Index of notation

$x+y$, $x.y$, $xy$, $x\top y$, $x\perp y$: I, § 1, no. 1. $X\top Y$, $X+Y$, $XY$ ($X$, $Y$ subsets): I, § 1, no. 1. $X\top a$, $a\top X$ ($X$ a subset, $a$ an element): I, § 1, no. 1. $\top_{\alpha\in A}x_{\alpha}$, $\top_{\alpha}x_{\alpha}$, $\top x_{\alpha}$, $\perp_{\alpha\in A}x$, $\perp_{\alpha}x_{\alpha}$, $\perp x_{\alpha}$, $\sum_{\alpha\in A}x_{\alpha}$, $\sum_{\alpha}x_{\alpha}$, $\sum x_{\alpha}$, $\prod_{\alpha\in A}x_{\alpha}$, $\prod_{\alpha}x_{\alpha}$, $\prod x_{\alpha}$: I, § 1, no. 2. $\top_{p\leq i\leq q}x_i$, $\top_{i=p}^{q}x_i$: I, § 1, no. 2. $x_p\top x_{p+1}\top\cdots\top x_q$: I, § 1, no. 3. $\top^n x$, $\perp x^n$, $nx$ ($n\in\mathbf{N}$): I, § 1, no. 3. $\top_{0\leq i<j\leq n}x_{ij}$, $\top_{i<j}x_{ij}$: I, § 1, no. 5. $\sum_{i=p}^{q}\sum_{j=r}^{s}x_{ij}$, $\sum_{j=r}^{s}\sum_{i=p}^{q}x_{ij}$: I, § 1, no. 5. $\top_{0\leq i_1<i_2<\cdots<i_p\leq n}x_{i_1i_2\cdots i_p}$, $\top_{i_1<i_2<\cdots<i_p}x_{i_1i_2\cdots i_p}$: I, § 1, no. 5. $0,1$: I, § 2, no. 1. $\gamma_a$, $\delta_a$, $\gamma(a)$, $\delta(a)$: I, § 2, no. 2. $E_S$ ($S$ a subset of a commutative monoid $E$): I, § 2, no. 4. $\mathbf{Z}$, $+$ (addition in $\mathbf{Z}$): I, § 2, no. 5. $\leq$ (order relation on $\mathbf{Z}$): I, § 2, no. 5. $\mathbf{N}^*$: I, § 2, no. 5. $\top^n$ (for $n\in\mathbf{Z}$): I, § 2, no. 7. $-x$, $x-y$, $x+y-z$, $x-y-z$, $x-y+z-t$: I, § 2, no. 8. $nx$ ($n\in\mathbf{Z}$): I, § 2, no. 8. $x^n$ ($n\in\mathbf{Z}$): I, § 2, no. 8. $\frac{1}{x}$, $\frac{x}{y}$, $x/y$: I, § 2, no. 8.

$\alpha.x, x.\alpha, x^\alpha$ ($\alpha$ an operator): I, § 3, no. 1. $\alpha \perp x, \alpha \perp X, \Xi \perp X$ ($\alpha$ an operator, $\Xi$ a set of operators): I, § 3, no. 1. $\mathfrak{S}_F$: I, § 4, no. 1. $(G:H), G/H$ (H a subgroup of G): I, § 4, no. 4. $x \equiv y$ (mod. H), $x \equiv y$ (H) (H a normal subgroup): I, § 4, no. 4. Ker $f$, Im $f$ ($f$ a group homomorphism): I, § 4, no. 5. $\prod_{i \in I} G_i \quad (G_i \text{ groups})$: I, § 4, no. 8. $G_1 \times_H G_2$: I, § 4, no. 8. $\prod_{i \in I} G_i \quad (G_i \text{ groups})$: I, § 4, no. 9. $x \equiv y$ (mod. $a$), $x \equiv y$ ($a$) ($a, x, y$ rational integers): I, § 4, no. 10. $v_p(a)$ ($p$ a prime number, $a$ a rational integer): I, § 4, no. 10. Aut(G), Int(G), Int(x) (G a group, $x \in G$): I, § 5, no. 3. $N_G(A), N(A)$ (G a group, $A \subset G$): I, § 5, no. 3. $C_G(A), C(A)$ (G a group, $A \subset G$): I, § 5, no. 3. $E/G, G|E$ (G a group operating on E): I, § 5, no. 4. $G|E/H$ (G, H groups operating on E by commuting actions): I, § 5, no. 4. $\mathfrak{S}_n$: I, § 5, no. 7. $\tau_{x,y}$ (transposition of support $\{x, y\}$): I, § 5, no. 7. $\varepsilon(\sigma), \varepsilon_\sigma$ ($\sigma$ a permutation): I, § 5, no. 7. $\mathscr{U}_E, \mathscr{U}_n$: I, § 5, no. 7. $F \xrightarrow{i} E \xrightarrow{\rho} G$ (E, F, G groups): I, § 6, no. 1. $F \times_\tau G, \mathscr{E}_\tau$ ($\tau$ a homomorphism of G into Aut(F)): I, § 6, no. 1. $gf$ ($f \in F, g \in G$): I, § 6, no. 1. $(f, g) \cdot_\tau (f', g')$ ($f, f'$ in F, $g, g'$ in G): I, § 6, no. 1. $(x, y), (A, B)$ ($x, y$ elements, A, B subsets of a group G): I, § 6, no. 2. $D(G)$: I, § 6, no. 2. $C^n(G)$: I, § 6, no. 3. $D^n(G)$: I, § 6, no. 4. $E^G$ (G a group operating on E): I, § 6, no. 5. $M_n(X), M(X)$ (X a set): I, § 7, no. 1. $l(w)$ ($w$ an element of M(X)): I, § 7, no. 1. $ww', w.w'$ ($w, w'$ elements of M(X)): I, § 7, no. 1. $l(w)$ ($w$ a word over X): I, § 7, no. 2. $ww', w.w'$ ($w, w'$ words over X): I, § 7, no. 2. $Mo(X)$ (X a set): I, § 7, no. 2. $l(\sigma)$ ($\sigma$ a decomposition): I, § 7, no. 3. $*G_i, G_1 * G_2$ ($G_1, G_2, G_i$ groups): I, § 7, no. 3. $\langle \tau_1, \ldots, \tau_n; r_1, \ldots, r_m \rangle$ ($\tau_j$ generators, $r_i$ relators): I, § 7, no. 6. $\langle \tau_1, \ldots, \tau_n; u_1 = v_1, \ldots, u_m = v_m \rangle$ ($\tau_j$ generators, $u_i, v_i$ elements of a free group): I, § 7, no. 6. $\mathbf{Z}^{(X)}, \mathbf{N}^{(X)}$ (X a set): I, § 7, no. 7.

$0, 1$ (elements of a ring): I, § 8, no. 1. $A^0$ (A a ring): I, § 8, no. 3. $(a)$ (a an element of A): I, § 8, no. 6. $\sum_{\lambda} a_\lambda$ (a_\lambda ideals): I, § 8, no. 6. $x \equiv y \pmod{a}$, $x \equiv y$ (a) (a an ideal): I, § 8, no. 7. $A/a$ (a a two-sided ideal): I, § 8, no. 7. $ab$ (a, b two-sided ideals): I, § 8, no. 9. $A[S^{-1}]$ (S a subset of a ring A): I, § 8, no. 12. $F_p$ (p a prime number): I, § 9, no. 1. $\mathbf{Q}$: I, § 9, no. 4. $\mathbf{Q}_+$: I, § 9, no. 4. |x|, sgn x (x a rational number): I, § 9, no. 4. $in(G)$: I, § 4, Exercise 13. $D_n$: I, § 6, Exercise 4. $Q$: I, § 6, Exercise 4. $A \approx B$: I, § 6, Exercise 39. $e(G)$: I, § 7, Exercise 39. $d_n(X)$: I, § 7, Exercise 39. $A_s, A_d$ (A a ring): II, § 1, no. 1. $\sum_{i \in I} x_i ((x_i)_{i \in I})$ a family of elements of a module of finite support: II, § 1, no. 1. $Hom_A(E, F), Hom(E, F)$, (E, F, A-modules): II, § 1, no. 2. $End_A(E), End(E), Aut(E), GL(E)$ (E an A-module): II, § 1, no. 2. $Hom_A(u, v), Hom(u, v)$ (u, v linear mappings): II, § 1, no. 2. $1_E$ (E a module): II, § 1, no. 2. $0$ (zero module): II, § 1, no. 3. $Ker u, Im u, Coim u, Coker u$ (u a linear mapping): II, § 1, no. 3. $\prod_i f_i$ (f_i: E_i \to F_i linear mappings): II, § 1, no. 5. $\bigoplus_{i \in I} E_i, E_p \oplus E_{p+1} \oplus \cdots \oplus E_q ((E_i)_{i \in I}$ a family of A-modules): II, § 1, no. 6. $\sum_{i \in I} f_i$ (f_i: E_i \to F_i linear mappings): II, § 1, no. 6. $\bigoplus_{i \in I} f_i, f_p \oplus f_{p+1} \oplus \cdots \oplus f_q$ (f_i: E_i \to F_i linear mappings): II, § 1, no. 6. $E^{(1)}$ (E a module): II, § 1, no. 6. $\bigoplus_{i \in I} M_i$ ((M_i)_{i \in I} a family of submodules): II, § 1, no 7. $long_A(M), long(M)$ (M an A-module of finite length): II, § 1, no. 10. $\delta_{st}$ (Kronecker symbol): II, § 1, no. 11. $\sum_{t \in T} \xi_t \cdot t$ (T a set, $\xi_t$ elements of a ring): II, § 1, no. 11.

$Ann(S), Ann(x)$ (S a subset of a module, x an element of a module): II, § 1, no. 12. $\rho_*(E), E_{[B]}$ (E an A-module, $\rho : B \to A$ a ring homomorphism): II, § 1, no. 13. $\rho_*(u)$ ($\rho : B \to A$ a ring homomorphism, u an A-linear mapping): II, § 1, no. 13. $E*$ (E a module): II, § 2, no. 3. $\langle x, x^* \rangle$ (x an element of a left module E, $x^*$ an element of its dual E*): II, § 2, no. 3. $\langle x^*, x \rangle$ (x an element of a right module E, $x^*$ an element of its dual E*): II, § 2, no. 3. $t_u$ (u a linear or semi-linear mapping): II, § 2, no. 5. $\tilde{u}$ (u an isomorphism): II, § 2, no. 5. $E \otimes_A F, E \otimes_A F$ (E a right A-module, F a left A-module): II, § 3, no. 1. $x \otimes y$ (x $\in$ E (a right module), y $\in$ F (a left module)): II, § 3, no. 1. $u \otimes v$ (u, v linear mappings): II, § 3, no. 2. $u \otimes v$ (u, v semi-linear mappings): II, § 3, no. 3. $s_{A_d}$ (A ring): II, § 3, no. 4. $\mathscr{L}_2(E, F; G)$ (E, F, G modules over a commutative ring): II, § 3, no. 5. $\bigotimes_{\lambda \in L} G_\lambda, \bigotimes_{\lambda \in L} x_\lambda$ (($G_\lambda$) a family of $\mathbf{Z}$-modules, $x_\lambda \in G_\lambda$ for all $\lambda$): II, § 3, no. 9. $\bigotimes_{\lambda \in L} v_\lambda$ ($v_\lambda : G_\lambda \to G'_\lambda$ $\mathbf{Z}$-linear mappings): II, § 3, no. 9. $\bigotimes_{(c,p,q)} G_\lambda, \bigotimes_{(c,p,q)} x_\lambda, \bigotimes_{(c)} x_\lambda$: II, § 3, no. 9. $\bigotimes_{(c)} v_\lambda$ ($v_\lambda$ $\mathbf{Z}$-linear mappings): II, § 3, no. 9. $E_1 \otimes_{A_1} E_2 \otimes_{A_2} E_3 \otimes \cdots \otimes_{A_{n-2}} E_{n-1} \otimes_{A_{n-1}} E_n$: II, § 3, no. 9. $x_1 \otimes x_2 \otimes \cdots \otimes x_n$: II, § 3, no. 9. $u_1 \otimes u_2 \otimes \cdots \otimes u_n$ ($u_i$ linear mappings): II, § 3, no. 9. $\mathscr{L}_n(E_1, \ldots, E_n; G)$ ($E_1, \ldots, E_n, G$ modules over a commutative ring): II, § 3, no. 9. $Tr(u)$ (u an endomorphism of a module over a commutative ring): II, § 4, no. 3. $\rho^*(E), E_{(B)}$ (E an A-module, $\rho : A \to B$ a ring homomorphism): II, § 5, no. 1. $\rho^*(u), u_{(B)}$ ($\rho : A \to B$ a ring homomorphism, u an A-module homomorphism): II, § 5, no. 1. $dim_K E, dim E, [E : K]$ (E a vector K-space): II, § 7, no. 2. $dim_A E, dim E$ (E an A-module any two bases of which are equipotent): II, § 7, no. 2. $codim_E F, codim F$ (F a vector subspace of a vector space E): II, § 7, no. 3. $rg(u)$ (u a linear mapping of vector spaces): II, § 7, no. 4. $rg(u)$ (u an element of a tensor product of vector spaces): II, § 7, no. 8. $dim_K E, dim E$ (E an affine space over a field K): II, § 9, no. 1. $a + t, t + a$ (a a point, t a translation of an affine space): II, § 9, no. 1.

$b - a$ ($a, b$ points of an affine space): II, § 9, no. 1.

$\sum_{c \in I} \lambda_i x_i$ (($x_i$)$_{i \in I}$ a family of points of an affine space, $(\lambda_i)_{i \in I}$ a family of scalars, of finite support, such that $\sum_i \lambda_i = 1$ or $\sum_i \lambda_i = 0$): II, § 9, no. 1.

$\mathbf{P}(V), \Delta(V)$ (V a vector space): II, § 9, no. 5. $\mathbf{P}_n(K), \Delta_n(K)$ (K a field): II, § 9, no. 5. $\dim_K \mathbf{P}(V), \dim \mathbf{P}(V)$ (V a vector K-space): II, § 9, no. 5. $\tilde{K}, \infty$ (K a field): II, § 9, no. 9. $\mathbf{PGL}(V), \mathbf{PGL}_n(K), \mathbf{PGL}(n, K)$ (K a field, V a vector space): II, § 9, no. 10. $^tM$ ($M$ a matrix): II, § 10, no. 1. $M' + M''$ ($M', M''$ matrices over a commutative group): II, § 10, no. 2. $f(M', M''), M'M''$ ($M', M''$ matrices): II, § 10, no. 2. $E_{ij}$ (matrix units): II, § 10, no. 3. $\sigma(M), M^\sigma$ ($M$ a matrix, $\sigma$ a ring homomorphism): II, § 10, no. 3. $M(x), x$ (x an element of a finitely generated free module): II, § 10, no. 4. $M(u)$ (u a homomorphism of a free module into a free module): II, § 10, no. 4. $M(x), M(u)$ (matrices relative to decompositions as direct sums): II, § 10, no. 5. $M(u)$ (u a semi-linear mapping): II, § 10, no. 6. $\mathbf{M}_n(A), I_n, 1_n$ (A a ring): II, § 10, no. 7. $\mathbf{GL}_n(A), \mathbf{GL}(n, A)$ (A a ring): II, § 10, no. 7. $^tX^{-1}$ (X an invertible square matrix): II, § 10, no. 7. $\operatorname{diag}(a_i)_{i \in I}, \operatorname{diag}(a_1, a_2, \ldots, a_n)$: II, § 10, no. 7. $X_1 \otimes X_2$ ($X_1, X_2$ matrices over a commutative ring): II, § 10, no. 10. $\operatorname{Tr}(X)$ (X a square matrix over a commutative ring): II, § 10, no. 11. $\operatorname{rg}(X)$ (X a matrix over a field): II, § 10, no. 12. $\deg(x)$ (x an element of a graded group): II, § 11, no. 1. $M(\lambda_0)$ (M a graded module, $\lambda_0$ an element of the monoid of degrees): II, § 11, no. 2. $\operatorname{Homgr}_A(M, N)$ (M, N graded modules over a graded ring A): II, § 11, no. 6. $\operatorname{Engr}_A(M), M^{*\operatorname{gr}}$ (M a graded module): II, § 11, no. 6. $M:N$ (M, N modules): II, § 1, Exercise 24.

$\left[\begin{smallmatrix} a & b \\ d & c \end{smallmatrix}\right]$ (a, b, c, d points on a projective line): II, § 9, Exercise 11.

$\mathbf{SL}(E)$ (E a vector space): II, § 10, Exercise 12. $\mathbf{PSL}(E)$ (E a vector space): II, § 10, Exercise 14. $x.y, xy$ (multiplication in an algebra): III, § 1, no. 1. $E^0$ (E an algebra): III, § 1, no. 1. $\operatorname{Hom}_{A-\operatorname{alg.}}(E, F)$ (E, F A-algebras): III, § 1, no. 1. $E/b$ (b a two-sided ideal of an algebra E): III, § 1, no. 2. $\hat{E}$ (E an algebra): III, § 1, no. 2. $\eta_c, \eta_E, \eta$ (E an algebra, c a unit): III, § 1, no. 3.

$T(u), N(u)$: III, § 2, no. 3. $\bar{u}$ : III, § 2, no. 4. $H$ : III, § 2, no. 5. $Lib_A(I), Libas_A(I), Libasc_A(I)$: III, § 2, no. 7. $U((x_i)_{i \in I})$ : III, § 2, no. 8. $A[(x_i)_{i \in I}], A[x_i]_{i \in I}$ : III, § 2, no. 9. $A[(X_i)_{i \in I}]$ : III, § 2, no. 9. $A[X_1, \ldots, X_n]$ : III, § 2, no. 9. $A[X], A[X, Y], A[X, Y, Z]$ : III, § 2, no. 9. $X^\nu$ ($\nu$ a multiindex): III, § 2, no. 9.

$\sum_s \xi_s e_s, \sum_s \xi_s . s$ (s elements of a monoid): III, § 2, no. 10. $A[[X_i]]_{i \in I}$ : III, § 2, no. 11. $\sum_\nu \alpha_\nu X^\nu$ : III, § 2, no. 11. $\omega(u), \omega_K(u)$ (u a formal power series): III, § 2, no. 11.

$\bigotimes_{i \in I} E_i, E_1 \otimes_A E_2 \otimes \cdots \otimes_A E_n, E_1 \otimes E_2 \otimes \cdots \otimes E_n$ (E_i A-algebras): III, § 4, no. 1. $E^{\otimes n}$ (E an algebra): III, § 4, no. 1.

$\bigotimes_{i \in I} E_i$ (I an infinite set, E_i algebras): III, § 4, no. 5.

$\bigotimes_{i \in I} u_i, \bigotimes_{i \in I} x_i$ (u_i algebra homomorphisms, x_i elements, I infinite): III, § 4, no. 5.

$\epsilon \bigotimes_{i \in I} E_i, \epsilon \bigotimes_{i \in I} f_i, \epsilon G^{\otimes n}$ ($E_i$, G graded algebras, $f_i$ graded algebra homomorphisms, $\epsilon$ a system of commutation factors): III, § 4, no. 7.

$g \bigotimes_{i \in I} E_i, E^g \otimes_A F, gG^{\otimes n}, g \bigotimes_{i \in I} f_i, f_1 \otimes f_2, gf^{\otimes n}$: III, § 4, no. 7.

$\bigotimes^n M, T^n(M), \mathrm{Tens}^n(M), T^n_A(M), T(M), \mathrm{Tens}(M), T_A(M)$ (M an A-module): III, § 5, no. 1. $T(u), T^n(u)$ (u a linear mapping): III, § 5, no. 2. $T_J^1(M), T_q^p(M)$ (M a module): III, § 5, no. 6. $\alpha_q^f(z), e_f^g$ : III, § 5, no. 6. $\alpha_{v_p}^{\lambda \cdots \mu}$ : III, § 5, no. 6. $c_j^i$: III, § 5, no. 6 $S(M), S_A(M), \mathrm{Sym}(M)$: III, § 6, no. 1. $\mathfrak{g}', \mathfrak{g}'_M, \mathfrak{g}'_n$: III, § 6, no. 1. $S^n(M), S^n(u), S(u)$ (M a module, u a linear mapping): III, § 6, nos. 1 and 2. $s.z$ : III, § 6, no. 3. $e^\alpha$ ($\alpha$ a multiindex): III, § 6, no. 6. $\wedge(M), \wedge_A(M), \mathrm{Alt}(M)$: III, § 7, no. 1. $\mathfrak{g}'', \mathfrak{g}''_M, \mathfrak{g}''_n$ : III, § 7, no. 1.

$\wedge^n(M)$: III, § 7, no. 1. $u \wedge v, x_1 \wedge x_2 \wedge \cdots \wedge x_n$: III, § 7, no. 1. $\wedge(u), \wedge^n(u)$ ($u$ a linear mapping): III, § 7, no. 2. $x_H$ (H a subset of $\{1, n\}$): III, § 7, no. 3. $u(x_1, \ldots, \hat{x}_j, \ldots, x_n)$: III, § 7, no. 4. $a.z$: III, § 7, no. 4. $A'_n(M), A''_n(M)$: III, § 7, no. 4. $e_J$: III, § 7, no. 8. $\rho_{J, K}$: III, § 7, no. 8. $\det(u)$ ($u$ an endomorphism): III, § 8, no. 1. $\det(x_1, x_2, \ldots, x_n)$ ($x_j$ vectors in an $n$-dimensional free A-module): III, § 8, no. 1. $\det(X)$ ($X$ a matrix): III, § 8, no. 3. $\det(\xi_{ij})_{1 \leq i \leq n, 1 \leq j \leq n}, \det(\xi_{ij})$: III, § 8, no. 3.

$\left|\begin{smallmatrix} \xi_{11} & \cdots & \xi_{1n} \\ \vdots & \ddots & \vdots \\ \xi_{n1} & \cdots & \xi_{nn} \end{smallmatrix}\right|$: III, § 8, no. 3.

$X_{H, K}, X^{H}$ ($X$ a matrix): III, § 8, nos 5 and 6. $\mathrm{SL}_n(A), \mathrm{SL}(n, A)$: III, § 8, no. 9. $p.x$ ($p \in A[X]$, $x$ an element of an A-module): III, § 8, no. 10. $M_u, M[X]$ ($M$ an A-module, $u$ an endomorphism): III, § 8, no. 10. $\chi_u(X)$ ($u$ an A-module endomorphism): III, § 8, no. 11. $\mathrm{Tr}_{M/K}(a), N_{M/K}(a), \mathrm{Pc}_{M/K}(a; X)$ ($A$ a K-algebra, $M$ an A-module, $a \in A$) : III, § 9, no. 1. $\mathrm{Tr}_{A/K}(a), N_{A/K}(a), \mathrm{Pc}_{A/K}(a; X)$ ($A$ a K-algebra, $a \in A$) : III, § 9, no. 3. $D_{A/K}(x_1, \ldots, x_n)$ ($x_j$ elements of a K-algebra A): III, § 9, no. 5. $[u, v]_g, [u, v]$ ($u, v$ elements of a graded algebra): III, § 10, no. 4. $P(D), P(d_1, \ldots, d_n)$ ($P$ a polynomial, $d_j$ derivations): III, § 10, no. 4. $\mathrm{ad}_e(a), \mathrm{ad}(a)$ ($a$ a homogeneous element of a graded algebra): III, § 10, no. 6. $D_K(B, F)$ ($B$ a K-algebra, $F$ a (B, B)-bimodule): III, § 10, no. 7. $D_{A, \rho}(B, F), D_A(B, F)$: III, § 10, no. 7. $D_s$ ($s$ an endomorphism): III, § 10, no. 9. $\Omega_K(A), d_{A/K}(x), dx$ ($x$ an element of a K-algebra A): III, § 10, no. 11. $D_iP, \partial P/\partial X_i$ ($P$ a polynomial): III, § 10, no. 11. $\Omega(u), \Omega_0(u)$ ($u$ a ring homomorphism): III, § 10, no. 12. $\Omega_u$ ($u$ a K-algebra homomorphism): III, § 10, no. 12. $M^{*gr}$ ($M$ a graded module): III, § 11. $u.v, u._mv$ ($u, v$ symmetric multilinear mappings): III, § 11, no. 2. $u \wedge v$ ($u, v$ alternating multilinear mappings): III, § 11, no. 2. $\theta_T, \theta_S, \theta_\Lambda$: III, § 11, no. 5.

$u \perp x, i(x)$ : III, § 11, no. 6. $x \perp u, i'(x)$ : III, § 11, no. 6. $x \perp u, i(u)$ : III, § 11, no. 7. $u \perp x, i'(u)$ : III, § 11, no. 7. $G_p(E), G_{n,p}(K)$ : III, § 11, no. 13. $a(x, y, z)$: III, Appendix, no. 1. $ME$: III, § 2, Exercise 13. $E * F$: III, § 5, Exercise 6. $R[a]$: III, § 6, Exercise 4. $K[X; \sigma, d]$: III, § 10, Exercise 3. $\tilde{X}$: III, § 11, Exercise 9.