Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49 (2008), No. 2, 399-408.

Projectivity and Flatness over the Endomorphism Ring of a Finitely Generated Comodule

T. Guédénon

1120 avenue Fournier, Québec, QC, G1V 2H8, Canada e-mail:guedenth@yahoo.ca

Abstract. Let k be a commutative ring, A a k-algebra, C an A-coring that is projective as a left A-module, ^{∗}C the dual ring of C and Λ a right C-comodule that is finitely generated as a left ^{∗}C-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring End^{C}(Λ). If C contains a grouplike element, we can replace Λ with A.

1. Introduction

Throughout the paper, k is a commutative ring. Let A be a k-algebra and C an A-coring. The left A-linear maps C → A have a ring structure. This allows one to define a dual ring ^{∗}C. Let Λ be a right C-comodule. Our starting point is the following: if C is finitely generated projective as a left A-module, then the category M^{C} of right C-comodules is isomorphic to _{^{∗}C}M, the category of left ^{∗}C-modules. Then [6] brings us necessary and sufficient conditions for an object of M^{C} to be projective (resp. flat) as a module over the endomorphism ring End^{C}(Λ) if Λ is finitely generated (resp. finitely presented) as a left ^{∗}C-module. In this paper we will generalize these results to the category M^{C}, where C is projective as a left A-module (not necessarily finitely generated). If C = A then M^{C} is the category of right A-modules, ^{∗}C = A and we recover the results from [6]. If C contains a grouplike element x, then A is a right C-comodule that is cyclic as a left ^{∗}Cmodule (see Lemma 2.5). So we can replace Λ with A in our main results; in this case, End^{C}(Λ) is the subring of (C, x)-coinvariants of A. Interesting examples

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2008 Heldermann Verlag

400 T. Guédénon: Projectivity and Flatness over …

of corings come from entwining structures. The language of corings allows us to generalize, unify and formulate more elegantly many results about generalized Hopf modules. This is due to the fact that Hopf modules and their generalizations (entwining modules, Doi-Hopf modules, relative Hopf modules, Yetter-Drinfeld modules, comodules over a coalgebra, etc. ...) are in fact comodules over a coring. Our techniques and methods are inspired from [5], [6] and [10].

2. Preliminary results

Let A be a k-algebra. An A-coring C is an (A, A)-bimodule together with two (A, A)-bimodule maps Δ_{C} : C → C ⊗_{A} C and ϵ_{C} : C → A such that the usual coassociativity and counit properties hold. For more details on corings, we refer to [1], [2], [3], [4], [8] and [9]. Let C be an A-coring. We use the notation of Sweedler-Heyneman but we will omit the symbol ∑ and the parentheses on subscripts. A right C-comodule is a right A-module M together with a right A-linear map ρ_{M,C} : M → M ⊗_{A} C; m ↦ m_{0} ⊗_{A} m_{1} such that

(id_{M} ⊗_{A} ϵ_{C}) ∘ ρ_{M,C} = id_{M}, and (id_{M} ⊗_{A} Δ_{C}) ∘ ρ_{M,C} = (ρ_{M,C} ⊗_{A} id_{C}) ∘ ρ_{M,C}.

A morphism of right C-comodules f : M → N is a right A-linear map such that

ρ_{N,C} ∘ f = (f ⊗_{A} id_{M}) ∘ ρ_{M,C}.

We denote the set of comodule morphisms between M and N by Hom^{C}(M, N). Denote by M^{C} the category formed by right C-comodules and comodule morphisms and by M the category of k-modules. By [4, 18.3], the category M^{C} has direct sums. We write ^{∗}C =_{A} Hom(_{A}C,_{A} A), the left dual ring of C. Then ^{∗}C is an associative ring with unit ϵ_{C} (see [4, 17.8]); the multiplication is defined by

f#g = f ∘ (id_{C} ⊗ g) ∘ Δ_{C} or equivalently f#g(c) = f(c_{1}g(c_{2}))

for all left A-linear maps f, g: C → A and c ∈ C; where Δ_{C}(c) = c_{1} ⊗_{A} c_{2}. We will denote by _{^{∗}C}M the category of left ^{∗}C-modules. Any right C-comodule M is a left ∗C-module: the action is defined by f.m = f(m_{0})m_{1} (see [4, 19.1]). Assume that C is projective as a left A-module. By [4, 18.14], M^{C} is a Grothendieck category and by [4, 19.3], it is a full subcategory of _{^{∗}C}M; i.e.,

Hom^{C}(M, N) =_{^{∗}C} Hom(M, N) for any M, N ∈ M^{C}.

As a consequence, an object of M^{C} that is projective in _{^{∗}C}M is projective in M^{C}.

Given two right C-comodules Λ and N, the k-module Hom^{C}(Λ, N) is a module over the endomorphism ring B = End^{C}(Λ) with the standard action

fb = f ∘ b; ∀f ∈ Hom^{C}(Λ, N), b ∈ B.

This defines a functor G = Hom^{C}(Λ, −) : M^{C} → M_{B}. The functor G has the left adjoint F = − ⊗_{B} Λ : M_{B} → M^{C}, where for any P ∈ M_{B}, F (P) = P ⊗_{B} Λ is a right C-comodule with the coaction ρ_{P,C} = id_{P} ⊗_{B} ρ_{Λ,C}; that is, there is a canonical isomorphism, for N ∈ M^{C} and P ∈ M_{B}

Hom^{C}(P ⊗_{B} Λ, N) → Hom_{B}(P, Hom^{C}(Λ, N)); f ↦ [p ↦ f(p ⊗ −)]

with inverse map g ↦ [p ⊗ λ ↦ g(p)(λ)].

The unit of the adjunction is given by

u_{N} : N → Hom^{C}(Λ, N ⊗_{B} Λ), n ↦ [λ ↦ n ⊗ λ]

for N ∈ M_{B} while the counit is the evaluation map

c_{M} : Hom^{C}(Λ, M) ⊗_{B} Λ → M, f ⊗ λ ↦ f(λ)

for M ∈ M^{C}. The adjointness property means that we have

G(c_{M}) ∘ u_{G(M)} = id_{G(M)}, c_{F (N)} ∘ F (u_{N}) = id_{F (N)}; M ∈ M^{C}, N ∈ M_{B} (⋆).

3. The main results

Let A be a k-algebra and C an A-coring. We keep the notations of the preceding sections. An object Λ ∈ M^{C} is called semi-Σ-projective if the functor Hom^{C}(Λ, −) : M^{C} → M sends an exact sequence of the form Λ^{(J)} → Λ^{(I)} → N → 0 to an exact sequence (see [11]). Obviously, a projective object in M^{C} is semi-Σ-projective.

Lemma 3.1. Assume that C is projective as a left A-module. Let Λ be a right C-comodule that is finitely generated as a left ^{∗}C-module, and let B = End^{C}(Λ). For every index set I,

(1) the natural map κ : B^{(I)} = Hom^{C}(Λ, Λ)^{(I)} → Hom^{C}(Λ, Λ^{(I)}) is an isomorphism;

(2) c_{Λ^{(I)}} is an isomorphism;

(3) u_{B^{(I)}} is an isomorphism;

(4) if Λ is semi-Σ-projective in M^{C}, then u is a natural isomorphism; in other words, the induction functor F = (−) ⊗_{B} Λ is fully faithful.

Proof. (1) is easy.

(2) It is straightforward to check that the canonical isomorphism B^{(I)} ⊗_{B} Λ ≃ Λ^{(I)} is nothing else than c_{Λ^{(I)}} ∘ (κ ⊗ id_{Λ}). It follows from (1) that κ ⊗ id_{Λ} is an isomorphism. So c_{Λ^{(I)}} is an isomorphism.

(3) Putting M = Λ^{(I)} in (⋆) and using (1), we find

Hom^{C}(Λ, c_{Λ^{(I)}}) ∘ u_{Hom^{C}(Λ,Λ^{(I)})} = id_{Hom^{C}(Λ,Λ^{(I)})}; i.e.,
Hom^{C}(Λ, c_{Λ^{(I)}}) ∘ u_{B^{(I)}} = id_{B^{(I)}}.

From (2), Hom^{C}(Λ, c_{Λ^{(I)}}) is an isomorphism, hence u_{B^{(I)}} is an isomorphism.

(4) Take a free resolution B^{(J)} → B^{(I)} → N → 0 of a right B-module N. Since u is natural, we have a commutative diagram

B^{(J)}  ⟶  B^{(I)}  ⟶  N⟶ 0
u_{B^{(J)}} ↓  u_{B^{(I)}} ↓  u_{N} ↓

GF(B^{(J)}) ⟶ GF(B^{(I)}) ⟶ GF(N) ⟶ 0

The top row is exact; the bottom row is exact, since

GF (B^{(I)}) = Hom^{C}(Λ, B^{(I)} ⊗_{B} Λ) = Hom^{C}(Λ, Λ^{(I)})

and Λ is semi-Σ-projective. By (3), u_{B^{(I)}} and u_{B^{(J)}} are isomorphisms; and it follows from the five lemma that u_{N} is an isomorphism. □

We can now give equivalent conditions for the projectivity and flatness of P ∈ M_{B}.

Theorem 3.2. Assume that C is projective as a left A-module. Let Λ be a right C-comodule that is finitely generated as a left ^{∗}C-module, and let B = End^{C}(Λ). For P ∈ M_{B}, we consider the following statements.

(1) P ⊗_{B} Λ is projective in M^{C} and u_{P} is injective;

(2) P is projective as a right B-module;

(3) P ⊗_{B} Λ is a direct summand in M^{C} of some Λ^{(I)}, and u_{P} is bijective;

(4) there exists Q ∈ M^{C} such that Q is a direct summand of some Λ^{(I)}, and P ≅ Hom^{C}(Λ, Q) in M_{B};

(5) P ⊗_{B} Λ is a direct summand in M^{C} of some Λ^{(I)}. Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇒ (5).

If Λ is semi-Σ-projective in M^{C}, then (5) ⇒ (3); if Λ is projective in M^{C}, then (3) ⇒ (1).

Proof. The proof uses Lemma 2.1 and is very similar to that of Theorem 2.2 in [6]. □

Over any ring A, a left module Λ is called finitely presented if there is an exact sequence A^{m} → A^{n} → Λ → 0 for some natural integers m and n. Clearly a finitely presented module is finitely generated.

Theorem 3.3. Assume that C is projective as a left A-module. Let Λ be a right C-comodule that is finitely presented as a left ^{∗}C-module, and let B = End^{C}(Λ). For P ∈ M_{B}, the following assertions are equivalent.

(1) P is flat as a right B-module; (2) P ⊗_{B} Λ = \underrightarrow{lim}Q_{i}, where Q_{i} ≅ Λ^{n_{i}} in M^{C} for some positive integer n_{i}, and

u_{P} is bijective;

(3) P ⊗_{B} Λ = \underrightarrow{lim}Q_{i}, where Q_{i} ∈ M^{C} is a direct summand of some Λ^{(I_{i})} in M^{C}, and u_{P} is bijective;

(4) there exists Q = \underrightarrow{lim}Q_{i} ∈ M^{C}, such that Q_{i} ≅ Λ^{n_{i}} for some positive integer n_{i} and Hom^{C}(Λ, Q) ≅ P in M_{B};

(5) there exists Q = \underrightarrow{lim}Q_{i} ∈ M^{C}, such that Q_{i} is a direct summand of some Λ^{(I_{i})} in M^{C}, and Hom^{C}(Λ, Q) ≅ P in M_{B}.

If Λ is semi-Σ-projective in M^{C}, these conditions are also equivalent to conditions (2) and (3), without the assumption that u_{P} is bijective.

Proof. The proof uses Lemma 2.1 and is very similar to that of Theorem 2.3 in [6]. □

Remark 3.4. Assume that C is projective and finitely generated as a left Amodule. By [4, 19.6], M^{coC} = _{^{∗}C}M. For every left ^{∗}C-module Λ, we have Λ^{coC} = ^{^{∗}C}Λ, the k-submodule of ^{∗}C-invariants of Λ. So End^{C}(Λ) = _{^{∗}C}End(Λ), the endomorphism ring of the left ^{∗}C-module Λ. Theorems 2.2 and 2.3 give necessary and sufficient conditions for projectivity and flatness of a module over _{^{∗}C}End(Λ). However these results are not new (see [6, Theorems 2.2 and 2.3]).

4. The coring contains a grouplike element

A grouplike element of C is an element x ∈ C such that Δ_{C}(x) = x ⊗_{A} x and ϵ_{C}(x) = 1_{A}. Assume that C contains a grouplike element x. By [4, 28.2], A is an object of M^{C}: the C-coaction is defined by ρ_{A,C}(a) = xa; ∀a ∈ A.

For any right C-comodule M, we call M^{coC,x} = {m ∈ M, ρ_{M,C}(m) = m⊗_{A}x} the k-submodule of (C, x)-coinvariants of M. Clearly, A^{coC,x} = {a ∈ A, ax = xa} is a subring of A called the subring of (C, x)-coinvariants.

We have Hom^{C}(A, M) = M^{coC,x} and End^{C}(A) = A^{coC,x}. Set B = A^{coC,x}. Then replacing Λ with A, the functors F and G become

F = (−) ⊗_{B} A : M_{B} → M^{C} and G = (−)^{coC,x} : M^{C} → M_{B}.

The unit of the adjunction pair (F, G) is given by

u_{N} : N → (N ⊗_{B} A)^{coC,x}; n ↦ (n ⊗ 1)

for every right B-module N while the counit is

c_{M} : M^{coC,x} ⊗_{B} A → M; m ⊗ a ↦ ma

for every right C-comodule M.

Lemma 4.1. Assume that C contains a grouplike element x. Then A is a cyclic left ^{∗}C-module under the action defined by f.a = f(xa) for all f ∈ ^{∗}C and a ∈ A.

Proof. Note that ϵ_{C}.a = ϵ_{C}(xa) = ϵ_{C}(x)a = 1_{A}a = a and Δ_{C}(xa) = x ⊗_{A} xa. Now for f, g ∈ ^{∗}C, we have (f#g).a=(f#g)(xa)=f(xg(xa))=f(x(g.a))=f.(g.a). □

From Theorems 2.2 and 2.3 we easily obtain the following results.

Theorem 4.2. Assume that C is projective as a left A-module and contains a grouplike element x. Set B = A^{coC,x}. For P ∈ M_{B}, we consider the following statements.

(1) P ⊗_{B} A is projective in M^{C} and u_{P} is injective;

(2) P is projective as a right B-module;

(3) P ⊗_{B} Λ is a direct summand in M^{C} of some A^{(I)}, and u_{P} is bijective;

(4) there exists Q ∈ M^{C} such that Q is a direct summand of some A^{(I)}, and P ≅ Hom^{C}(A, Q) in M_{B};

(5) P ⊗_{B} A is a direct summand in M^{C} of some A^{(I)}. Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇒ (5).

If A is semi-Σ-projective in M^{C}, then (5) ⇒ (3); if A is projective in M^{C}, then (3) ⇒ (1).

Theorem 4.3. Assume that C is projective as a left A-module and contains a grouplike element x, and that A is finitely presented as a left ^{∗}C-module. Set B = A^{coC,x}. For P ∈ M_{B}, the following assertions are equivalent.

(1) P is flat as a right B-module;

(2) P ⊗_{B} A = \underrightarrow{lim}Q_{i}, where Q_{i} ≅ A^{n_{i}} in M^{C} for some positive integer n_{i}, and u_{P} is bijective;

(3) P ⊗_{B} A = \underrightarrow{lim}Q_{i}, where Q_{i} ∈ M^{C} is a direct summand of some A^{(I_{i})} in M^{C}, and u_{P} is bijective;

(4) there exists Q = \underrightarrow{lim}Q_{i} ∈ M^{C}, such that Q_{i} ≅ A^{n_{i}} for some positive integer n_{i} and Hom^{C}(A, Q) ≅ P in M_{B};

(5) there exists Q = \underrightarrow{lim}Q_{i} ∈ M^{C}, such that Q_{i} is a direct summand of some A^{(I_{i})} in M^{C}, and Hom^{C}(A, Q) ≅ P in M_{B}.

If A is semi-Σ-projective in M^{C}, these conditions are also equivalent to conditions (2) and (3), without the assumption that u_{P} is bijective.

The following lemma will enable us to improve Theorems 3.2 and 3.3 if P is a right A-module (for example, if P is a right C-comodule).

Lemma 4.4. Assume that C contains a grouplike element x. Let N be a right A-module. Then u_{N} is an injection.

Proof. By [4, 18.9(1)], C is a right C-comodule with the coaction Δ_{C}; and N ⊗_{A} C is a right C-comodule with the coaction id_{N} ⊗ Δ_{C}. By [4, 18.10(1)], the k-linear map

φ : (N ⊗_{A} C)^{coC,x} → N; n ⊗ c ↦ nϵ_{C}(c)

is an isomorphism; i.e., G(W) ≅ N, where W = N ⊗_{A} C. Replace M with W in (⋆), then we get G(c_{W}) ∘ u_{N} = id_{N}. So u_{N} is an injection. □

Remark 4.5. By Lemma 3.4, u_{P} is an injection for every P ∈ M_{A}. So in the particular case where P is a right A-module, the injection assumption in Theorem 3.2 is superfluous. So we can replace “bijection” with “surjection” in Theorems 3.2 and 3.3. In other words, we get necessary and sufficient conditions for a right A-module P to be projective (resp. flat) as a right B-module.

5. The main example

Interesting examples of corings come from entwining structures. A right-right entwining structure over k is a triple (A, C, ψ) consisting of a k-algebra A, a coalgebra C over k (with comultiplication Δ_{C} and counit ϵ_{C}) and a k-module map ψ : C ⊗ A → A ⊗ C satisfying the following conditions (see [4, 32.1]):

ψ ∘ (id_{C} ⊗ μ) = (μ ⊗ id_{C}) ∘ (id_{A} ⊗ ψ) ∘ (ψ ⊗ id_{A})
(id_{A} ⊗ Δ_{C}) ∘ ψ = (ψ ⊗ id_{C}) ∘ (id_{C} ⊗ ψ) ∘ (Δ_{C} ⊗ id_{A})
(id_{A} ⊗ ϵ_{C}) ∘ ψ = ϵ_{C} ⊗ id_{A}
ψ ∘ (id_{C} ⊗ ι) = ι ⊗ id_{C},

where μ is the multiplication of A and ι is its unit.

If we set ψ(c ⊗ a) = a_{α} ⊗ c^{α}, these relations are respectively equivalent to

(aa′)_{α} ⊗ c^{α} = a_{α}a′_{β} ⊗ c^{αβ}
a_{α} ⊗ Δ_{C}(c^{α}) = a_{αβ} ⊗ c_{1}^{β} ⊗ c_{2}^{α}
a_{α}ϵ_{C}(c^{α}) = ϵ_{C}(c)a
1_{α} ⊗ c^{α} = 1 ⊗ c.

The map ψ is called an entwining map, and A and C are said to be entwined by ψ. By [4, 32.1], C = A ⊗ C is an A-coring with A-multiplications a′(a ⊗ c)a′′ = a′aψ(c ⊗ a′′), coproduct

Δ_{C} : A ⊗_{k} C → A ⊗_{k} C ⊗_{A} A ⊗_{k} C ≅ A ⊗_{k} C ⊗_{k} C; a ⊗ c ↦ a ⊗ Δ_{C}(c)

and counit ϵ_{C}(a ⊗ c) = aϵ_{C}(c). The category M_{A}^{C}(ψ) of right-right (A, C, ψ)- entwined modules (see [4, 32.4] for the definition) is isomorphic to M^{C}. If k is a field, C is projective as a left A-module. An element c ∈ C is called a grouplike element if Δ_{C}(c) = c ⊗ c and ϵ_{C}(c) = 1. If C contains a grouplike element c, then the coring A ⊗_{k} C contains x = 1_{A} ⊗ c as a grouplike element. For more information about entwining structures we refer to [1], [2], [4] and [8].

Many entwining structures come from Doi-Hopf data.

Let k be a field, H a Hopf k-algebra with a bijective antipode S_{H} and C a coalgebra over k. We say that C is a right H-module coalgebra if C is a right H-module, and Δ_{C} and ϵ_{C} are right H-linear. A right H-comodule A that is also an algebra is called a right H-comodule algebra, if the unit and the multiplication are right H-colinear.

Definition 5.1. Let A be a right H-comodule algebra and C a right H-module coalgebra. According to [7], we call the triple (H, A, C) a right-right Doi-Hopf datum.

By [4, 33.4], any right-right Doi-Hopf datum (H, A, C) gives rise to an entwining structure (A, C, ψ): the map ψ is defined by ψ(c ⊗ a) = a_{[0]} ⊗ (c ↼ a_{[1]}), where ρ_{A,H}(a) = a_{[0]} ⊗ a_{[1]} and ↼ denotes the H-action on C. By [4, 33.4], the category M^{C} is isomorphic to M(H)_{A}^{C}, the category of right-right (H, A, C)-Doi-Hopf modules.

Let us give some examples of Doi-Hopf data. In the examples, H is a Hopf algebra with a bijective antipode.

Examples 5.2. (i) Let I be a Hopf ideal of H. For any right H-comodule algebra A, the triple (H, A, H/I), where the H-action on H/I is given by (h′ + I).h = (h′h) + I is a right-right Doi-Hopf datum. The coring C = A ⊗ (H/I) contains 1_{A}⊗(1_{H}+I) as a grouplike element. If I = 0, the category of right-right (H, A, H)- Doi-Hopf modules M(H)_{A}^{H} is nothing else than the category M_{A}^{H} of right-right relative (A, H)-Hopf modules. The coring C = A ⊗ H contains x = 1_{A} ⊗ 1_{H} as a grouplike element. For every right-right relative (A, H)-Hopf module Λ, Λ^{coC,x} is the vector subspace Λ^{coH} of H-coinvariants of Λ; and A^{coC,x} is the subring A^{coH} of H-coinvariants of A. In the particular case A = H, we obtain the category of Hopf modules. Note that our results on projectivity are not new when Λ = A and I = 0 (see [5, Theorem 2.1]).

(ii) Let A be an (H, H)-bicomodule algebra. We know from [8, page 182] that the triple (H^{op} ⊗ H, A, H) is a right-right Doi-Hopf datum. The coaction of H^{op} ⊗ H on A is given by

ρ_{H^{op}⊗H,H}(a) = a_{[0]} ⊗ S_{H}(a_{[−1]}) ⊗ a_{[1]},

and the action of H^{op} ⊗ H on H is given by

l ↼ (h ⊗ h′) = hlh′, for all h, h′, l ∈ H.

The coring C = A ⊗ H contains x = 1_{A} ⊗ 1_{H} as a grouplike element. The corresponding category of right-right (H^{op}⊗H, A, H)-Doi-Hopf modules M(H^{op} ⊗ H)_{H}^{H} is nothing else than the category YD(H, H)_{A}^{H} of right-right Yetter-Drinfeld modules. For every right-right Yetter-Drinfeld module Λ, Λ^{coC,x} is the vector subspace Λ^{coH} of H-coinvariants of Λ; and A^{coC,x} is the subring A^{coH} of H-coinvariants of A. In the particular case A = H, we obtain the category of classical Yetter-Drinfeld modules, also named crossed modules or quantum Yang-Baxter modules.

(iii) Let us consider the right-right Doi-Hopf datum (H, A, H), where the coaction of H on A is trivial, and the H-action on H is given by right multiplication. The corresponding category of right-right (H, A, H)-Doi-Hopf modules M(H)_{A}^{H} is nothing else than the category L_{A}^{H} of right-right generalized Long dimodules [8, page 193]. The coring C = A⊗H contains x = 1_{A}⊗1_{H} as a grouplike element, and for every Long dimodule Λ, Λ^{coC,x} is the vector subspace Λ^{coH} of H-coinvariants of Λ; and A^{coC,x} is the subring A^{coH} of H-coinvariants of A. In the particular case A = H, we obtain the category of classical Long dimodules.

(iv) Let H = k be the trivial Hopf algebra. For any algebra A and any coalgebra C, the triple (k, A, C) is a right-right Doi-Hopf datum. If A = k (see [8, page 49]), the category of right-right (k, k, C)-Doi-Hopf modules M(k)_{k}^{C} is nothing else than the category M^{C} of right C-comodules.

(v) For any right H-module coalgebra C, the triple (H, H, C) is a right-right Doi-Hopf datum.

Acknowledgements. This paper was written during my visit to the Mathematics Section of the International Centre for Theoretical Physics (ICTP). I gratefully acknowledge both the kind invitation and the hospitality extended to me.

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Received April 18, 2007