Corollary 5.5. If \( Y \) is a nonsingular projectively normal subvariety of \( P = \) \( {\mathbb{P}}_{k}^{n} \), and if \( {H}^{1}\left( {{\mathcal{O}}_{Y}\left( \nu \right) }\right) = {H}^{1}\left( {{\mathcal{T}}_{Y}\left( \nu \right) }\right) = 0 \) for all \( \nu \in \mathbb{Z} \), then the affine cone \( X \) ove... | Proof. Indeed, taking into account (5.4.1), the theorem implies \( {T}_{B/k}^{1} = 0 \) . | No |
Let \( Y \) be a closed subscheme of \( X = {\mathbb{P}}_{k}^{n} \). Assume that there are no local obstructions to deformations of \( Y \), and that \( {H}^{1}\left( {Y,{\mathcal{N}}_{Y/X}}\right) = 0 \). Then the Hilbert scheme \( H \) is nonsingular at the point \( y \) corresponding to \( Y \). | Proof. According to the infinitesimal lifting property (4.6), to show that \( H \) is nonsingular at \( Y \), it is sufficient to show that for any local Artin ring \( C \) over \( k \) and a morphism \( f : \operatorname{Spec}C \rightarrow H \) sending the closed point to \( y \), and for any surjection of local Artin... | Yes |
Theorem 6.4. In the above situation:\n\n(a) There is an obstruction \( \delta \in {H}^{2}\left( {J{ \otimes }_{C}{\mathcal{O}}_{X}}\right) \) whose vanishing is a necessary and sufficient condition for the existence of \( {\mathcal{L}}^{\prime } \) on \( {X}^{\prime } \) .\n\n(b) If an \( {\mathcal{L}}^{\prime } \) exi... | Proof. As in the proof of (2.6) the exact sequence\n\n\[ 0 \rightarrow J \otimes {\mathcal{O}}_{X} \rightarrow {\mathcal{O}}_{{X}^{\prime }} \rightarrow {\mathcal{O}}_{X} \rightarrow 0 \]\n\ngives rise to an exact sequence of abelian groups\n\n\[ 0 \rightarrow J \otimes {\mathcal{O}}_{X} \rightarrow {\mathcal{O}}_{{X}^... | Yes |
Theorem 7.1. Let \( X,\mathcal{F} \) be as above, and assume that \( {\mathcal{F}}_{0} \) is locally free on \( {X}_{0} \) . Let \( {\mathcal{A}}_{0} = \mathcal{H} \) om \( \left( {{\mathcal{F}}_{0},{\mathcal{F}}_{0}}\right) \) be the sheaf of endomorphisms of \( {\mathcal{F}}_{0} \) (also sometimes written \( \left. {... | Proof. (a) If \( {\mathcal{F}}^{\prime } \) is an extension of \( \mathcal{F} \), because of flatness there is an exact sequence\n\n\[ 0 \rightarrow J{ \otimes }_{k}{\mathcal{F}}_{0} \rightarrow {\mathcal{F}}^{\prime } \rightarrow \mathcal{F} \rightarrow 0 \]\n\nIf \( \sigma \in \operatorname{Aut}\left( {{\mathcal{F}}^... | Yes |
Theorem 7.2. Given \( {X}_{0},{\mathcal{E}}_{0} \rightarrow {\mathcal{F}}_{0} \rightarrow 0 \) in the situation as above, assuming \( {\mathcal{E}}_{0} \) locally free, and \( \operatorname{hd}{\mathcal{F}}_{0} \leq 1 \), we have:\n\n(a) There is an obstruction in \( {H}^{1}\left( {{X}_{0}, J{ \otimes }_{k}\operatornam... | Proof. (a) Given \( \mathcal{E} \rightarrow \mathcal{F} \rightarrow 0 \), because of the hypothesis \( \operatorname{hd}{\mathcal{F}}_{0} \leq 1 \), the kernel \( Q \) will be locally free. Therefore on a small open set \( {U}_{i} \) it can be lifted to a locally free subsheaf \( {Q}_{i}^{\prime } \) of \( {\mathcal{E}... | Yes |
In the same situation as (7.1), instead of assuming \( {\mathcal{F}}_{0} \) locally free, we will assume \( \operatorname{hd}{\mathcal{F}}_{0} \leq 1 \) and \( {X}^{\prime } \) projective. Then:\n\n(a) If an extension \( {\mathcal{F}}^{\prime } \) of \( \mathcal{F} \) over \( {X}^{\prime } \) exists, then \( \operatorn... | (a) The same as (7.1), since that step did not use the hypothesis \( {\mathcal{F}}_{0} \) locally free, noting that \( {\operatorname{Ext}}^{0}\left( {{\mathcal{F}}_{0},{\mathcal{F}}_{0}}\right) = {H}^{0}\left( {{X}_{0},\mathcal{E}{nd}{\mathcal{F}}_{0}}\right) \) . | Yes |
Theorem 8.1 (Hilbert, Burch). Let \( A \) be a regular local ring of dimension \( n \) . Let \( B = A/I \) be a Cohen-Macaulay quotient of codimension 2. Then there is an \( r \times \left( {r + 1}\right) \) matrix \( \varphi \) of elements of \( A \) whose \( r \times r \) minors \( {f}_{1},\ldots ,{f}_{r + 1} \) mini... | Proof. We make use of the theorem that if \( M \) is a finitely generated module over a regular local ring \( A \), then depth \( M + \operatorname{hd}M = n \), where \( \operatorname{hd}M \) is the homological dimension of \( n\left\lbrack {{104},{19.1}}\right\rbrack \) . Thus hd \( B = 2 \) as an \( A \) -module. If ... | Yes |
Proposition 8.2. Let \( X \) be a smooth scheme over a field \( k \), and let \( Y \subseteq X \) be a closed Cohen-Macaulay subscheme of codimension 2. Then for each point \( y \in Y \) there is an open affine neighborhood \( U \) of \( y \) in \( X \) and there is a matrix \( \varphi \) of regular functions on \( U \... | Proof. We apply (8.1) to the local ring \( {\mathcal{O}}_{y, X} \) and its quotient \( {\mathcal{O}}_{y, Y} \) . This gives a matrix \( \varphi \) of elements of \( {\mathcal{O}}_{y, X} \) . These elements are all defined on some open affine neighborhood \( U \) of \( y \) and so determine a complex\n\n\[ {\mathcal{O}}... | Yes |
Theorem 8.3 (Schaps [144]). In the above situation we have:\n\n(a) There is an \( r \times \left( {r + 1}\right) \) matrix \( \varphi \) of elements of \( A \) whose \( r \times r \) minors \( {f}_{i} \) generate \( I \) and give a resolution\n\n\[ 0 \rightarrow {A}^{r}\overset{\varphi }{ \rightarrow }{A}^{r + 1}\overs... | Proof. We start with the proof of (b), assuming (a). Let \( {\varphi }^{\prime } \) be any lifting of \( \varphi \) . Then we can consider the complex\n\n\[ {L}_{ \bullet }^{\prime } : {A}^{\prime r}\overset{{\varphi }^{\prime }}{ \rightarrow }{A}^{\prime r + 1}\overset{{f}^{\prime }}{ \rightarrow }{A}^{\prime } \]\n\n... | No |
Lemma 8.4. Let \( A \) be a \( C \) -algebra flat over \( C \), with \( A{ \otimes }_{C}k \) normal. Let \( Z \subseteq X = \operatorname{Spec}A \) be a subset of codimension \( \geq 2 \) . Then \( {H}^{0}\left( {X - Z,{\mathcal{O}}_{X}}\right) = A \) . | Proof. By induction on length \( C \), the case of length 1 being known, since then \( A \) is normal. The result follows inductively, using the sheaf sequence associated to the exact sequence of modules\n\n\[ 0 \rightarrow {A}^{\prime }{ \otimes }_{{C}^{\prime }}J \rightarrow {A}^{\prime } \rightarrow {A}^{\prime }{ \... | No |
Corollary 8.5. In the situation of (6.2), deformations of a closed subscheme \( {Y}_{0} \) of a scheme \( {X}_{0} \), assume that \( {X}_{0} \) is nonsingular and that \( {Y}_{0} \) is Cohen-Macaulay of codimension 2. Then the obstructions to deforming \( Y \) to a closed subscheme \( {Y}^{\prime } \subseteq {X}^{\prim... | Proof. Indeed,(8.2) and (8.3) tell us that deformations of \( Y \) over \( C \) exist on small enough affine open subsets of \( {X}_{0} \), so (6.2) applies. | No |
Proposition 8.6. Let \( Y \) be a closed subscheme of \( X = {\mathbb{P}}_{k}^{n} \) . If \( \dim Y = 0 \) , then \( Y \) is ACM. If \( \dim Y \geq 1 \), the following conditions are equivalent:\n\n(i) \( Y \) is \( {ACM} \) .\n\n(ii) \( R \rightarrow {H}_{ * }^{0}\left( {\mathcal{O}}_{Y}\right) \) is surjective, and \... | Proof. (Here we use the notation, for any coherent sheaf \( \mathcal{F} \) on \( X,{H}_{ * }^{i}\left( \mathcal{F}\right) = \) \( {\bigoplus }_{l \in \mathbb{Z}}{H}^{i}\left( {X,\mathcal{F}\left( l\right) }\right) \) .) Let \( \mathfrak{m} = \left( {{x}_{0},\ldots ,{x}_{n}}\right) \) be the irrelevant prime ideal of \(... | Yes |
Proposition 8.7. Let \( Y \) be an ACM closed subscheme of \( X = {\mathbb{P}}_{k}^{n} \) of codimension 2. Then there is an \( r \times \left( {r + 1}\right) \) matrix \( \varphi \) of homogeneous elements of \( R \) whose \( r \times r \) minors \( {f}_{i} \) minimally generate \( {I}_{Y} \), giving rise to a resolut... | Proof. Since \( R/{I}_{Y} \) is Cohen-Macaulay and a quotient of codimension 2 of \( R \), it has homological dimension 2 over \( R \) . The proof then follows exactly as in the proof of the local case (8.1), using the graded analogues of depth and homological dimension. | No |
Theorem 8.9 (Ellingsrud [27]). Let \( {Y}_{0} \) be an ACM closed subscheme of codimension 2 of \( {X}_{0} = {\mathbb{P}}_{k}^{n} \), and assume \( \dim {Y}_{0} \geq 1 \) . Using notation (6.1), suppose we are given a closed subscheme \( Y \) of \( X = {\mathbb{P}}_{C}^{n} \), flat over \( C \) and with \( Y{ \times }_... | Proof. Since \( {Y}_{0} \) is ACM of dimension \( \geq 1 \), it follows that \( {R}_{0}/{I}_{0} \) is Cohen-Macaulay of dimension \( \geq 2 \) . In particular, it has depth \( \geq 2 \), so we can apply (8.8) and thus reduce to studying deformations of \( {R}_{0}/{I}_{0} \) . Then using (8.7) we can adapt the proof of ... | No |
Corollary 8.10. The Hilbert scheme at a point corresponding to a codimension 2 ACM closed subscheme \( Y \subseteq {\mathbb{P}}_{k}^{n} \) is smooth. | Proof. If \( n = 2 \) and \( Y \) is a zero-scheme, then (8.5) tells us that deformations extend, since there is no \( {H}^{1} \) on \( Y \) . If \( \dim Y \geq 1 \), then (8.9) tells us similarly that deformations always extend. The infinitesimal lifting property (4.6) implies that the Hilbert scheme is smooth, as in ... | No |
Theorem 8.11. For every \( n > 0 \), the Hilbert scheme \( {\operatorname{Hilb}}^{n}\left( {\mathbb{P}}_{k}^{2}\right) \), parametrizing zero-dimensional subschemes of length \( n \) of \( {\mathbb{P}}^{2} \), is irreducible. | Proof. By induction on \( n \), the case \( n = 1 \) being trivial. There is one obvious component, containing the sets of \( n \) distinct points, that is irreducible of dimension \( {2n} \) . Thus it will be sufficient to show that any zero-dimensional subscheme \( Z \) of \( {\mathbb{P}}^{2} \) is a limit of a flat ... | No |
Lemma 8.12. Let \( \mathfrak{a} \subseteq A = k\left\lbrack {x, y}\right\rbrack \) be an ideal of finite colength \( n \) such that \( Z = \operatorname{Spec}\left( {A/\mathfrak{a}}\right) \) has support at the origin \( \left( {0,0}\right) \) . Then there is an ideal \( {\mathfrak{a}}_{t} \subseteq A\left\lbrack t\rig... | Proof of Lemma. Choose \( f \in \mathfrak{a} \) of minimal order \( s \), that is, \( f \in {\mathfrak{m}}^{s} - {\mathfrak{m}}^{s + 1} \) with \( s \) minimal, where \( \mathfrak{m} = \left( {x, y}\right) \) . Then, by a linear change of coordinates, we may assume that the leading form of \( f \) is \( {f}_{0} = {x}^{... | Yes |
Corollary 8.13. \( {\operatorname{Hilb}}^{n}{\mathbb{P}}^{2} \) is smooth and irreducible. | Proof. Combine (8.10) and (8.11). | No |
Proposition 9.1. Let \( A \) be a local Cohen-Macaulay ring, let \( {a}_{1},\ldots ,{a}_{r} \) be elements of \( A \), let \( I = \left( {{a}_{1},\ldots ,{a}_{r}}\right) \), and let \( B = A/I \) . The following conditions are equivalent:\n\n(i) \( {a}_{1},\ldots ,{a}_{r} \) is a regular sequence in \( A \) .\n\n(ii) \... | Proof. \( \left\lbrack {{104},{16.5}}\right\rbrack \) . | No |
Theorem 9.2. Using notation (6.1), suppose we are given \( {A}^{\prime } \) flat over \( {C}^{\prime } \) such that \( {A}_{0} = {A}^{\prime }{ \otimes }_{{C}^{\prime }}k \) is a local Cohen-Macaulay ring. Suppose also that \( B = A/I \), a quotient of \( A = {A}^{\prime }{ \times }_{{C}^{\prime }}C \), flat over \( C ... | Proof. The proof follows the plan of proof of (8.3) except that it is simpler.\n\nFor (b), suppose we are given the situation of (a) and let \( {a}_{1}^{\prime },\ldots ,{a}_{r}^{\prime } \) be liftings of the \( {a}_{i} \) . Then we get an exact sequence of Koszul complexes\n\n\[ 0 \rightarrow {K}_{ \bullet }\left( {{... | Yes |
If \( Y \) is a locally complete intersection subscheme of \( {\mathbb{P}}^{n} \), then obstructions to deforming \( Y \) as a subscheme of \( {\mathbb{P}}^{n} \) lie in \( {H}^{1}\left( {\mathcal{N}}_{Y}\right) \). | Combine (9.2) with (6.2) and (4.3). This proves (1.1c). | No |
Corollary 9.5. If \( {Y}_{0} \subseteq {X}_{0} = {\mathbb{P}}_{k}^{n} \) is a complete intersection, the Hilbert scheme at the corresponding point is smooth. | Proof. If \( \dim {Y}_{0} \geq 1 \), the result follows from (9.4), as in the proof of (8.10). If \( \dim {Y}_{0} = 0 \), then \( {Y}_{0} \) is contained in an affine \( n \) -space \( {\mathbb{A}}^{n} \), and we can use (9.3) together with the fact that a zero-scheme has no \( {H}^{1} \). | Yes |
Theorem 9.6. Let \( A \) be a regular local ring, and let \( B = A/I \) be a quotient that is Gorenstein and of codimension 3. Then there is a skew-symmetric matrix \( \varphi \) of odd order \( n \) of elements of \( A \) whose \( P \) faffians \( {f}_{i} \) generate the ideal I and that gives rise to a resolution\n\n... | Using techniques analogous to those in the Cohen-Macaulay codimension 2 case, one can show that deformations of \( B \) always extend, and have resolutions of the same type. We leave the details to the reader. | No |
Theorem 10.2. In the above situation:\n\n(a) There are three successive obstructions to be overcome for the existence of an extension \( {X}^{\prime } \) of \( X \) over \( {C}^{\prime } \), lying in \( {H}^{0}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}}^{2} \otimes J}\right) ,{H}^{1}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}}^... | Proof. (a) Suppose we are given \( X \) . For each open affine subset \( {U}_{i} \subseteq X \) there is an obstruction lying in \( {H}^{0}\left( {{U}_{i},{\mathcal{T}}_{{U}_{i}}^{2} \otimes J}\right) \) for the existence of a deformation \( {U}_{i}^{\prime } \) over \( {U}_{i} \), by (10.1). These patch together to gi... | Yes |
Corollary 10.3. If \( {X}_{0} \) is nonsingular, then\n\n(a) There is just one obstruction in \( {H}^{2}\left( {{X}_{0},{\mathcal{T}}_{{X}_{0}} \otimes J}\right) \) for the existence of an extension \( {X}^{\prime } \) of \( X \) over \( {C}^{\prime } \) .\n\n(b) If such extensions exist, their equivalence classes form... | Proof. In this case the sheaves \( {\mathcal{T}}_{{X}_{0}}^{1} \) and \( {\mathcal{T}}_{{X}_{0}}^{2} \) are zero (4.9). | No |
Proposition 10.4. Let \( {Y}_{0} \) be a closed subscheme of \( {X}_{0} \) over \( k \), and let \( X, Y \) , \( {X}^{\prime }, C,{C}^{\prime }, J \) be as in (6.2). Then there is an obstruction \( \beta \in {H}^{0}\left( {{Y}_{0},{\mathcal{T}}_{{Y}_{0}/k}^{2}{ \otimes }_{k}J}\right) \) for the local existence of exten... | Proof. If we examine the proof of (6.2), we see that what was missing was the existence of an affine covering of \( {Y}_{0} \) where local extensions exist. Since for affine schemes abstract and embedded obstructions are the same (Ex. 10.1), the obstruction for each open affine subset lies in the corresponding \( {T}^{... | No |
We have already seen a typical example of an obstruction theory in studying the Hilbert scheme. Let \( {Y}_{0} \) be a closed subscheme of \( {X}_{0} = \) \( {\mathbb{P}}_{k}^{n} \), and assume that \( {Y}_{0} \) has no local obstructions to its deformations (e.g., \( {Y}_{0} \) is nonsingular, or locally complete inte... | To lift \( u \) to a homomorphism \( {u}^{\prime } : A \rightarrow {C}^{\prime } \) corresponds to extending the deformation \( Y \) to a deformation \( {Y}^{\prime } \) over \( {C}^{\prime } \) . Thus if we take \( V = {H}^{1}\left( {{Y}_{0},{\mathcal{N}}_{{Y}_{0}/{X}_{0}}}\right) \), we have an obstruction theory for... | Yes |
Theorem 11.1. Let \( A \) be a local ring that can be written as a quotient of a regular local ring \( P \) by an ideal \( I \subseteq {\mathfrak{m}}_{P}^{2} \), and let \( \left( {V,\varphi }\right) \) be an obstruction theory for \( A \) . Then there is a natural inclusion of \( {V}_{A} \) (11.0.2) into \( V \) . In ... | Proof. Note first that we cannot expect to get the exact number of generators for \( I \), because if \( \left( {V,\varphi }\right) \) is an obstruction theory, any bigger vector space \( {V}^{\prime } \) containing \( V \) will also be one.\n\nWe apply the obstruction theory \( V \) to a particular case. Take \( 0 \ri... | Yes |
Corollary 11.2. Let \( \left( {A,\mathfrak{m}}\right) \) be a local ring that can be written as a quotient of a regular local ring \( P \) of dimension \( n = \dim \mathfrak{m}/{\mathfrak{m}}^{2} \) . If \( A \) has an obstruction theory in a vector space \( V \), then \( \dim A \geq n - \dim V \) . Furthermore, if equ... | Proof. Indeed, \( \dim A \geq \dim P - \# \) generators of \( I \), and equality makes \( A \) a local complete intersection ring by definition. | No |
Theorem 11.3. Let \( Y \) be a locally complete intersection subscheme of the projective space \( X = {\mathbb{P}}_{k}^{n} \) . Then the dimension of the Hilbert scheme \( H \) at the point \( y \in H \) corresponding to \( Y \) is at least \( {h}^{0}\left( {Y,\mathcal{N}}\right) - {h}^{1}\left( {Y,\mathcal{N}}\right) ... | Proof. Let \( A \) be the local ring of \( y \) on \( H \) . Then \( V = {H}^{1}\left( {Y,\mathcal{N}}\right) \) gives an obstruction theory for \( A \) (11.0.1). On the other hand, \( A \) has embedding dimension equal to \( {h}^{0}\left( {Y,\mathcal{N}}\right) \) by (2.4). Hence the result follows from (11.2). | Yes |
A nonsingular 3-fold with obstructed deformations. In the same paper, Mumford observed that the above example, by blowing up the curve, produces a 3-fold with obstructed deformations. We outline the argument. | Let \( C \subseteq {\mathbb{P}}^{3} \) be a nonsingular curve. Let \( f : X \rightarrow {\mathbb{P}}^{3} \) be obtained by blowing up \( C \) . Let \( E \subseteq X \) be the exceptional divisor. Then \( f : E \rightarrow C \) is the projective space bundle \( \mathbb{P}\left( {\mathcal{I}/{\mathcal{I}}^{2}}\right) \) ... | Yes |
We would like to show that there exists a morphism \( S \rightarrow T \), i.e., a homomorphism \( k\left\lbrack t\right\rbrack \overset{\varphi }{ \rightarrow }k\left\lbrack \left\lbrack s\right\rbrack \right\rbrack \) given by a power series \( \varphi \left( t\right) = T\left( s\right) \) with \( T\left( 0\right) = 0... | To do this, it will be sufficient to find functions \( X\left( {x, y, s}\right) \) and \( Y\left( {x, y, s}\right) \) reducing to \( x \) and \( y \) for \( s = 0 \), and a unit \( U\left( {x, y, s}\right) \) reducing to 1 for \( s = 0 \) , such that\n\n\[ U\left( {{XY} - T}\right) = g\left( {x, y, s}\right) \]\n\n(*) ... | Yes |
Lemma 14.2. Let \( F\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a polynomial or power series. Let \( {h}_{1},\ldots ,{h}_{n} \) be new variables. Then\n\n\[ F\left( {{x}_{1} + {h}_{1},\ldots ,{x}_{n} + {h}_{n}}\right) \equiv F\left( {{x}_{1},\ldots ,{x}_{n}}\right) + \mathop{\sum }\limits_{{i = 1}}^{n}{h}_{i}\frac{\p... | The proof of the lemma is elementary, and we leave to the reader the simple verification of the claim made above, applying the lemma to the function \( {UF}\left( {X, Y,{T}_{1},\ldots ,{T}_{r}}\right) \) . | No |
Let us study the cusp defined by \( f\left( {x, y}\right) = {y}^{2} - {x}^{3} \) . The partial derivatives are \( {2y} \) and \( 3{x}^{2} \), so (assuming char \( k \neq 2,3 \) ) we can take \( 1, x \) as a basis for \( R/J \), and the versal deformation is defined by \( F\left( {x, y, t, u}\right) = \) \( {y}^{2} - {x... | For general values of \( t, u \), the nearby curve will be nonsingular, but for special nonzero values of \( t, u \) it may be singular. Indeed, if we set \( F,{F}_{x} \), and \( {F}_{y} \) equal to zero, we find a singular point at \( t = - 2{x}^{3}, u = 3{x}^{2} \) . Hence there are singularities in the fiber over po... | Yes |
Proposition 15.1. If \( F \) is a functor from \( \mathcal{C} \) to (Sets) and \( R \) is a complete local \( k \) -algebra with residue field \( k \), then there is a natural bijection between the set \( \widehat{F}\left( R\right) \) of formal families \( \left\{ {{\xi }_{n} \mid {\xi }_{n} \in F\left( {R/{\mathfrak{m... | Thus, if \( F \) is pro-representable, there is an isomorphism \( \xi : {h}_{R} \rightarrow F \) for some \( R \), and we can think of \( \xi \) as an element of \( \widehat{F}\left( R\right) \) . We say that the pair \( \left( {R,\xi }\right) \) pro-represents the functor \( F \) . One can verify easily that if \( F \... | No |
Proposition 15.2. Let \( \left( {R,\xi }\right) \) be a formal family of the functor \( F \) .\n\n(a) If \( \left( {R,\xi }\right) \) is a versal family, then for any other formal family \( \left( {S,\eta }\right) \), there is a ring homomorphism \( f : R \rightarrow S \) such that the induced map \( \widehat{F}\left( ... | Proof. (a) Let \( \left( {R,\xi }\right) \) be a versal family, and let \( \left( {S,\eta }\right) \) be any formal family. Then by definition we have a strongly surjective morphism of functors \( \varphi \) : \( {h}_{R} \rightarrow F \), determined by \( \xi \) . For each \( n \), we have an element \( {\eta }_{n} \in... | Yes |
Suppose that \( \mathcal{F} \) is a globally defined contravariant functor from (Sch \( /k \) ) to (Sets). For example, think of the functor Hilb, which to each scheme \( S/k \) associates the set of closed subschemes of \( {\mathbb{P}}_{S}^{n} \), flat over \( S \). Given a particular element \( {X}_{0} \in \mathcal{F... | If the global functor \( \mathcal{F} \) is representable, then the local functor \( F \) will be pro-representable (23.3). Thus pro-representability of the local functor is a necessary condition for representability of the global functor. | Yes |
The converse of (15.2.1) is false: the local functor may be pro-representable when the global functor is not representable. Take for example deformations of \( {\mathbb{P}}^{1} \) . It is easy to see that this functor is not representable (25.2.1). | But since all local deformations over Artin rings are trivial (5.3.1), (Ex. 10.3), the local functor is pro-represented by the ring \( k \) . | No |
For an example of a functor with no versal family, we note that if \( \left( {R,\xi }\right) \) is a versal family for the functor \( F \), then the map \( \operatorname{Hom}\left( {R, D}\right) \rightarrow \) \( F\left( D\right) \) is surjective, so \( F\left( D\right) \) is a quotient of a finite-dimensional vector s... | For example, let \( B = k\left\lbrack {x, y, z}\right\rbrack /\left( {xy}\right) \) . Then \( {T}_{B/k}^{1} = k\left\lbrack z\right\rbrack \) . The trouble is that \( B \) does not have isolated singularities. | Yes |
Proposition 16.1. If \( F \) has a versal family, then:\n\n(a) \( F\left( k\right) \) has just one element.\n\n(b) For any morphisms \( {A}^{\prime } \rightarrow A \) and \( {A}^{\prime \prime } \rightarrow A \) in \( \mathcal{C} \), the natural map\n\n\[ F\left( {{A}^{\prime }{ \times }_{A}{A}^{\prime \prime }}\right)... | Proof. (a) Since \( \operatorname{Hom}\left( {R, k}\right) \rightarrow F\left( k\right) \) is surjective, and \( \operatorname{Hom}\left( {R, k}\right) \) has just one element, so does \( F\left( k\right) \) .\n\n(b) Given elements \( {\eta }^{\prime } \in F\left( {A}^{\prime }\right) \) and \( {\eta }^{\prime \prime }... | Yes |
Theorem 16.2 (Schlessinger’s criterion). The functor \( F : \mathcal{C} \rightarrow \) (Sets) has a miniversal family if and only if:\n\n\( \left( {H}_{0}\right) F\left( k\right) \) has just one element.\n\n\( \left( {H}_{1}\right) F\left( {{A}^{\prime }{ \times }_{A}{A}^{\prime \prime }}\right) \rightarrow F\left( {A}... | Proof. The necessity of conditions \( \left( {H}_{i}\right) \) has been seen in (16.1).\n\nSo now let \( F \) be a functor satisfying conditions \( {H}_{0},{H}_{1},{H}_{2},{H}_{3} \) . First we will construct a ring \( R \) and a morphism \( {h}_{R} \rightarrow F \) . Then we will show that it has the versal family pro... | Yes |
Lemma 16.3. Let \( A,{A}^{\prime },{A}^{\prime \prime } \) be abelian groups, with maps \( {A}^{\prime } \rightarrow A,{A}^{\prime \prime } \rightarrow A \) . In the diagram\n\n\[ \n0 \rightarrow \ker {u}^{\prime } \rightarrow {A}^{\prime }{ \times }_{A}{A}^{\prime \prime }\overset{{u}^{\prime }}{ \rightarrow }{A}^{\pr... | Proof. Immediate diagram chasing. | No |
Proposition 16.4. Let \( A,{A}^{\prime },{A}^{\prime \prime } \) be rings with maps as before, and let \( {A}^{ * } = \) \( {A}^{\prime }{ \times }_{A}{A}^{\prime \prime } \) . Let \( M,{M}^{\prime },{M}^{\prime \prime } \) be modules over \( A,{A}^{\prime },{A}^{\prime \prime } \) respectively, with compatible maps \(... | Proof. (a) Since \( {A}^{\prime \prime } \rightarrow A \) is surjective and \( {M}^{\prime \prime }{ \otimes }_{{A}^{\prime \prime }}A = M \), it follows that \( {M}^{\prime \prime } \rightarrow M \) is surjective. Then by Lemma 16.3, \( {M}^{ * } \rightarrow {M}^{\prime } \) is surjective, and hence \( {M}^{ * }{ \oti... | No |
Theorem 17.1. For a given closed subscheme \( {X}_{0} \subseteq {\mathbb{P}}_{k}^{n} \), the local Hilb functor \( F \) is pro-representable. | Proof. We apply Schlessinger’s criterion (16.2). Condition \( \left( {H}_{0}\right) \) says that \( F\left( k\right) \) should have just one element, which it does, namely \( {X}_{0} \) itself.\n\nCondition \( \left( {H}_{1}\right) \) says that for every small extension \( {A}^{\prime \prime } \rightarrow A \), and any... | Yes |
Theorem 17.2. Assume \( {X}_{0} \) is projective over \( k \) and that \( {H}^{0}\left( {{X}_{0},{\mathcal{O}}_{{X}_{0}}}\right) = k \) . Then the local Picard functor for a given invertible sheaf \( {\mathcal{L}}_{0} \) on \( {X}_{0} \) is pro-representable. | Proof. We apply Schlessinger’s criterion. \( F\left( k\right) \) consists of the one element \( {\mathcal{L}}_{0} \), so \( \left( {H}_{0}\right) \) is satisfied. For \( \left( {H}_{1}\right) \), let invertible sheaves \( {\mathcal{L}}^{\prime } \) on \( {X}^{\prime } \) and \( {\mathcal{L}}^{\prime \prime } \) on \( {... | Yes |
Theorem 18.4. Suppose the hypotheses of (18.1) satisfied. Then:\n\n(a) The crude local functor \( {F}_{1} \) has a versal family.\n\n(b) \( {F}_{1} \) has a miniversal family if and only if in addition, Aut \( X \rightarrow \operatorname{Aut}{X}_{0} \) is surjective for each flat family \( X \) over the dual numbers \(... | Proof. (a) The map \( F \rightarrow {F}_{1} \) is strongly surjective, so a miniversal family for \( F \) gives a versal family for \( {F}_{1} \) .\n\n(b) and (c) are proved by arguments similar to those above (Ex. 18.1). | No |
Let us take \( {X}_{0} \) to be the affine scheme \( \operatorname{Spec}k\left\lbrack {x, y}\right\rbrack /\left\lbrack {xy}\right\rbrack \) . This is the node that was discussed previously (14.0.1). (a) It is easy to check that the automorphisms of \( {X}_{0} \) are of two types:\n\n\[ \text{(1)}\left\{ \begin{array}{... | To satisfy this equation, we find that \( {ab} = 1, f = x{f}_{1} \) , \( g = y{g}_{1} \), and \( h = a{g}_{1} + b{f}_{1} \) . Thus the lifted automorphism is of the form\n\n\[ {x}^{\prime } = \left( {a + t{f}_{1}}\right) x \]\n\n\[ {y}^{\prime } = \left( {b + t{g}_{1}}\right) y \]\n\nsubject to the condition \( {ab} = ... | Yes |
Example 18.4.2 (Pointed elliptic curves). Let \( {X}_{0} \) be a nonsingular projective curve of genus 1 over \( k \), and let \( {P}_{0} \) be a fixed point. Assume char \( k \neq 2,3 \) . We consider two functors associated to the pair \( \left( {{X}_{0},{P}_{0}}\right) \) . One, \( F\left( A\right) \) , consists of ... | Repeating the analysis of \( \left\lbrack {{57},\mathrm{{IV}},{4.7}}\right\rbrack \) we find that any family of pointed curves \( \left( {X, P}\right) \) over the dual numbers \( D \) has an equation\n\n\[ \n{y}^{2} = x\left( {x - 1}\right) \left( {x - \lambda }\right) \n\]\n\nwith \( \lambda \in D \), and that the gro... | Yes |
Theorem 19.1. In the above situation, assume that \( {X}_{0} \) is projective. Then the functor \( F \) has a miniversal family. | Proof. We apply Schlessinger's criterion (16.2), the proof being similar to the case of deformations of schemes (18.1).\n\n\( \left( {H}_{0}\right) \;F\left( k\right) \) has just one element \( {\mathcal{F}}_{0}\overset{\text{ id }}{ \rightarrow }{\mathcal{F}}_{0} \) .\n\n\( \left( {H}_{1}\right) \) Given \( {\mathcal{... | Yes |
Theorem 19.2. Assume \( {X}_{0} \) projective as above, but now assume in addition that \( {\mathcal{F}}_{0} \) is simple, i.e., \( {H}^{0}\left( {\mathcal{E}{nd}{\mathcal{F}}_{0}}\right) = k \) . Then the functors \( F \) and \( {F}_{1} \) are equal and pro-representable. | Proof. As in the case of deformations of schemes (18.2) and (18.4), it is merely a matter of showing that \( \operatorname{Aut}{\mathcal{F}}^{\prime } \rightarrow \operatorname{Aut}\mathcal{F} \) is surjective for any \( {\mathcal{F}}^{\prime } \rightarrow \mathcal{F} \) . We have assumed \( {\mathcal{F}}_{0} \) simple... | Yes |
Theorem 19.3. Let \( {X}_{0} \) be a projective scheme over \( k \), and let \( {\mathcal{E}}_{0} \rightarrow {\mathcal{F}}_{0} \rightarrow 0 \) be a surjective map of coherent sheaves. For any local Artin \( k \) -algebra \( A \), let \( X = {X}_{0}{ \times }_{k}A \), and let \( \mathcal{E} = {\mathcal{E}}_{0}{ \times... | Proof. Conditions \( \left( {H}_{0}\right) ,\left( {H}_{1}\right) ,\left( {H}_{2}\right) \) of Schlessinger’s criterion are verified as in the previous proof. The tangent space \( {t}_{F} \) is \( {H}^{0}\left( {{X}_{0},\mathcal{H}{om}\left( {{Q}_{0},{\mathcal{F}}_{0}}\right) }\right) \), which is finite-dimensional, s... | Yes |
Example 19.3.2 (Deformations of \( \mathcal{O}\left( {-1}\right) \oplus \mathcal{O}\left( 1\right) \) on \( {\mathbb{P}}_{k}^{1} \) ). Over any Artin ring \( A \), we can construct a coherent sheaf \( \mathcal{F} \) on \( {\mathbb{P}}_{A}^{1} \) as an extension\n\n\[ 0 \rightarrow {\mathcal{O}}_{{\mathbb{P}}_{A}^{1}}\l... | (a) First we take \( A = D = k\left\lbrack t\right\rbrack /\left( {t}^{2}\right) \) the dual numbers, and let the sheaf \( \mathcal{F} \) be defined by \( f = t \) . Then \( \mathcal{F}{ \otimes }_{D}k \) is the trivial extension \( {\mathcal{F}}_{0} \) . Furthermore, since \( \delta : A \rightarrow A \) is multiplicat... | Yes |
Proposition 20.1. Let \( f : {F}_{1} \rightarrow {F}_{2} \) be a morphism of functors on Artin rings. Assume that \( {F}_{1} \) and \( {F}_{2} \) both have versal families corresponding to complete local rings \( {R}_{1},{R}_{2} \) . Then there is a morphism of schemes \( \bar{f} \) : Spec \( {R}_{1} \rightarrow \) \( ... | Proof. Consider the inverse system \( \left( {{R}_{1}/{\mathfrak{m}}^{n}}\right) \) . The natural maps \( {R}_{1} \rightarrow {R}_{1}/{\mathfrak{m}}^{n} \) induce elements \( {\xi }_{n} \in {F}_{1}\left( {{R}_{1}/{\mathfrak{m}}^{n}}\right) \) forming a compatible sequence. By \( f \) we get a compatible sequence \( f\l... | Yes |
Proposition 20.2. Suppose that \( X = {X}_{0} \) is a nonsingular subscheme of \( {\mathbb{P}}^{n} \) . Then the exact sequence\n\n\[ \n{\left. 0 \rightarrow {\mathcal{T}}_{X} \rightarrow {\mathcal{T}}_{{\mathbb{P}}^{n}}\right| }_{X} \rightarrow {\mathcal{N}}_{X/{\mathbb{P}}^{n}} \rightarrow 0 \n\]\n\ngives rise to an ... | Proof. The only thing to prove is the identification of \( {\delta }^{0} \) and \( {\delta }^{1} \) with the corresponding properties of the functors \( {F}_{1} \) and \( {F}_{2} \), and this we leave to the reader. | No |
Let us apply this proposition to the case of a nonsingular surface \( X \) of degree \( d \geq 2 \) in \( {\mathbb{P}}^{3} \). | Restricting the Euler sequence on \( {\mathbb{P}}^{3} \) to \( X \) we obtain\n\n\[ \n{\left. 0 \rightarrow {\mathcal{O}}_{X} \rightarrow {\mathcal{O}}_{X}{\left( 1\right) }^{4} \rightarrow {\mathcal{T}}_{{\mathbb{P}}^{3}}\right| }_{X} \rightarrow 0 \n\]\n\nFrom the cohomology of this sequence we obtain \( {h}^{0}\left... | No |
Lemma 20.3. Let \( f \in k\left\lbrack {{x}_{0},\ldots ,{x}_{n}}\right\rbrack \) be a homogeneous polynomial of degree \( d \geq 3 \) whose zero scheme is a nonsingular hypersurface in \( {\mathbb{P}}^{n} \) and assume that char \( k \nmid d \) . Let \( {f}_{i}, i = 0,\ldots, n \), be the partial derivatives of \( f \)... | Proof. Since the zero scheme of \( f \) is nonsingular, the subset of \( {\mathbb{P}}^{n} \) defined by \( \left( {f,{f}_{0},\ldots ,{f}_{n}}\right) \) is empty. The Euler relation \( d \cdot f = \sum {x}_{i}{f}_{i} \) shows that this ideal is the same as the ideal \( \left( {{f}_{0},\ldots ,{f}_{n}}\right) \) . Theref... | Yes |
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