Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 7. If \( G \) is the internal direct product of \( {N}_{1},\ldots ,{N}_{n} \) then \( G \) is isomorphic to \( \times {}_{i}{N}_{i} \) . On the other hand \( { \times }_{i}{N}_{i} \) is the internal direct product of the standard embeddings of the \( {N}_{i} \) . | Proof: Exercise. | No |
Lemma 9. In a modular lattice, the map \( u \mapsto u \sqcap y \) is a lattice isomorphism from \( \left\lbrack {x \sqcup y/x}\right\rbrack \) to \( \left\lbrack {y/x \sqcap y}\right\rbrack \) , with inverse \( v \mapsto x \sqcup v \) . | Proof: In any lattice, \( u \mapsto u \sqcap y \) is readily verified to be a meet preserving map between the two sublattices, and \( v \mapsto x \sqcup v \) a join preserving map in the opposite direction. It therefore suffices to show that the maps are inverse to each other (since being order preserving they then pre... | Yes |
Lemma 10. In a modular lattice, suppose \( a \leq A, b \leq B \) . Then \( a \sqcup \left( {A \cap B}\right) /a \sqcup \left( {A \cap b}\right) \) and \( A \cap B/\left( {A \cap b}\right) \sqcup \left( {a \cap B}\right) \) are perspective. | Proof: The proof of lemma 2 goes through; that \( \alpha \cap \beta = \left( {a \cap B}\right) \sqcup \left( {A \cap b}\right) \) follows by modularity. | No |
Lemma 12. If \( y \leq x \) then \( l\left( {x/y}\right) = l\left( x\right) - l\left( y\right) \) . Also \( l\left( {x \sqcup y}\right) = l\left( x\right) + l\left( y\right) - l\left( {x \sqcap y}\right) \) . | Proof: There is a maximal chain from \( y \) to 0 containing \( x \) by corollary 6, and the first claim follows. The second claim follows because \( l\left( {x \sqcup u/x}\right) = l\left( {y/x \sqcap y}\right) \) by lemma 9 . | No |
Lemma 24. If \( R \) is Artinian then \( \operatorname{Rad}\left( R\right) \) is nilpotent. | Proof: Write \( I \) for \( \operatorname{Rad}\left( R\right) \) ; the sequence \( I,\left\lbrack {I}^{2}\right\rbrack ,\ldots \) is eventually constant, say \( \left\lbrack {I}^{j}\right\rbrack = \left\lbrack {I}^{n}\right\rbrack \) for \( j \geq n \) . Write \( J \) for \( \left\lbrack {I}^{n}\right\rbrack \), and su... | Yes |
Theorem 26. \( R \) is semisimple iff \( R \) is Artinian and \( \operatorname{Rad}\left( R\right) = 0 \) . | Proof: One direction was already observed. Conversely suppose \( R \) is Artinian and \( \operatorname{Rad}\left( R\right) = 0 \) . If \( L \) is a left ideal then it is not nilpotent, by the hypothesis that \( \operatorname{Rad}\left( R\right) = 0 \) . By lemma \( {25L} \) contains an idempotent \( e \) . If \( L \) i... | No |
Theorem 5. Suppose \( \left\{ {A}_{j}\right\} \) is a set of subsets of some set \( X \), each equipped with a topology. Suppose \( \left\{ {A}_{j}\right\} \) is closed under intersection, \( {A}_{j} \cap {A}_{k} \) has the topology inherited from either \( {A}_{j} \) or \( {A}_{k} \), and \( {A}_{j} \cap {A}_{k} \) is... | Proof: Suppose \( K \subseteq {A}_{j} \) is closed in \( {A}_{j} \) ; then by the hypotheses \( K \cap {A}_{k} \) is closed in \( {A}_{k} \) for all \( k \), whence \( K \) is closed in \( X \) . In particular \( {A}_{j} \) is closed in \( X \) . The argument for open intersections is similar. | Yes |
Theorem 6 (Lindelof’s theorem). A second countable space is a Lindelof space. | Proof: Let \( B \) be a countable base and \( C \) a cover of the space \( X \) . For each \( x \in X \) choose a set \( S \in B \) containing \( x \) and a subset of a set of \( C \) . For each such \( S \), of which there are only countably many, choose a superset in \( C \) . The resulting subcollection of \( C \) i... | Yes |
Corollary 7. In a second countable space any uncountable set \( S \) has a limit point. | Proof: If \( S \) has no limit point then for each \( x \in S \) we may choose a basic open set containing \( x \) but no other point of \( S \) . | No |
Theorem 8 (Urysohn’s lemma). \( X \) is normal iff, given disjoint closed sets \( A, B \) there is a continuous function \( f : X \mapsto \left\lbrack {0,1}\right\rbrack \) such that \( f\left( w\right) = 0 \) for \( w \in A \) and \( f\left( w\right) = 1 \) for \( w \in B \) . | Proof: Suppose \( X \) is normal; for each rational \( r \) of the form \( p/{2}^{n} \) for \( n \geq 0 \) and \( 0 \leq p \leq {2}^{n} \), a set \( {A}_{r} \) will be defined, open if \( r \neq 0 \) . Let \( {A}_{0} = A \) and \( {A}_{1} = {B}^{c} \) . Inductively, if \( t = \left( {{2p} + 1}\right) /{2}^{n + 1} \) le... | Yes |
Corollary 9 (Tietze extension theorem). If \( X \) is a normal space, \( K \) is a closed subset, and \( f : K \mapsto \left\lbrack {-1,1}\right\rbrack \) be a continuous function. Then there is a continuous function \( g : X \mapsto \left\lbrack {-1,1}\right\rbrack \) such that \( g \upharpoonright K = f \) . | Proof: Let \( {K}_{0} = \{ x : f\left( x\right) \leq - 1/3\} ,{L}_{0} = \{ x : f\left( x\right) \geq 1/3\} \) . Then \( {K}_{0},{L}_{0} \) are closed and disjoint, so by Urysohn’s lemma there is a continuous function \( {g}_{0} : X \mapsto \left\lbrack {-1/3,1/3}\right\rbrack \) which is \( - 1/3 \) on \( {K}_{0} \) an... | No |
Lemma 10 (shrinking lemma). Suppose \( \left\{ {{U}_{j} : j \in J}\right\} \) is a point finite open cover of a normal space \( X \) . Then there is an open cover \( \left\{ {V}_{j}\right\} \) where \( {V}_{j}^{\mathrm{{cl}}} \subseteq {U}_{j} \) . | Proof: Say that a partial function \( \phi \) on \( J \) is a partial shrinking if \( \phi \left( j\right) \) is an open set whose closure is contained in \( {U}_{j} \), and the sets \( \phi \left( j\right) \) for \( j \in \operatorname{Dom}\left( \phi \right) \), together with the sets \( {U}_{j} \) for \( j \notin \o... | Yes |
Corollary 11. Under the same hypotheses, there are functions \( {f}_{j} : {U}_{j} \mapsto \left\lbrack {0,1}\right\rbrack \), such that \( {f}_{j}\left( x\right) = 0 \) if \( x \notin {U}_{j} \), and for each \( x \in X,\mathop{\sum }\limits_{j}{f}_{j}\left( x\right) = 1 \) (where by hypothesis the sum is finite). | Proof: By the shrinking lemma and Urysohn’s lemma, choose a function \( {g}_{j} \) which is 1 on a closed subset of \( {U}_{j} \) and 0 outside \( {U}_{j} \) . Then set \( {f}_{j}\left( x\right) = {g}_{j}\left( x\right) /\mathop{\sum }\limits_{j}{g}_{j}\left( x\right) \) . | Yes |
Lemma 16. If \( C \) is a locally finite cover of a connected space then \( C \) is countable (or finite). | Proof: Let \( {D}_{0} \) be any member of \( C \), and let \( {D}_{n + 1} \) be \( {D}_{n} \) with the sets of \( C \) which intersect sets of \( {D}_{n} \) added. Let \( D = \mathop{\bigcup }\limits_{n}{D}_{n} \), and let \( Y = \bigcup D \) . \( Y \) is certainly open. If \( U \in C \) intersects \( Y \) then it is i... | No |
Theorem 17. Suppose \( X \) is a locally compact Hausdorff space.\na. If \( X \) is second countable then \( X \) is \( \sigma \) -compact.\nb. If \( X \) is \( \sigma \) -compact \( X \) is paracompact.\nc. If \( X \) is connected and paracompact then \( X \) is \( \sigma \) -compact. | Proof: For part a, let \( {U}_{i} \) be a countable base; we claim that the \( {U}_{i} \) for which \( {U}_{i}^{\text{cl }} \) is compact are still a base, and taking the closures of these yields the theorem. Let \( V \) be open and \( x \in V \) . Then since \( X \) is locally compact Hausdorff there is a compact neig... | Yes |
Theorem 23 (Tychanoff’s theorem). The product of a collection of compact spaces is compact. | The sets \( K \times \left( {{ \times }_{k \neq j}{X}_{j}}\right), K \subseteq {X}_{j}, K \) closed, form a subbase for the closed sets in the product topology. Let \( C \) be a collection of these with the finite intersection property. Let \( {C}_{i} = \left\{ {{\pi }_{i}\left( K\right) : K \in C}\right\} \) where \( ... | Yes |
Lemma 28. Suppose \( \left\langle {{X}_{i},{d}_{i}}\right\rangle, i \geq 1 \), is a countable family of pseudo-metric spaces. The binary function on \( { \times }_{i}{X}_{i} \) given by\n\n\[ d\left( {\left\langle {x}_{i}\right\rangle ,\left\langle {y}_{i}\right\rangle }\right) = \mathop{\sum }\limits_{i}\min \left( {1... | Proof: First note that \( \min \left( {1, d\left( {x, y}\right) }\right) \) is a pseudo-metric (exercise 10), and has the same topology since the open balls of radius less than 1 are a base for the topology of either function. We leave the verification that \( d \) is a pseudo-metric to exercise 10 . Write \( x \) for ... | No |
a. The map \( j : X \mapsto {X}^{\text{cmpl }} \) induced by \( {j}_{1} \) is an isometry. | For part a, clearly \( d\left( {{j}_{1}\left( x\right) ,{j}_{1}\left( y\right) }\right) = d\left( {x, y}\right) \) . | No |
Lemma 1. If, for a family \( {\alpha }_{i} : c \mapsto F\left( i\right) \), for each \( a{\operatorname{Hom}}^{a} \) maps the family to a limit cone, then \( \langle c,\alpha \rangle \) is a limit for \( F \) . Dually if for a family \( {\alpha }_{i} : F\left( i\right) \mapsto c \), for each \( b{\operatorname{Hom}}_{b... | Proof: Suppose \( \left\{ {{\alpha }_{i} : i \in J}\right\} \) is a family and \( \mu : F\left( i\right) \mapsto F\left( j\right) \) . By hypothesis (in fact using only that \( {\operatorname{Hom}}^{c} \) maps the family to a cone), \( {\mu }^{L}{\alpha }_{i}^{L} = {\alpha }_{j}^{L} \), so \( \mu {\alpha }_{i} = {\mu }... | Yes |
Lemma 3. Suppose \( \left\langle {{F}_{i},{G}_{i},{\psi }_{i}}\right\rangle, i = 1,2 \), are adjunctions from \( A \) to \( C \), and \( \alpha \) is a natural transformation from \( {G}_{1} \) to \( {G}_{2} \) . Then there is a unique natural transformation \( \beta \) from \( {F}_{2} \) to \( {F}_{1} \) such that the... | Proof: The function\n\n\[ \n{\beta }_{ac}^{\prime } = {\psi }_{2ac}^{-1}{\alpha }_{c}^{L}{\psi }_{1ac} \n\]\n\nis the unique one making the diagram commutative. One verifies that the \( {\beta }_{ac}^{\prime } \) are the components of a natural transformation from \( \operatorname{Hom}\left( {{F}_{1}\left( a\right), c}... | Yes |
Theorem 4. Suppose \( G \) is a functor from \( {B}^{\mathrm{{op}}} \times C \) to \( A \), and each functor \( G\left( {b, - }\right) \) has a left adjoint \( {F}_{b} \) . Then there is a unique functor \( F : A \times B \mapsto C \) agreeing with the \( {F}_{b} \), such that the equivalence (1) holds. The dual statem... | Proof: The object function of \( F \) is clear; the arrow function must be specified. Suppose \( f : {b}_{2} \mapsto {b}_{1} \), and let \( {F}_{i} = {F}_{{b}_{i}},{G}_{i} = G\left( {{b}_{i}, - }\right) \), and \( {\alpha }_{c} = G\left( {f,{\iota }_{c}}\right) \) . By (1),(2) commutes; so by lemma 3 natural transforma... | Yes |
Lemma 2. If \( {g}_{i} : a \mapsto {b}_{i} \) is a pullback for \( {f}_{i} : {b}_{i} \mapsto c \) and \( {f}_{1} \) is epic, then \( {g}_{2} \) is epic and \( {f}_{1}^{K} \equiv {g}_{1}{g}_{2}^{K} \) . Dually if \( {g}_{i} : {b}_{i} \mapsto a \) is a pushout for \( {f}_{i} : c \mapsto {b}_{i} \) and \( {f}_{1} \) is mo... | Proof: Let \( {p}_{i} \) and \( {m}_{i} \) be the projections and injections for \( {b}_{1} \times {b}_{2} \) . Let \( \phi = {f}_{1}{p}_{1} - {f}_{2}{p}_{2} \), and \( k = \operatorname{Ker}\left( \phi \right) \) , so that \( {g}_{i} \) may be taken as \( {p}_{i}k \) . Now, \( \phi {m}_{1} = {f}_{1} \), so \( \phi \) ... | Yes |
Lemma 3. \( \operatorname{Im}\left( f\right) \equiv \operatorname{Ker}\left( g\right) \) iff \( {gf} = 0 \), and if \( {gx} = 0 \) then \( {x\epsilon } = {ft} \) for some \( t \) and epic \( \epsilon \) . Dually \( \operatorname{Coim}\left( f\right) \equiv \operatorname{Coker}\left( g\right) \) iff \( {fg} = 0 \), and ... | Proof: Write \( f \) as a coimage-image pair \( {me} \) . If \( \operatorname{Ker}\left( g\right) = m \) then \( {gm} = 0 \), so \( {gf} = 0 \) ; and if \( {gx} = 0 \) then \( x = {m\theta } \) for some \( \theta \), so by pullback \( {x\epsilon } = {m\theta \epsilon } = {met} = {ft} \) for some \( t \) and epic \( \ep... | Yes |
Theorem 7 (NINE LEMMA). Given a diagram in an Abelian category\n\n\n\nwith exact columns and bottom two rows, there exist unique \( {m}_{0} \) and \( {e}_{0} \) making the diagram commutative; further the top row ext... | Proof: Exercise 5. | No |
Theorem 8 (Middle nine lemma). Suppose that, in the diagram of theorem 7 short exact sequences are given in the first and third rows, and arrows \( m, e \) in the middle row. Then the middle row extends to a short exact sequence. | Proof: Exercise 6. | No |
Lemma 16. If given a left ideal \( I \subseteq R \) and a linear map \( f : I \mapsto A \), there is a linear map \( {f}^{\prime } : R \mapsto A \) extending \( f \), then \( A \) is injective. | Proof: Suppose \( C \) is a submodule of \( B \), and \( f : C \mapsto A \) . By a typical application of Zorn’s lemma (see for example theorem 15.4) it suffices to show that, for a submodule \( D \) with \( C \subseteq D \), and \( x \in B - D \), a function \( g : D \mapsto A \) can be extended to the submodule gener... | Yes |
Theorem 26. Suppose \( R \) is an integral domain, and \( M \) an \( R \) -module. Let \( Q \) denote \( R/{R}_{ \neq } \), and let \( J \) denote \( Q/R \) . Then \( \operatorname{Tor}\left( M\right) \cong {\operatorname{Tor}}_{1}\left( {M, J}\right) \) . | Proof: \( R \) and \( Q \) are both flat, so from the long exact sequence for \( 0 \rightarrow R \rightarrow Q \rightarrow J \rightarrow 0,{\operatorname{Tor}}_{n}\left( {M, J}\right) = 0 \) for \( n \geq 2 \) . If \( M \) is torsion then \( M \otimes Q = 0 \) (if \( {rm} = 0 \) for \( r \neq 0 \) then \( m \otimes {r}... | No |
Corollary 3. Any two algebraic closures of a field \( F \) are isomorphic, by an isomorphism fixing \( F \) . | Proof: Let \( {K}_{1} \) and \( {K}_{2} \) be two algebraic closures. By the theorem there is an embedding \( \sigma : {K}_{1} \mapsto {K}_{2} \) fixing \( F \) . Further \( \sigma \left\lbrack {K}_{1}\right\rbrack \) is algebraically closed and \( {K}_{2} \) is algebraic over it, so \( \sigma \left\lbrack {K}_{1}\righ... | Yes |
Lemma 4. Suppose \( E \supseteq F \) is an extension of fields. Let \( S \subseteq E \) be an algebraically independent subset of size \( k > 0 \), and let \( T \) be a subset of size \( k + 1 \) . If \( S \cup \{ t\} \) is algebraically dependent for each \( t \in T \) then \( T \) is algebraically dependent. | Proof: Suppose \( \{ 1,\ldots, k\} \) is the disjoint union of \( {I}_{s} \) and \( {I}_{t} \) ; we claim that every \( {s}_{i} \) and \( {t}_{i} \) is algebraic over \( F\left( {\left\{ {{s}_{i} : i \in {I}_{s}}\right\} \cup \left\{ {{s}_{i} : i \in {I}_{t}}\right\} }\right) \) . The claim is proved by induction on \(... | No |
Lemma 8. Suppose \( \\left| x\\right| \) satisfies the restrictions of lemma 7. Then \( \\left| x\\right| \) satisfies the triangle inequality iff \( \\left| x\\right| \\leq 1 \) implies \( \\left| {x + 1}\\right| \\leq 2 \) . | Proof: If the triangle inequality holds and \( \\left| x\\right| \\leq 1 \) then \( \\left| {x + 1}\\right| \\leq 2 \) . For the converse, supposing \( \\left| x\\right| \\leq \\left| y\\right| \), \( \\left| {x + y}\\right| = \\left| y\\right| \\left| {x/y + 1}\\right| \\leq 2\\left| y\\right| = 2\\max \\left( {\\left... | Yes |
Lemma 12. An absolute value \( X \) is non-Archimedean iff \( \left| x\right| \leq 1 \) for all \( x \) in the prime field of \( K \) . | Proof: One direction follows by lemma 11. The proof in the other direction proceeds similarly to that of the converse direction of lemma 10; using the binomial theorem and the hypothesis one obtains \( {\left| x + y\right| }^{n} \leq \) \( \left( {n + 1}\right) {\left( \max \left( \left| x\right| ,\left| y\right| \righ... | Yes |
Corollary 17. Suppose \( A, B, C, D \) are commutative rings.\na. If \( B \) is a finitely generated \( A \) -module then \( B \supseteq A \) is integral.\nb. If \( B \supseteq A \) is integral and \( B \) is finitely generated over \( A \) as a ring then \( B \) is a finitely generated \( A \) -module.\nc. If \( C \su... | Proof: For part a, let \( M = B \) in the theorem. For part b, first note that if \( B \) is a finitely generated \( A \) -module, say by \( {u}_{1},\ldots ,{u}_{m} \), and \( C \) is a finitely generated \( B \) -module, say by \( {v}_{1},\ldots ,{v}_{n} \), then \( C \) is a finitely generated \( A \) -module, by \( ... | Yes |
Lemma 29. Suppose \( {N}_{i} \) is a \( P \) -primary submodule for \( 1 \leq i \leq r \) and \( N = { \cap }_{1 \leq i \leq r}{N}_{i} \) ; then \( N \) is \( P \) -primary. | Proof: By hypothesis \( \operatorname{Rad}\left( {\operatorname{Ann}\left( {M/{N}_{i}}\right) }\right) = P \) for all \( i \), so by theorem 26 and the preceding paragraph \( \operatorname{Rad}\left( {\operatorname{Ann}\left( {M/N}\right) }\right) = P \) . Suppose \( {ax} \in N \) and \( x \notin N \) ; then for some \... | Yes |
Lemma 30. Suppose \( N \) is a \( P \) -primary submodule, and \( x \in M \) . Then \( \operatorname{Rad}\left( \left( {N : x}\right) \right) = R \) if \( x \in N \), else \( P \) . | Proof: If \( x \in N \) then clearly \( \left( {N : x}\right) = R \) ; suppose \( x \notin N \) . If \( a \in P \) then \( {a}^{n} \in \operatorname{Ann}\left( {M/N}\right) \) for some \( n \), so \( {a}^{n} \in \left( {N : x}\right) \), so \( a \in \operatorname{Rad}\left( \left( {N : x}\right) \right) \) . If \( a \n... | Yes |
Corollary 35. If \( F \) is algebraically closed an ideal \( I \) in \( F\left\lbrack \mathbf{x}\right\rbrack \) is maximal iff it is the ideal generated by \( \left\{ {{x}_{1} - {a}_{1},\ldots ,{x}_{n} - {a}_{n}}\right\} \) for some \( {a}_{1},\ldots ,{a}_{n} \in F \) . | Proof: One direction is proved above. Let \( I \) be maximal, so that \( F\left\lbrack \mathbf{x}\right\rbrack /I \) is a field. By the theorem, this field is an algebraic extension of \( F \), and since \( F \) is algebraically closed it equals \( F \) . Lemma 33 then shows that it has the specified form. | Yes |
Corollary 36 (weak Nullstellensatz). If \( F \) is algebraically closed and \( I \) is a proper ideal in \( F\left\lbrack \mathbf{x}\right\rbrack \) then \( V\left( I\right) \) is nonempty. | Proof: Choose a maximal ideal \( J \) containing \( I \), and suppose it is generated by \( \left\{ {{x}_{1} - {a}_{1},\ldots ,{x}_{n} - {a}_{n}}\right\} \) . Then \( \mathbf{a} \in V\left( I\right) \) where \( \mathbf{a} = \left\langle {{a}_{1},\ldots ,{a}_{n}}\right\rangle \) . | Yes |
Theorem 37 (Nullstellensatz). Suppose \( F \) is algebraically closed, and \( I \) is an ideal in \( F\\left\\lbrack \\mathbf{x}\\right\\rbrack \) . Suppose \( p \\in F\\left\\lbrack \\mathbf{x}\\right\\rbrack \) vanishes on \( V\\left( I\\right) \) . Then \( {p}^{m} \\in I \) for some integer \( m \), that is, \( p \\... | Proof: Suppose \( {p}_{1},\\ldots ,{p}_{t} \) generate \( I \), and let \( J \) be the ideal in \( F\\left\\lbrack {\\mathbf{x}, y}\\right\\rbrack \) generated by \( {p}_{1},\\ldots ,{p}_{t},1 - {yp} \) (this is called Rabinowitsch’s trick). \( J \) is readily verified to have an empty zero set, whence it is the entire... | Yes |
Lemma 47. Suppose \( x, d \in {\mathcal{H}}_{h} \) and \( d \neq 0 \) . Then there are \( q, r \in \mathcal{H} \) with \( x = {qd} + r \) and \( Q\left( r\right) < Q\left( d\right) \) . | Proof: First we show the theorem when \( d \) is an integer \( m \) . Let \( \zeta = \left( {1 + i + j + k}\right) /2 \) . Every \( x \in {\mathcal{H}}_{h} \) can be written uniquely as \( {x}_{0}^{\prime }\zeta + {x}_{1}^{\prime }i + {x}_{2}^{\prime }j + {x}_{3}^{\prime }k \), indeed \( {x}_{0}^{\prime } = 2{x}_{0} \)... | Yes |
Lemma 48. If \( p \) if an odd prime then for some \( r, s \) with \( 0 \leq r, s < p,1 + {r}^{2} + {s}^{2} \equiv 0{\;\operatorname{mod}\;p} \) . | Proof: The numbers \( {x}^{2} \) with \( 0 \leq x \leq \left( {p + 1}\right) /2 \) are incongruent \( {\;\operatorname{mod}\;p} \), and likewise the numbers \( 1 - {y}^{2} \) with \( 0 \leq y \leq \left( {p + 1}\right) /2 \) . Therefore \( {x}^{2} \equiv 1 - {y}^{2}{\;\operatorname{mod}\;p} \) for some such \( x, y \) ... | No |
Corollary 4. Suppose \( D \) is a complete lattice and \( G \) is an order preserving map from \( D \) to a poset \( C \) . Then \( G \) has a left adjoint iff it preserves meets. | Proof: It has already been shown that a right adjoint preserves meets. For the converse, let \( F\left( c\right) = \) \( \sqcap \{ d \in D : c \leq G\left( d\right) \} \) . Since \( G \) preserves meets \( {GF}\left( c\right) = \sqcap \{ G\left( d\right) : c \leq G\left( d\right) \} \), whence \( {GF}\left( c\right) \g... | Yes |
Lemma 11. Suppose \( F \) is a functor from \( A \) to \( B, G \) is a functor from \( B \) to \( A \), and \( {FG} \) and \( {GF} \) are naturally equivalent to the identity functor. Then \( F \) and \( G \) are faithful, full, and both left and right adjoints to each other. | Proof: Let \( \alpha \) be the natural equivalence in \( A \) from \( {GF} \) to the identity. Suppose \( f : {a}_{1} \mapsto {a}_{2} \) is an arrow in \( A \) ; the equation \( f{\alpha }_{{a}_{1}} = {\alpha }_{{a}_{2}}{GF}\left( f\right) \) shows that \( f \) is determined by \( F\left( f\right) \), i.e., \( F \) is ... | Yes |
Lemma 12. Suppose \( A, B \in {\operatorname{Lat}}_{B} \) and \( f : A \mapsto B \) is a morphism. Let \( {\beta }_{a}^{A} \) be as above.\na. For \( a, b \in A,{\beta }_{a \sqcap b} = {\beta }_{a} \cap {\beta }_{b} \) . | Proof: For part \( \mathrm{a}, f \in {\beta }_{a \sqcap b} \) iff \( f\left( {a \sqcap b}\right) = 1 \) iff \( f\left( a\right) \sqcap f\left( b\right) = 1 \) iff \( f\left( a\right) = 1 \) and \( f\left( b\right) = 1 \) iff \( f \in {\beta }_{a} \cap {\beta }_{b} \) . | Yes |
Lemma 13. If \( A \in {\operatorname{DLat}}_{B} \), an open set in \( \operatorname{Spec}\left( A\right) \) is compact iff it is \( {\beta }_{a} \) for some \( a \in A \) . | Proof: Suppose \( {\beta }_{a} \subseteq { \cup }_{b \in S}{\beta }_{b} \), i.e., if \( f\left( a\right) = 1 \) then for some \( b \in {Sf}\left( b\right) = 1 \) . Let \( I \) be the ideal generated by \( S \) and let \( F \) be the filter \( {a}^{ \geq } \) . By lemma 11.16, if \( I \cap F = \varnothing \) then there ... | Yes |
Lemma 25. Let \( L \) be a complete lattice, and \( T \subseteq \operatorname{Pt}\left( L\right) \) . Then \( T \) is compact iff \( {\sigma }_{T} \) is Scott open. | Proof: First, note that in a lattice, a subset \( U \) is Scott open iff it is \( \geq \) -closed, and whenever \( \sqcup S \in U \) then for some finite \( F \subseteq S, \sqcup F \in U \) . Second, recall that \( { \cup }_{x \in S}{\beta }_{x} = {\beta }_{\sqcup S} \) ; whence \( T \subseteq { \cup }_{x \in S}{\beta ... | Yes |
Lemma 29. Let \( L \) be a complete lattice, and let \( \prec \) be \( { \prec }_{T} \) where \( T \) is the Scott topology.\na. If \( x \prec y \) then \( x \ll y \). | Proof: Suppose \( y \in U \) where \( U \) is Scott open and \( U \subseteq {x}^{ \geq } \), and suppose \( y \leq \sqcup S \) . Then \( \sqcup S \in U \), so \( \sqcup F \in U \) for some finite \( F \subseteq S \), and \( x \leq \sqcup F \) . This proves part a. | Yes |
Lemma 30. A topology \( T \) on \( \operatorname{Hom}\left( {Y, Z}\right) \) is in \( {\mathcal{T}}_{2} \) iff it makes \( {\varepsilon }_{YZ} \) continuous. | Proof: For any topology \( T\bar{\varepsilon } \), being the identity, is continuous from \( \operatorname{Hom}\left( {Y, Z}\right) \) to \( \operatorname{Hom}\left( {Y, Z}\right) \), both equipped with \( T \) . If \( T \in {\mathcal{T}}_{2} \) it follows that \( \varepsilon \) is continuous. Conversely suppose \( T \... | No |
Lemma 31. \( {\mathcal{T}}_{1} \) is closed down, \( {\mathcal{T}}_{2} \) is closed up, and any topology in \( {\mathcal{T}}_{1} \) is weaker than any topology in \( {\mathcal{T}}_{2} \) . In particular, there is at most one exponential topology. | Proof: If \( \bar{g} \) is continuous when \( \operatorname{Hom}\left( {Y, Z}\right) \) is equipped with \( T \), and \( {T}^{\prime } \subseteq T \), then \( \operatorname{Hom}\left( {Y, Z}\right) \) is continuous when equipped with \( {T}^{\prime } \) ; thus, \( {\mathcal{T}}_{1} \) is closed down. If \( \bar{g} \) i... | Yes |
Lemma 33. Let \( S \) denote the Scott topology on \( \Omega \left( Y\right) \), and \( T \) any other topology.\na. \( S \in {\mathcal{T}}_{1} \). | Proof: Suppose \( W \subseteq \times Y \) is open, \( x \in W \), and \( O \) is a Scott open neighborhood of \( {W}_{x} \) . To prove part a, we will find an open neighborhood \( U \) of \( x \) such that \( {W}_{u} \in O \) for all \( u \in U \) . For each \( y \) with \( \langle x, y\rangle \in {W}_{x} \) choose ope... | Yes |
Corollary 35. A sober space is exponentiable iff it is strongly locally compact. A Hausdorff space is exponentiable iff it is locally compact. For a sober exponentiable space \( Y \) the exponential topology on \( \operatorname{Hom}\left( {Y, Z}\right) \) is the compact-open topology. | Proof: The first two claims follows by lemma 28. For the third, as observed above the compact-open topology is \( {T}^{ * } \) where \( T \) has the set of \( {K}^{u} \) for \( K \) compact as a subbase; so it suffices to show that these are a base for the Scott topology. Given a set \( O \) open in the Scott topology ... | Yes |
Lemma 1. Suppose \( X \) is a real Hilbert space, \( S \subseteq X \) is a nonempty closed convex subset, and \( x \in X - S \) . Then there is a unique point \( y \in S \) such that \( \left| {y - x}\right| = \inf \{ \left| {w - x}\right| : w \in S\} \) . | Proof: By translating \( S \) we may suppose that \( x = 0 \) . Let \( d = \inf \{ \left| w\right| : w \in S\} \), and let \( {w}_{n} \in S \) be such that \( \left| {w}_{n}\right| \) converges to \( d \) . Since \( S \) is convex, \( \left| {\left( {{w}_{n} + {w}_{m}}\right) /2}\right| \in S \), so \( \left| {{w}_{n} ... | Yes |
Lemma 2. Suppose \( X \) is a real Hilbert space, \( S \subseteq X \) is a nonempty closed convex subset and \( x \in X - S \) . Then \( z \in S \) is the unique point which minimizes \( \left| {y - x}\right| \) iff for all \( y \in S,\left( {x - z}\right) \cdot \left( {y - z}\right) \leq 0 \) . | Proof: By the convexity of \( S, z \) is the closest point iff for each \( y \in S \), for each \( t \in \left\lbrack {0,1}\right\rbrack {\left| x - \left( z + t\left( y - z\right) \right) \right| }^{2} \geq \) \( {\left| x - z\right| }^{2} \) . By algebra the requirement for \( y \) holds iff for each \( t \in \left\l... | Yes |
Corollary 8. A set \( S \subseteq {\mathcal{R}}^{n} \) is a polyhedral cone iff \( S = \operatorname{Ccone}\left( {{a}_{1},\ldots ,{a}_{m}}\right) \) for some \( {a}_{1},\ldots ,{a}_{m} \). | Proof: Suppose \( S = \operatorname{Ccone}\left( {{a}_{1},\ldots ,{a}_{m}}\right) \). By restricting to a subspace we can assume \( A \) has full rank. Let \( T \) be the set of halfspaces containing the \( {a}_{i} \), which are a halfspace of the hyperplane spanned by \( n - 1 \) linearly independent \( {a}_{i} \); cl... | Yes |
Corollary 11. A polyhedron is bounded iff it is the convex hull of a finite set of points. | Proof: If a polyhedron \( S \) is bounded, the cone as in the theorem must be \( \varnothing \) . Conversely by the theorem the convex hull of a finite set of points is a polyhedron, and it is clearly bounded. | Yes |
Lemma 4. Let \( S \subseteq {\mathcal{R}}^{n} \), and let \( \left\{ {S}_{i}\right\} \) be a (finite or) countable disjoint collection of sets with \( S = \cup {S}_{i} \) . Then\n\na. \( \sum {\mu }_{I}\left( {S}_{i}\right) \leq {\mu }_{I}\left( S\right) \),\n\nb. \( \sum {\mu }^{ * }\left( {S}_{i}\right) \geq {\mu }^{... | Proof: For part a it suffices to show that if \( c < {\mu }_{1} = \sum {\mu }_{I}\left( {S}_{i}\right) \) then \( c < {\mu }_{I}\left( S\right) \) . If \( {\mu }_{1} \) is finite there are packings \( {P}_{i} \) of \( {S}_{i} \) with \( \nu \left( {P}_{i}\right) \geq \left( {c/{\mu }_{1}}\right) {\mu }_{I}\left( {S}_{i... | Yes |
Theorem 6. Suppose \( T \) is a linear transformation of determinant \( \lambda \neq 0 \), and \( S \subseteq {\mathcal{R}}^{n} \) is Lebesgue measurable. Then \( T\left\lbrack S\right\rbrack \) is Lebesgue measurable, and \( \mu \left( {T\left\lbrack S\right\rbrack }\right) = \left| \lambda \right| \mu \left( S\right)... | Proof: The theorem is clear for \( T \) a diagonal or permutation matrix and \( S \) a cell, whence for any measurable \( S \) . It thus suffices to show the theorem for a matrix of an elementary operation. For such a matrix in \( {\mathcal{R}}^{2} \), a cell \( C \) may be dissected into rectangles and triangles whose... | No |
Corollary 10. Suppose \( S \subseteq {\mathcal{R}}^{n} \) is bounded and convex, \( S = - S, L \) is a full rank lattice, and \( \mu \left( S\right) > \) \( {2}^{n}\det \left( L\right) \) ; then \( S \) contains a nonzero point of \( L \) . | Proof: Let \( S = G\left\lbrack T\right\rbrack \) where \( G \) is a generator matrix for \( L \) . Then \( T \) satisfies the hypotheses of the theorem, and the image of a nonzero point of \( {\mathcal{Z}}^{n} \) is a nonzero point of \( L \) . | No |
Corollary 11. Let \( G \) be an \( n \times n \) real matrix with entries \( {g}_{ij} \) and \( \det \left( G\right) \neq 0 \), and let \( {\zeta }_{i}\left( x\right) \) be the linear form \( \mathop{\sum }\limits_{j}{g}_{ij}{x}_{j} \) . Suppose \( {\lambda }_{i} \) for \( 1 \leq i \leq n \) are positive real numbers w... | Proof: Let \( S = \left\{ {x : \left| {x}_{i}\right| < {\lambda }_{i}}\right. \) for \( \left. {1 \leq i \leq n}\right\} \) and apply corollary 10 . | Yes |
Lemma 23. For a Coxeter group \( G \), and \( w \in G \), if \( l\left( {w{\rho }_{r}}\right) = l\left( w\right) + 1 \) then \( \bar{w}\left( {e}_{r}\right) \in \) Plc. | Proof: Suppose \( w, r \) is a counterexample with \( l\left( w\right) \) least. Since \( l\left( {e}_{r}\right) = 1, l\left( w\right) = 1 \) . For some \( {w}_{1} \) and \( s, w = {w}_{1}{\rho }_{s} \) and \( l\left( {w}_{1}\right) = l\left( w\right) - 1 \) . Inductively, if \( l\left( {{w}_{i}{\rho }_{t}}\right) = l\... | Yes |
Theorem 24. For a Coxeter group \( G \), the homomorphism \( w \mapsto \bar{w} \) from \( G \) to \( \bar{G} \) is an isomorphism. | Proof: It is surjective by definition. If \( \bar{w} = 1 \) and \( w \neq 1 \) then for some \( r, w = {w}^{\prime }{\rho }_{r} \) and \( l\left( w\right) = l\left( {w}^{\prime }\right) + 1 \).\n\nBy lemma \( {23},{\bar{w}}^{\prime }\left( {e}_{r}\right) \in \) Plc. But \( - {\bar{w}}^{\prime }\left( {e}_{r}\right) = {... | Yes |
Lemma 9. Let \( f : X \mapsto Y \) be a vector space homomorphism spaces in \( F \) -NLS. Then \( f \) is continuous iff there is a real number \( r > 0 \) such that \( \left| {f\left( x\right) }\right| \leq r\left| x\right| \) . | Proof: It was observed in section 3 that \( f \) is continuous iff it is continuous at 0 . If \( r \) exists then \( f \) is continuous at 0 ; indeed, given \( \epsilon \) if \( \left| x\right| < \epsilon /r \) then \( \left| {f\left( x\right) }\right| < \epsilon \) (nontriviality is not required for this direction). F... | Yes |
Corollary 10. Two norms \( {\left| x\right| }_{1} \) and \( {\left| x\right| }_{2} \) in a vector space \( X \) induce the same topology iff there are real numbers \( {r}_{1} > 0 \) and \( {r}_{2} > 0 \) such that \( {\left| x\right| }_{1} < {r}_{1}{\left| x\right| }_{2} \) and \( {\left| x\right| }_{2} < {r}_{2}{\left... | Proof: The identity function is continuous from norm 1 to norm 2 iff \( {r}_{1} \) exists, and from norm 2 to norm 1 iff \( {r}_{2} \) exists. | No |
Lemma 11. Suppose \( Y \subseteq X \) is a subspace. Let \( \left| {x + Y}\right| = \inf \{ \left| {x + y}\right| : y \in Y\} \) . a. \( \left| {x + Y}\right| \) is a pseudo-norm on \( X/Y \) . | Proof: Letting \( y \) vary over \( Y \) , \[ \left| {{ax} + Y}\right| = \inf \{ \left| {{ax} + y}\right| \} = \inf \{ \left| {{ax} + {ay}}\right| \} = \left| a\right| \inf \{ \left| {x + y}\right| \} = \left| a\right| \left| {x + Y}\right| . \] Also \[ \left| {{x}_{1} + {x}_{2} + Y}\right| = \inf \left\{ \left| {{x}_{... | Yes |
Lemma 13. Suppose \( X \) is a normed linear space and \( f : X \mapsto F \) is a linear functional whose kernel is not all of \( X \) . Then \( f \) is continuous iff \( \operatorname{Ker}\left( f\right) \) is closed. Further in this case \( X \) is homeomorphic to the product of \( \operatorname{Ker}\left( f\right) \... | Proof: The first claim follows by lemma 5. In the proof of of b implies c, open balls may be used rather than balanced sets, noting that \( f\left\lbrack {B}_{0, r}\right\rbrack \) is bounded if it is not all of \( F \) ; also lemma 9 may be appealed to. Now, if \( w \notin \operatorname{Ker}\left( f\right) \) then \( ... | Yes |
Corollary 27. Suppose \( \langle L, R,\alpha \rangle \) is an adjunction. If \( R \) is a functor to groups then \( L \) is a functor to cogroups, and conversely. | Proof: If \( R \) is a functor to groups then given \( H,\operatorname{Hom}\left( {H, R\left( -\right) }\right) \) is a group, and so \( \operatorname{Hom}\left( {L\left( H\right) , - }\right) \) is. Remaining details are left to the reader. | No |
Lemma 2. Let \( X \) be a nonempty topological space and \( S \) a nonempty subset.\na. If \( X \) is irreducible and \( S \) is open then \( S \) is irreducible. | Proof: For part a, if \( {U}_{1},{U}_{2} \) are disjoint open subsets of \( S \) they are disjoint open subsets of \( X \) . | No |
Lemma 9. Suppose \( F \) is a field, \( p \in F\left\lbrack x\right\rbrack \), and \( e \) is greater than the degree of any variable in \( p \) . For \( 1 \leq i < n \) let \( {y}_{i} = {x}_{i} - {x}_{n}^{{e}^{i}} \) . Then \( p = q\left( {{y}_{1},\ldots ,{y}_{n - 1},{x}_{n}}\right) \) where \( q \) is monic as a poly... | Proof: If \( {x}_{1}^{{d}_{1}}\cdots {x}_{n}^{{d}_{n}} \) is a monomial of \( p \), when written in terms of \( {y}_{1},\ldots ,{y}_{n - 1},{x}_{n} \) its highest degree term is \( {x}_{n}^{f} \) where \( f = {d}_{n} + {d}_{1}e + \cdots + {d}_{n - 1}{e}^{n - 1} \) . The \( {d}_{i} \) are just the digits in the \( e \) ... | Yes |
Lemma 11. Suppose \( A \) is a finitely generated \( F \) -algebra over a field \( F \), and an integral domain. If \( P \) is a nontrivial prime ideal of \( A \) then \( \operatorname{Trdeg}\left( {A/P}\right) < \operatorname{Trdeg}\left( A\right) \) . | Proof: Suppose \( {a}_{1} + P,\ldots ,{a}_{n} + P \) are algebraically independent, and \( b \in P \) is nonzero. Suppose \( p\left( {{a}_{1},\ldots ,{a}_{n}, b}\right) = 0 \) ; then \( p\left( {{a}_{1}, + P\ldots ,{a}_{n} + P,0}\right) = 0 \) in \( A/P \), so \( p\left( {{x}_{1},\ldots ,{x}_{n},0}\right) \) must be th... | No |
Lemma 12. Suppose \( F \) is a field; then \( \operatorname{Kdim}\left( {F\left\lbrack x\right\rbrack }\right) = n \) . | Proof: For \( 0 \leq i \leq n \) let \( {P}_{i} \) be the ideal generated by \( {x}_{1},\ldots ,{x}_{i}.{P}_{i} \) is a prime ideal, because \( p \in {P}_{i} \) iff, as a polynomial with coefficients in \( F\left\lbrack {{x}_{i + 1},\ldots ,{x}_{n}}\right\rbrack \), the constant term is 0 . Further these ideals form a ... | Yes |
Lemma 14. Suppose \( R \) is a Noetherian ring. Then there are only finitely many minimal prime ideals in \( R \) . | Proof: Let \( S \) be the set of ideals \( I \) such that \( R/I \) has infinitely many minimal prime ideals. Since \( R \) is Noetherian, if \( S \) is nonempty then it contains a maximal element \( I \) . We may thus assume that \( R \) is such that for any ideal \( I \) in \( R \), there are only finitely many prime... | No |
Corollary 16. Suppose \( A \) is a finitely generated \( F \) -algebra over a field \( F \), and an integral domain. Then all maximal chains of prime ideals have length \( \operatorname{Trdeg}\left( A\right) \) . | Proof: The proof is by induction on \( d = \operatorname{Trdeg}\left( A\right) \) . If \( d = 0 \), then \( A \) is an integral extension of \( F \), so by theorem 20.21.e \( A \) is a field, and the only prime ideal is \( \{ 0\} \) . Suppose \( {P}_{0} \subset \cdots {P}_{e} \) is a chain of prime ideals. It follows b... | No |
Lemma 24. Suppose \( R \) is a \( \mathcal{N} \) -graded commutative ring and \( M \) is a \( \mathcal{Z} \) -graded \( R \) -module. Suppose \( \operatorname{Ann}\left( x\right) \) is a prime ideal \( P \) . Then \( P \) is homogeneous. | Proof: Let \( x = {x}_{1} + \cdots + {x}_{s} \) where the \( {x}_{i} \) are homogeneous and \( {x}_{1} \) is of least degree. We prove the lemma by induction on \( s \) . If \( r \in P \), write \( r = {r}_{1} + \cdots + {r}_{t} \) where the \( {r}_{i} \) are homogeneous and of increasing degree. Then \( {r}_{1}{x}_{1}... | Yes |
Lemma 25. Suppose \( R \) is a Noetherian commutative ring and \( M \) is a finitely generated \( R \)-module. Then there is a chain \( 0 = {M}_{0} \subset \cdots \subset {M}_{m} = M \) of submodules, such that for \( i > 0{M}_{i}/{M}_{i - 1} \) is isomorphic to \( R/{P}_{i} \) for some prime ideal \( {P}_{i} \). If \(... | Proof: As noted in section 8.5, \( M \) is Noetherian. By lemma 23 there is an \( x \in M \) such that \( \operatorname{Ann}\left( x\right) \) is a prime ideal \( P \). As noted in section 8.1, the submodule \( {M}_{1} = {Rx} \) is isomorphic to \( R/P \). Continuing inductively with \( M/{M}_{1} \), using the correspo... | No |
Lemma 32. For \( S \subseteq {\mathcal{N}}^{n}, S \) has only finitely many minimal elements in the product order. | Proof: Since the minimal elements of \( S \) and those of \( {S}^{ \geq } \) are the same, we may assume that \( S \) is \( \geq \) -closed. If \( n = 1 \) the claim follows because \( \mathcal{N} \) is well-ordered. For \( n > 1 \), let \( {S}_{i} = \left\{ {\left\langle {{u}_{1},\ldots ,{u}_{n - 1}}\right\rangle : \l... | Yes |
Lemma 2. Suppose \( A \) is an integrally closed integral domain, \( K \) is its field of fractions, \( L \) is a finite separable extension of \( K \), and \( B \) is the integral closure of \( A \) in \( L \). a. For \( b \in B \), the coefficients of the irreducible polynomial of \( b \) over \( K \), and the norm a... | Proof: For part a, since the quantities in question are in the ring generated over \( A \) by the conjugates of \( b \) they are all integral over \( A \), and in \( K \). Since \( A \) is integrally closed, they are in \( A \). For part b, choose a basis \( {e}_{1},\ldots ,{e}_{n} \) for \( L \) over \( K \). By multi... | Yes |
Lemma 4. Suppose \( L \supseteq K \) is a finite extension of dimension \( n \) . If \( {l}^{\prime } = {Tl} \) for an \( n \times n \) matrix \( T \) over \( K \) and column vectors \( l,{l}^{\prime } \) then \( \operatorname{Disc}\left( {{l}_{1}^{\prime },\ldots ,{l}_{n}^{\prime }}\right) = \det {\left( T\right) }^{2... | Proof: \( \;\operatorname{Disc}\left( {{l}_{1}^{\prime },\ldots ,{l}_{n}^{\prime }}\right) = \det \left( N\right) \) where\n\n\[ \n{N}_{ij} = \operatorname{Tr}\left( {{l}_{i}^{\prime }{l}_{j}^{\prime }}\right) = \operatorname{Tr}\left( {\mathop{\sum }\limits_{{rs}}{T}_{ir}{T}_{js}{l}_{r}{l}_{s}}\right) = \mathop{\sum }... | Yes |
Lemma 15. Suppose \( R \) is a Dedekind domain, \( F \) is the field of fractions, \( O \) is a valuation ring in \( F \) containing \( R, M \) is the maximal ideal of \( O \), and \( P = M \cap R \). Then \( O = {R}_{P} \). | Proof: Let \( U = O - M \). If \( s \in R - P \) then \( s \in O - M = U \); this shows that \( {R}_{P} \subseteq O \). If \( P = \{ 0\} \) then \( F = O \) follows. Otherwise, \( {P}_{P} \subseteq \left\lbrack {MO}\right\rbrack = M \). That is, we have valuation rings \( {O}_{2} \subseteq {O}_{1} \), with maximal idea... | Yes |
Lemma 21. Suppose \( A \) is a Dedekind domain, \( P \) is a prime ideal, and \( e > 1 \) . Then the \( A \) -module \( {P}^{e - 1}/{P}^{e} \) is isomorphic to \( A/P \) . | Proof: Since \( {P}^{e - 1} \supset {P}^{e} \), there is an element \( a \in {P}^{e - 1} - {P}^{e} \) . The map \( x \mapsto {ax} + {P}^{e} \) from \( A \) to \( {P}^{e - 1}/{P}^{e} \) is an \( A \) -module homomorphism. It is surjective, because \( {Aa} + {P}^{e} \) contains \( {P}^{e} \) and is contained in \( {P}^{e... | Yes |
Corollary 24. Given an a.i.r. \( A \), let \( {\beta }_{A} \) be as in theorem 23. Then every ideal class of \( A \) contains an ideal \( I \) with \( \operatorname{Inrm}\left( I\right) \leq {\beta }_{A} \) . | Proof: Let \( C \) be the class, let \( J \) be an ideal in the inverse class, and choose \( x \in J \) such that \( \mathrm{N}\left( x\right) \leq \) \( {\beta }_{A}\operatorname{Inrm}\left( J\right) \) . Then \( J \supseteq {Ax} \), so \( \left\lbrack {IJ}\right\rbrack = {Ax} \) for some \( I \), and \( I \in C \) . ... | Yes |
Lemma 28. If \( p \) is a polynomial whose coefficients are algebraic numbers then there is an a.n.f. \( K \) containing the coefficients, such that \( p = c{p}^{\prime } \) where \( c \in K \) and \( {p}^{\prime } \in {O}_{K} \) is primitive. In any such \( {O}_{K}, c \) and \( {p}^{\prime } \) are unique up to units. | Proof: Suppose \( p = \mathop{\sum }\limits_{{i = 0}}^{n}\left( {{a}_{i}/{b}_{i}}\right) {x}^{i} \), and let \( F \) be an a.n.f. containing the coefficients. Let \( e = {b}_{1}\cdots {b}_{n} \) ; then \( {ep} \in {O}_{F}\left\lbrack x\right\rbrack \) . Let \( I \subseteq {O}_{F} \) be the ideal generated by the coeffi... | No |
Lemma 5. Suppose \( R \) is a field, and \( {p}_{1} = q{p}_{2} + {p}_{3} \), where \( {n}_{1} \geq {n}_{2} > {n}_{3} \) and \( {p}_{3} \neq 0 \). a. For \( 0 \leq i < {n}_{3},{S}_{i}\left( {{p}_{1},{p}_{2}}\right) = {\left( -1\right) }^{\left( {{n}_{1} - i}\right) \left( {{n}_{2} - i}\right) }{\pi }_{2}^{{n}_{1} - {n}_... | Proof: The argument is a generalization of the argument for \( i = 0 \) given above. Let \( N \) be the copy of Sylvester’s matrix for \( {p}_{2} \) and \( {p}_{3} \) in the lower right corner of \( {M}^{\prime \prime } \). Then \( {N}_{i} \) for \( 0 \leq i < {n}_{3} \) is obtained from \( N \) by deleting the rows of... | Yes |
Corollary 7. In Collins’ reduced method, \( {p}_{i} = {\rho }_{i}{S}_{{n}_{i - 1} - 1}\left( {{p}_{1},{p}_{2}}\right) \) for \( 3 \leq i \leq k \), where \( {\rho }_{i} \in R \) . In fact, | \[ {\rho }_{i} = {\left( -1\right) }^{{n}_{i - 2} - {n}_{i - 1} + 1}{\Pi }_{l = 3}^{i - 1}{\left( -1\right) }^{\left( {{n}_{l - 1} - {n}_{i - 1} + 1}\right) \left( {{n}_{l - 2} - {n}_{i - 1} + 1}\right) }{\pi }_{l - 1}^{\left( {{n}_{l - 2} - {n}_{l - 1} + 1}\right) \left( {{n}_{l - 1} - {n}_{l} - 1}\right) } \] Proof: ... | Yes |
Lemma 10. Suppose \( a, b \in F, a < b, p \in F\left\lbrack x\right\rbrack \), and \( p\left( a\right) p\left( b\right) < 0 \) . Then \( p\left( c\right) = 0 \) for some \( c \) with \( a < c < b \) . | Proof: Assume \( p \) is monic, and factors into irreducible factors as \( \left( {x - {r}_{1}}\right) \cdots \left( {x - {r}_{k}}\right) {q}_{1}\left( x\right) \cdots {q}_{l}\left( x\right) \) where \( {r}_{1} \leq \cdots \leq {r}_{k} \) and \( {q}_{j} \) is an irreducible quadratic. It is readily shown that \( {q}_{j... | No |
Lemma 16. Suppose \( L \subseteq {\mathcal{R}}^{n} \) is a lattice, \( {b}_{1},\ldots ,{b}_{n} \) is an LLL-reduced basis, and \( {x}_{1},\ldots ,{x}_{t} \in L \) are linearly independent. Then for \( 1 \leq j \leq t,\left| {b}_{j}\right| \leq {2}^{\left( {n - 1}\right) /2}\max \left\{ {{x}_{1},\ldots ,{x}_{t}}\right\}... | Proof: For \( 1 \leq i \leq n \) and \( 1 \leq j \leq t \) there are \( {r}_{ij} \in \mathcal{Z}\left( {{\widehat{r}}_{ij} \in \mathcal{R}}\right) \) such that \( {x}_{j} = \mathop{\sum }\limits_{i}{r}_{ij}{b}_{i}\left( {{\widehat{x}}_{j} = \mathop{\sum }\limits_{i}{\widehat{r}}_{ij}{\widehat{b}}_{i}}\right) \) , where... | Yes |
a. If the rows or columns of a matrix are permuted then the determinant is unchanged if the permutation is even, and multiplied by -1 if the permutation is odd. | Proof: Part a follows by the remarks above concerning permutation matrices. | No |
Theorem 1.1.6. Let \( p\left( t\right) \) be a given polynomial of degree \( k \) . If \( \lambda, x \) is an eigenvalue-eigenvector pair of \( A \in {M}_{n} \), then \( p\left( \lambda \right), x \) is an eigenvalue-eigenvector pair of \( p\left( A\right) \) . Conversely, if \( k \geq 1 \) and if \( \mu \) is an eigen... | Proof. We have\n\n\[ p\left( A\right) x = {a}_{k}{A}^{k}x + {a}_{k - 1}{A}^{k - 1}x + \cdots + {a}_{1}{Ax} + {a}_{0}x,\;{a}_{k} \neq 0 \]\n\nand \( {A}^{j}x = {A}^{j - 1}{Ax} = {A}^{j - 1}{\lambda x} = \lambda {A}^{j - 1}x = \cdots = {\lambda }^{j}x \) by repeated application of the eigenvalue-eigenvector equation. Thu... | Yes |
Theorem 1.1.9. Let \( A \in {M}_{n} \) be given. Then \( A \) has an eigenvalue. In fact, for each given nonzero \( y \in {\mathbf{C}}^{n} \), there is a polynomial \( g\left( t\right) \) of degree at most \( n - 1 \) such that \( g\left( A\right) y \) is an eigenvector of \( A \) . | Proof. Let \( m \) be the least integer \( k \) such that the vectors \( y,{Ay},{A}^{2}y,\ldots ,{A}^{k}y \) are linearly dependent. Then \( m \geq 1 \) since \( y \neq 0 \), and \( m \leq n \) since any \( n + 1 \) vectors in \( {\mathbf{C}}^{n} \) are linearly dependent. Let \( {a}_{0},{a}_{1},\ldots ,{a}_{m} \) be s... | Yes |
What are the eigenvalues and determinant of \( I + x{y}^{ * } \) ? | Using (0.8.5.11) and the fact that \( \operatorname{adj}\left( {\alpha I}\right) = {\alpha }^{n - 1}I \), we compute\n\n\[ \n{p}_{I + x{y}^{ * }}\left( t\right) = \det \left( {{tI} - \left( {I + x{y}^{ * }}\right) }\right) = \det \left( {\left( {t - 1}\right) I - x{y}^{ * }}\right) \n\]\n\n\[ \n= \det \left( {\left( {t... | Yes |
Let \( x, y \in {\mathbf{C}}^{n}, x \neq 0 \), and \( A \in {M}_{n} \) . Suppose that \( {Ax} = {\lambda x} \) and let the eigenvalues of \( A \) be \( \lambda ,{\lambda }_{2},\ldots ,{\lambda }_{n} \) . What are the eigenvalues of \( A + x{y}^{ * } \) ? | First observe that \( \left( {t - \lambda }\right) x = \left( {{tI} - A}\right) x \) implies that \( \left( {t - \lambda }\right) \operatorname{adj}\left( {{tI} - A}\right) x = \operatorname{adj}\left( {{tI} - A}\right) \left( {{tI} - A}\right) x = \det \left( {{tI} - A}\right) x \), that is,\n\n\[ \left( {t - \lambda ... | Yes |
Theorem 1.2.17. Let \( A \in {M}_{n} \) . There is some \( \delta > 0 \) such that \( A + {\varepsilon I} \) is nonsingular whenever \( \varepsilon \in \mathbf{C} \) and \( 0 < \left| \varepsilon \right| < \delta \) . | Proof. Observation 1.1.8 ensures that \( \lambda \in \sigma \left( A\right) \) if and only if \( \lambda + \varepsilon \in \sigma \left( {A + {\varepsilon I}}\right) \) . Therefore, \( 0 \in \sigma \left( {A + {\varepsilon I}}\right) \) if and only if \( \lambda + \varepsilon = 0 \) for some \( \lambda \in \sigma \left... | Yes |
Theorem 1.2.18. Let \( A \in {M}_{n} \) and suppose that \( \lambda \in \sigma \left( A\right) \) has algebraic multiplicity \( k \) . Then \( \operatorname{rank}\left( {A - {\lambda I}}\right) \geq n - k \) with equality for \( k = 1 \) . | Proof. Apply the preceding observation to the characteristic polynomial \( {p}_{A}\left( t\right) \) of a matrix \( A \in {M}_{n} \) that has an eigenvalue \( \lambda \) with multiplicity \( k \geq 1 \) . If we let \( B = A - \) \( {\lambda I} \), then zero is an eigenvalue of \( B \) with multiplicity \( k \) and henc... | Yes |
Theorem 1.3.3. Let \( A, B \in {M}_{n} \) . If \( B \) is similar to \( A \), then \( A \) and \( B \) have the same characteristic polynomial. | Proof. Compute\n\n\[ \n{p}_{B}\left( t\right) = \det \left( {{tI} - B}\right) \]\n\n\[ \n= \det \left( {t{S}^{-1}S - {S}^{-1}{AS}}\right) = \det \left( {{S}^{-1}\left( {{tI} - A}\right) S}\right) \]\n\n\[ \n= \det {S}^{-1}\det \left( {{tI} - A}\right) \det S = {\left( \det S\right) }^{-1}\left( {\det S}\right) \det \le... | Yes |
Corollary 1.3.4. Let \( A, B \in {M}_{n} \) and suppose that \( A \) is similar to \( B \) . Then\n\n(a) A and B have the same eigenvalues.\n\n(b) If \( B \) is a diagonal matrix, its main diagonal entries are the eigenvalues of \( A \) .\n\n(c) \( B = 0 \) (a diagonal matrix) if and only if \( A = 0 \) .\n\n(d) \( B =... | Exercise. Verify the assertions in the preceding corollary. | No |
Theorem 1.3.7. Let \( A \in {M}_{n} \) be given. Then \( A \) is similar to a block matrix of the form\n\n\[ \left\lbrack \begin{matrix} \Lambda & C \\ 0 & D \end{matrix}\right\rbrack ,\;\Lambda = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{k}}\right), D \in {M}_{n - k},1 \leq k < n \]\n\nif and only ... | Proof. Suppose that \( k < n \), the \( n \) -vectors \( {x}^{\left( 1\right) },\ldots ,{x}^{\left( k\right) } \) are linearly independent, and \( A{x}^{\left( i\right) } = {\lambda }_{i}{x}^{\left( i\right) } \) for each \( i = 1,\ldots, k \) . Let \( \Lambda = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambd... | Yes |
Lemma 1.3.8. Let \( {\lambda }_{1},\ldots ,{\lambda }_{k} \) be \( k \geq 2 \) distinct eigenvalues of \( A \in {M}_{n} \) (that is, \( {\lambda }_{i} \neq {\lambda }_{j} \) if \( i \neq j \) and \( 1 \leq i, j \leq k \) ), and suppose that \( {x}^{\left( i\right) } \) is an eigenvector associated with \( {\lambda }_{i... | Proof. Suppose that there are complex scalars \( {\alpha }_{1},\ldots {\alpha }_{k} \) such that \( {\alpha }_{1}{x}^{\left( 1\right) } + {\alpha }_{2}{x}^{\left( 2\right) } + \cdots + {\alpha }_{r}{x}^{\left( r\right) } = 0 \) . Let \( {B}_{1} = \left( {A - {\lambda }_{2}I}\right) \left( {A - {\lambda }_{3}I}\right) \... | Yes |
Theorem 1.3.9. If \( A \in {M}_{n} \) has \( n \) distinct eigenvalues, then \( A \) is diagonalizable. | Proof. Let \( {x}^{\left( i\right) } \) be an eigenvector associated with the eigenvalue \( {\lambda }_{i} \) for each \( i = \) \( 1,\ldots, n \) . Since all the eigenvalues are distinct, Lemma 1.3.8 ensures that the vectors \( {x}^{\left( 1\right) },\ldots ,{x}^{\left( n\right) } \) are linearly independent. Theorem ... | Yes |
Lemma 1.3.10. Let \( {B}_{1} \in {M}_{{n}_{1}},\ldots ,{B}_{d} \in {M}_{{n}_{d}} \) be given and let \( B \) be the direct sum\n\n\[ B = \left\lbrack \begin{matrix} {B}_{1} & & 0 \\ & \ddots & \\ 0 & & {B}_{d} \end{matrix}\right\rbrack = {B}_{1} \oplus \cdots \oplus {B}_{d} \]\n\nThen \( B \) is diagonalizable if and o... | Proof. If for each \( i = 1,\ldots, d \) there is a nonsingular \( {S}_{i} \in {M}_{{n}_{i}} \) such that \( {S}_{i}^{-1}{B}_{i}{S}_{i} \) is diagonal, and if we define \( S = {S}_{1} \oplus \cdots \oplus {S}_{d} \), then one checks that \( {S}^{-1}{BS} \) is diagonal.\n\nFor the converse, we proceed by induction. Ther... | Yes |
Theorem 1.3.12. Let \( A, B \in {M}_{n} \) be diagonalizable. Then \( A \) and \( B \) commute if and only if they are simultaneously diagonalizable. | Proof. Assume that \( A \) and \( B \) commute, perform a similarity transformation on both \( A \) and \( B \) that diagonalizes \( A \) (but not necessarily \( B \) ) and groups together any repeated eigenvalues of \( A \) . If \( {\mu }_{1},\ldots ,{\mu }_{d} \) are the distinct eigenvalues of \( A \) and \( {n}_{1}... | Yes |
Lemma 1.3.19. Let \( \mathcal{F} \subset {M}_{n} \) be a commuting family. Then some nonzero vector in \( {\mathbf{C}}^{n} \) is an eigenvector of every \( A \in \mathcal{F} \) . | Proof. There is always a nonzero \( \mathcal{F} \) -invariant subspace, namely, \( {\mathbf{C}}^{n} \) . Let \( m = \) \( \min \left\{ {\dim V : V}\right. \) is a nonzero \( \mathcal{F} \) -invariant subspace of \( \left. {\mathbf{C}}^{n}\right\} \) and let \( W \) be any given \( \mathcal{F} \) -invariant subspace suc... | Yes |
Theorem 1.3.21. Let \( \mathcal{F} \subset {M}_{n} \) be a family of diagonalizable matrices. Then \( \mathcal{F} \) is a commuting family if and only if it is a simultaneously diagonalizable family. Moreover, for any given \( {A}_{0} \in \mathcal{F} \) and for any given ordering \( {\lambda }_{1},\ldots ,{\lambda }_{n... | Proof. If \( \mathcal{F} \) is simultaneously diagonalizable, then it is a commuting family by a previous exercise. We prove the converse by induction on \( n \) . If \( n = 1 \), there is nothing to prove since every family is both commuting and diagonal. Let us suppose that \( n \geq 2 \) and that, for each \( k = 1,... | Yes |
Theorem 1.3.22. Suppose that \( A \in {M}_{m, n} \) and \( B \in {M}_{n, m} \) with \( m \leq n \) . Then the \( n \) eigenvalues of \( {BA} \) are the \( m \) eigenvalues of \( {AB} \) together with \( n - m \) zeroes; that is, \( {p}_{BA}\left( t\right) = {t}^{n - m}{p}_{AB}\left( t\right) \) . If \( m = n \) and at ... | Proof. A computation reveals that\n\n\[ \left\lbrack \begin{matrix} {I}_{m} & - A \\ 0 & {I}_{n} \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {AB} & 0 \\ B & {0}_{n} \end{matrix}\right\rbrack \left\lbrack \begin{matrix} {I}_{m} & A \\ 0 & {I}_{n} \end{matrix}\right\rbrack = \left\lbrack \begin{matrix} {0}_{m} ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.