Printer Friendly

Algebraic reduction and rigidity for Hilbert modules.

1. Introduction. Following the advent of quantum mechanics and the influence that had on mathematics, operator theorists studied mostly self-adjoint phenomena. This changed during the last two decades. One source of motivation for this change were the results obtained on the shift operator in the rich context of function theory and harmonic analysis. The theorem of Beurling [7] characterizing the invariant subspaces of the unilateral shift was the major impetus. Extensions of this idea led to many important works by other investigators.

In one direction, the older idea of a kernel Hilbert space of analytic functions formed the setting and many interesting results were obtained. The article of Shields [17] provides a good interim overview of this development. Although some of the results were couched in the language of several complex variables, the subject largely retained the flavor and the heuristics of one complex variable. The other direction concerned the study of invariant subspaces for other Hilbert spaces of holomorphic functions. Again, although the domains were multi-variable, the point of view was one variable. The same was true for much of operator theory. But the groundwork was being laid by various authors, cf. Taylor [18], for example, for a study of several variable operator theory. Nevertheless, such a theory lacked a foundation and a point of view.

In an attempt to provide a needed framework for multi-variable operator theory, the first author developed a module formulation which was systematically presented in [12]. In this module context, the result of Beurling shows that all submodules of the Hardy module for the disk algebra are isomorphic [9]. In generalizing this result to several variables, one was somewhat surprised to find that the analogous result was false for the polydisk algebra. Earlier examples of this phenomena [6], [1], [2] were subsumed in the Rigidity Theorem [12], [13]. This result applied to submodules obtained from the closure of polynomial ideals and was based on results and ideas from algebraic geometry involving localization. (See [14] for related results which apply to more general submodules.)

In the present work, we use these localization techniques to obtain general rigidity results. In particular, under mild restrictions, we show that the submodules obtained from the closure of ideals are equivalent if and only if the ideals coincide. The algebra consists of holomorphic functions and is assumed to be Noetherian, while the basic module is the closure of this algebra with respect to a suitable Banach space norm. In addition, the closure is assumed to satisfy conditions for point evaluation in analogy to those for a kernel Hilbert space. Finally, the ideals must not be principal or even locally principal, and must satisfy certain other technical hypotheses.

In section 2, we discuss algebraic reduction and then apply it to our problem in section 3. Here we have used algebraic localization, while in [12] we based our localization on the module tensor product. In section 4, we relate these two notions of localization. Finally, in section 5, we specialize our results by giving several concrete examples to which the main theorem applies. As a corollary we obtain essentially all the earlier rigidity results.

The authors wish to express their thanks to Michael Artin for a tutorial on the work of Grothendieck and its relevance to the present investigation.

2. Algebraic reduction. In this section we prove a series of relative closure theorems. These can be regarded as generalizations of the bijective correspondence of [3] between the subspaces of [H.sup.2]([D.sup.n]) of finite codimension which are invariant under multiplication by coordinate functions on the one hand and the ideals in the polynomial ring C[[z.sub.1],...,[z.sub.n]] of finite codimension whose zero sets are contained in the polydisk, [D.sup.n], on the other. We establish terminology in this section which will be used throughout the paper.

Let G be a nonempty bounded open subset of [C.sup.n], let Hol(G) denote the ring of analytic functions on G, and let X be a Banach space contained in Hol(G). We call X a reproducing G-space if X contains the constant function 1 and if for each w in G the evaluation functional, [E.sub.w](f) = f(w), is a bounded linear functional on X.

Let k = ([k.sub.1],...,[k.sub.n]) be an n-tuple of nonnegative integers (a multi-index), set [absolute value of k] = [k.sub.1] + ... + [k.sub.n] and define

[Mathematical Expression Omitted]

The following result is well known. We give its proof for the sake of completeness.

LEMMA 2.1. If X is a reproducing G-space, then [Mathematical Expression Omitted] is a bounded linear functional on X for every multi-index k and every w in G.

Proof. Let [X.sup.*] denote the dual of X and consider the map [Phi](w) = [E.sub.w] from G into [X.sup.*]. This map is easily seen to be [weak.sup.*]-analytic and hence norm analytic. But clearly,

[Mathematical Expression Omitted],

from which the result follows.

Now let R be a (complex) subalgebra of Hol(G) containing 1, and let X be a reproducing G-space. We call X a reproducing R-module on G, if f [center dot] X is contained in X for every f in R. Note that, by a simple application of the closed graph theorem, the operator [T.sub.f] defined to be multiplication by f is bounded on X for each f in R. Note also that R [subset] X follows from the fact that 1 is in X.

For any subset I of R we let [I] denote the closure of I in X. If I is an ideal in R, then [I] is a closed R-submodule of X, and if Y is a closed R-submodule of X, then Y [intersection] R is a closed ideal in R. We call an ideal I of R contracted if I = [I] [intersection] R. We are interested in the identification of the contracted ideals of R. We will begin with the codimension 1 case.

Let x be a point in the maximal ideal space of R and let [M.sub.x] = {f [element of] R [where] f(x) = 0} denote the corresponding maximal ideal. Since R is contained in Hol(G), every point in G corresponds to a maximal ideal in R. We call x a virtual point of X provided that the homomorphism f [approaches] f(x) defined on R extends to a bounded linear functional on X. Since X is a reproducing G-space, every point in G is a virtual point. We let vp(X) denote the collection of virtual points.

LEMMA 2.2. Let X be a reproducing R-module and let x be in the maximal ideal space of R. Then x is a virtual point of X if and only if [M.sub.x] is contracted.

Proof. Let x be a virtual point and let [Phi] : [R] [approaches] C be the bounded linear functional that extends f [approaches] f(x). Since [M.sub.x] [subset] [[M.sub.x]] [intersection] R [subset] (ker [Phi]) [intersection] R = [M.sub.x], we see that [M.sub.x] is contracted. Conversely, if x is not a virtual point, then we have a sequence {[g.sub.n]} [subset] R, [g.sub.n](x) = 1 with [[[g.sub.n]]] [less than or equal to] 1/n. But then 1 - [g.sub.n] is in [M.sub.x]; hence 1 is in [[M.sub.x]]. This shows that [M.sub.x] is not contracted and we have a contradiction.

LEMMA 2.3. Let X be a reproducing R-module and let I, J be ideals in R.

(a) If I and J are contracted, then I [intersection] J is contracted.

(b) If I is contracted, I [subset] J, and [dim.sub.C] (J/I) is finite, then J is contracted.

(c) If Y is a closed R-submodule of X, then Y [intersection] R is a contracted ideal.

Proof. (a) follows from I [intersection] J [subset] [I [intersection] J] [intersection] R [subset] ([I] [intersection] R) [intersection] ([J] [intersection] R) [subset] I [intersection] J.

To prove (b), write J = I + F with dim (F) = dim (J/I). Then we have J [subset] [I] + F [subset] [J]. But since [I] + F is closed, we have [I] + F = [J]. Hence, [J] [intersection] R = ([I] + F) [intersection] R [subset] ([I] [intersection] R) + F = J. Thus J is contracted.

(c) is obvious.

The above result can also be found in [12, Lemma 6.7].

If I is an ideal in R, then we let Z(I) denote the zero set of I, that is, the set of points x in the maximal ideal space of R such that f(x) = 0 holds for all f in I.

PROPOSITION 2.4. Let X be a reproducing R-module with [R] = X, let Y be a closed R-submodule with [dim.sub.C] (X/Y) finite, and set I = Y [intersection] R. Then Y = [I], [dim.sub.C] (R/I) = [dim.sub.C] (X/Y) and Z(I) is contained in vp(X).

Proof. Let [Pi] : X [approaches] X/Y denote the quotient map. Since [dim.sub.C] (X/Y) is finite, and R is dense in X, [Pi](R) = X/Y. But I = (ker [Pi]) [intersection] R and so, [dim.sub.C] (R/I) = [dim.sub.C] (X/Y).

Let F be a finite dimensional subspace of R so that I + F = R and [dim.sub.C] (F) = [dim.sub.C] (R/I). Since [I] + F is closed and contains R, [I] + F = X. By using [I] [subset] Y and a simple dimension count, [I] = Y.

Finally, if x is in Z(I), then the maximal ideal [M.sub.x] [contains] I, [dim.sub.C] ([M.sub.x]/I) is finite and I is contracted, so that [M.sub.x] is contracted by Lemma 2.3 (b). By Lemma 2.2, x is a virtual point.

In the above generality, there is little more that can be said about the closed subspaces of finite codimension. (Of course, every ideal I in R of finite codimension over C gives rise to the closed R-submodule [I].) In the case of [H.sup.2]([D.sup.n]), Ahern and Clark [3] proved the converse of Proposition 2.4, namely, that every ideal I in C[[z.sub.l],...,[z.sub.n]] of finite codimension is contracted whenever Z(I) is contained in vp([H.sup.2]([D.sup.n])) = [D.sup.n]. Thus, the correspondence I [approaches] [I] between these ideals and the closed R-submodules of finite codimension in [H.sup.2]([D.sup.n]) is bijective. The following example shows that when vp(X) [not equal to] G, then the correspondence need not be bijective. That is, there can exist ideals of finite codimension with Z(I) [subset] vp(X) such that I [not equal to] [I] [intersection] R.

EXAMPLE 2.5. Let R = C[z] be the ring of polynomials in one variable and let X = A(D) be the disc algebra. Clearly, X is a reproducing R-module on D. However, since every function in A(D) is continuous on the closed disc [D.sup.-], we see that vp(X) = [D.sup.-]. Thus, for x in [Delta]D, we see that the maximal ideal [M.sub.x] is contracted. It is not difficult to see that [Mathematical Expression Omitted] is not contracted. If it were, then by arguments similar to those in the proof of Lemma 2.2, it would follow that the linear map: f [approaches] f[prime](x) defined on R extends to a bounded linear functional on X. In fact, we have [Mathematical Expression Omitted].

For the next result we need to recall some concepts from commutative algebra. If R is a Noetherian ring, then every ideal I in R has a (irredundant) primary decomposition I = [[intersection].sub.1[less than or equal to]j[less than or equal to]m][I.sub.j], where each [I.sub.j] is [P.sub.j]-primary for some prime ideal [P.sub.j]. While the set {[I.sub.j] [where] 1 [less than or equal to] j [less than or equal to] m} is not uniquely determined by I, the set {[P.sub.j] [where] 1 [less than or equal to] j [less than or equal to] m} is, and these are called the associated primes. We note that Z(I) = [[union].sub.1[less than or equal to]j[less than or equal to]m]Z([P.sub.j]) and we call each Z([P.sub.j]) = Z ([I.sub.j]) an algebraic component of Z(I).

If S is a multiplicatively closed subset of R, then we may form the ring of quotients (or simply the quotient ring when there is no chance of confusion),

[S.sup.-1] (R) = {p/q [where] P [element of] R, q [element of] S}.

If I is an ideal in R, then [S.sup.-1] (I) = {p/q [where] P [element of] I, q [element of] S} is the ideal generated by I in [S.sup.-1] (R) and we set S(I) = [S.sup.-1](I) [intersection] R. If [M.sub.x] is a maximal ideal in R, then [S.sub.x] = R\[M.sub.x] is multiplicatively closed and we call [Mathematical Expression Omitted] the localization of R at x. This construction makes sense for any prime ideal in place of the maximal ideal [M.sub.x]. When the pair R and [S.sup.-1](R) is understood, [S.sup.-1](I) and S(I) are also written as [I.sup.e] and [I.sup.ec], respectively; see [19, I, p. 218].

Let G be a nonempty bounded open subset of [C.sup.n] and let R be a (complex) subalgebra of Hol(G) containing 1. We will call R analytically admissible provided that R is Noetherian and that for each x in G and each natural number j, the maximal ideal [M.sub.x] = {f [element of] R [where] f(x) = 0} has the property that

[Mathematical Expression Omitted].

It is easy to see that [Mathematical Expression Omitted] is always contained in the right hand side.

EXAMPLE 2.6. For any G, the algebra of polynomials is admissible as is the algebra of rational functions with poles off the closure [G.sup.-] of G. On the other hand, if [Phi] is a transcendental analytic function on D with [Phi](0) = 0, then the subalgebra R of Hol(D) generated by z and [Phi](z) is not analytically admissible. To see this, note that since R is the algebra of polynomials in z and [Phi](z), the maximal ideal [M.sub.0] consists of those functions of the form p(z, [Phi](z)) = [[Sigma].sub.i,j][a.sub.i,j][z.sup.i][Phi][(z).sup.j] with [a.sub.0,0] = 0. Thus, [Mathematical Expression Omitted] consists of those functions with [a.sub.0,0] = [a.sub.0,1] = 0, while [Mathematical Expression Omitted] if and only if [a.sub.1,0] + [a.sub.0,1][Phi][prime](0) = 0.

THEOREM 2.7. Let G be a nonempty bounded open subset of [C.sup.n], X be a reproducing R-module on G with R analytically admissible, and I be an ideal in R. If every algebraic component of Z(I) has a nonempty intersection with G, then I is contracted

Proof. If J = [I] [intersection] R, then J is an ideal in R which contains I. We must prove that J = I.

Fix x in G and fix a nonnegative integer j. Since [Mathematical Expression Omitted] is a finite family of continuous linear functionals on X and J is contained in the closure of I, it follows for p in J that there exists q in I with

[Mathematical Expression Omitted], for all k with [absolute value of k] [less than] j.

Since R is analytically admissible, this implies that p - q is in [Mathematical Expression Omitted] so [Mathematical Expression Omitted] holds for all j [greater than or equal to] 0.

By the Artin-Rees lemma [4, Corollary 10.10], there is an integer k [greater than or equal to] 0 such that [Mathematical Expression Omitted] for j [greater than or equal to] k. In particular, setting j = k + 1, we see that J [subset] I + [M.sub.x] [center dot] J. Passing to the quotient ring [Mathematical Expression Omitted], this implies that

[Mathematical Expression Omitted],

and so by Nakayama's lemma [4, Corollary 2.7], [Mathematical Expression Omitted]. Hence,

[Mathematical Expression Omitted].

Now let I = [[intersection].sub.1[less than or equal to]j[less than or equal to]m][I.sub.j] be the primary decomposition of I with {[P.sub.j] [where] 1 [less than or equal to] j [less than or equal to] m} the associated primes. Since Z([P.sub.j]) [intersection] G is nonempty by hypothesis, we deduce from [4, Proposition 4.9] that [Mathematical Expression Omitted] or R (according to x [element of] Z([P.sub.j]) [intersection] G or not) because [I.sub.j] is primary, and hence [Mathematical Expression Omitted] for each x in Z([P.sub.j]) [intersection] G. Therefore

[Mathematical Expression Omitted].

This proves the assertion.

We are now in a position to give a complete generalization of the result in [3], and of some of the results in [5].

COROLLARY 2.8. Let G be a bounded open subset of [C.sup.n], let X be a reproducing R-module on G with R analytically admissible, and assume [R] = X. If vp(X) = G, then the maps I [approaches] [I] and Y [approaches] Y [intersection] R define bijective correspondence between the ideals I in R of finite codimension with Z(I) [subset] G and the closed R-submodules Y of X of finite codimension. These maps are mutual inverses and preserve codimension.

In particular, we see that when R is analytically admissible and vp(X) = G, then the contracted ideals of finite codimension in R are precisely those ideals I with Z(I) [subset] G. For "natural" function spaces it would be interesting to classify the contracted ideals. In particular, for [H.sup.2]([D.sup.n]) we conjecture that the contracted ideals in C[[z.sub.1],...,[z.sub.n]] are precisely the ideals given by Theorem 2.7, that is, those ideals such that every algebraic component of their zero set meets [D.sup.n] nontrivially. Many examples are known of Bergman spaces for which vp([B.sup.2](G)) [not equal to] G. However, for Bergman spaces with vp([B.sup.2](G)) = G, little is known about the classification of contracted ideals. On sets G with flat boundary sides, one might be able to produce a contracted prime ideal P whose zero set is only tangent to [Delta]G. Some related results have recently been obtained in [16].

It would also be valuable to understand the closure properties of the contracted ideals under various algebraic operations. Example 2.5 shows that if I and J are contracted, then I [center dot] J is not necessarily contracted. However, we do not know if such an example exists if vp(X) = G (G open!). We also do not know if the associated primes of a contracted ideal are necessarily contracted.

One result along these lines is the following. It will be used in section 3.

PROPOSITION 2.9. Let G be a bounded open subset of [C.sup.n] and X be a reproducing R-module on G with R analytically admissible. If I is a contracted ideal and J is an ideal of finite codimension in R with Z(J) [subset] G, then I [center dot] J is contracted.

Proof. By Theorem 2.7, [J.sup.k] is contracted for all k [greater than or equal to] 0. Applying the Artin-Rees lemma [4, Corollary 10.10] again, we have that for some k [greater than or equal to] 0,

[J.sup.k+1] [intersection] I = J [center dot] ([J.sup.k] [intersection] I) [subset] J [center dot] I.

By Lemma 2.3 (a), [J.sup.k+1] [intersection] I is contracted. Since [dim.sub.C] {J [center dot] I/([J.sup.k+1] [intersection] I)} is finite, we conclude that J [center dot] I is contracted by Lemma 2.3 (b).

The contractedness we discussed above is a concept of relative closedness of subspaces with respect to some topology. It can be discussed in a more general context. We include here some general theory about contractedness which adds more perspective to the techniques we used above and which should have further application.

Let M be a linear topological Hausdorff space with topology T. A subspace I of M is said to be contracted with respect to T if I is a closed subspace of M. If we denote the completion of M with respect to the given topology T by [Mathematical Expression Omitted], then M can be considered as a subspace of [Mathematical Expression Omitted]. A subspace I of M being contracted is equivalent to the following condition:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the closure of I in [Mathematical Expression Omitted].

The following lemma states an obvious but useful property of contractedness.

LEMMA 2.10. If [T.sub.1] and [T.sub.2] are two Hausdorff linear topologies on M with [T.sub.1] [subset] [T.sub.2], then a linear subspace I being contracted with respect to [T.sub.1] implies that I is contracted with respect to [T.sub.2].

We now focus on the particular case when M is a Noetherian ring and use the topology determined by an ideal. If A is an ideal in M, then the A-adic topology on M is defined by the powers of A. Thus the closure of a subset E of M in T is defined to be

[Mathematical Expression Omitted].

LEMMA 2.11. Let M be a commutative Noetherian ring and A be an ideal in M. An ideal I of M is contracted with respect to the A-adic topology T if and only if A + P [not equal to] M holds for every associated prime ideal P of I.

Proof. The result follows from Krull's Theorem, see [19, I, p. 217, Theorem 12[prime]].

If [[intersection].sub.1[less than or equal to]i[less than or equal to]m][Q.sub.i] is an irredundant intersection of primary ideal [Q.sub.i] with associated prime ideals [P.sub.i], 1 [less than or equal to] i [less than or equal to] m, then the closure of I is equal to the intersection of those [Q.sub.i] with [Q.sub.i] + A [not equal to] M. From this description, we see that there are many A-adic topologies for which I is closed. For example, we can take A to be [A.sub.1] ... [A.sub.m], where [A.sub.i] is any maximal ideal containing [P.sub.i], 1 [less than or equal to] i [less than or equal to] m. Moreover, by Lemma 2.10, I will be closed with respect to any topology of M that is stronger than T.

We now discuss the space of all continuous linear functionals on (M, T), where T is the A-adic topology for some ideal A of finite codimension in M. From the general theory of locally convex topological spaces, a linear functional f on M is T-continuous if and only if there exists a natural number m such that f is zero on [A.sup.m]. This means that there is a linear functional F on (the finite dimension space) M/[A.sup.m] such that the following diagram is commutative:

[Mathematical Expression Omitted]

Thus, if we give M the weak topology [T.sup.#] determined by the set of all continuous linear functionals on (M, T), then [T.sup.#] is weaker than T but (M, T) and (M, [T.sup.#]) have the same convex closed sets. In particular, a subspace I is contracted in (M, T) if and only if it is contracted in (M, [T.sup.#]). As a result, Lemma 2.11 remains true if T is replaced by [T.sup.#]. We note that convergence in [T.sup.#] is described as follows:

[a.sub.[Lambda]] [approaches] a if and only if [pr.sub.m]([a.sub.[Lambda]]) = [pr.sub.m](a) holds in the Euclidean topology for each m.

As a consequence of the preceding discussion, if K is a locally convex linear topology on M for which {[A.sup.n]} is closed, then K is stronger than T, so I is contracted with respect to K if it is so with respect to T. In Theorem 2.7, put M = R and pick one point from each algebraic component of the zero set of I. Let {[x.sub.1],...,[x.sub.m]} be this set. If [A.sub.i] is the maximal ideal in R associated to [x.sub.i], then A = [A.sub.1] ... [A.sub.m] satisfies the condition described in Lemma 2.11, so I is contracted with respect to T. Moreover, we have,

[Mathematical Expression Omitted].

If we put a Banach space topology K on R which makes the completion X into a reproducing R-module on G, then the [Mathematical Expression Omitted] are continuous functionals by Lemma 2.1. Thus [A.sup.n] is closed in R with respect to the Banach space topology K. This gives another proof of Theorem 2.7.

From the above discussion, we see that the Banach space topology on R is much stronger than necessary to guarantee that I be contracted. Our digression on contractedness suggests the possibility of further generalizations of our results.

3. The rigidity theorems. In this section we prove our main rigidity theorems. We begin first with some preliminaries, and then prove the algebraic rigidity theorems.

We assume that all rings are commutative and contain 1. If R is Noetherian and contains a unique maximal ideal M (i.e., R is a Noetherian local ring), then we let [Mathematical Expression Omitted] denote the M-adic completion [4, Chapter 10] of R. If E is a finitely generated R-module, then [Mathematical Expression Omitted] denotes the M-adic completion of E which is canonically isomorphic to [Mathematical Expression Omitted].

Let P be a prime ideal in the ring R. The height of P is defined to be the maximal length r of any properly increasing chain of prime ideals

[P.sub.0] [subset] ... [subset] [P.sub.r] = P.

When R is Noetherian, every prime ideal has finite height and the height of an arbitrary ideal is defined to be the minimum of the heights of its associated prime ideals. The Krull dimension of R is the supremum of the heights of the proper ideals of R. When R = C[[z.sub.1],...,[Z.sub.n]], the height r of an ideal I satisfies [dim.sub.C] (Z(I)) = n - r and the Krull dimension of R is n. A Noetherian local ring R with maximal ideal M is regular if the Krull dimension of R is the same as the R/M vector space dimension of M/[M.sup.2].

PROPOSITION 3.1. Let R be a Noetherian regular local ring. Let I and J be ideals in R of height at least 2. If I and J are isomorphic as R-modules, then I = J.

Proof. By a theorem of Auslander-Buchsbaum [19, II, Appendix 7], R is an UFD. Let K denote the quotient field of R, i.e. K = [S.sup.-1](R) with S = R\{0}. If [Phi] : I [approaches] J denotes the isomorphism, then [Phi] extends to an isomorphism [S.sup.-1]([Phi]) : [S.sup.-1](I) [approaches] [S.sup.-1](J). Since [S.sup.-1](I) = [S.sup.-1](J) = K, we can find x in K so that [S.sup.-1]([Phi]) is multiplication by x. Restricting to I, we have xI = J.

Write x = p/q with p, q in R. Since R is an UFD, we may assume that p and q have gcd 1. Thus, pI = qJ [subset] qR. Since R is an UFD with gcd(p, q) = 1, it follows that I [subset] qR. The prime ideals associated to a principal proper ideal in a Noetherian UFD have height at most 1. Thus, if qR [not equal to] R, then I [subset] P for some prime ideal P of height 1. Since one of the prime ideals associated to I must be contained in P [4, Proposition 4.6], I has height at most 1. This contradiction shows that qR = R. By symmetry, pR = R. Thus p and q are both units and I = J as desired.

LEMMA 3.2. Let R be a Noetherian regular local ring with maximal ideal M and let I, J be ideals of height at least 2. For each k, assume we are given module isomorphisms [[Phi].sub.k] : I/[M.sup.k] I [approaches] J/[M.sup.k]J such that the following diagram commutes,

[Mathematical Expression Omitted]

Then I = J.

Proof. The assumptions imply that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are isomorphic as [Mathematical Expression Omitted]-modules. Since [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are ideals of height at least 2, [Mathematical Expression Omitted] = [Mathematical Expression Omitted] follows from Proposition 3.1. Combining Krull's Theorem [19, I, p. 216, Theorem 12], [19, II, p. 257, Corollary 2] and Lemma 2.11, we have [Mathematical Expression Omitted].

Remark. In the case of interest to us, the isomorphism between I and J can in fact be shown without requiring the commutativity of the diagrams. However, the proof is considerably more demanding. This will be done in Proposition 3.5. In essence, we need to use results of Grothendieck [15] which extend the results quoted from Zariski and Samuel [19].

We return now to the setting of section 2. Let G denote a nonempty bounded open set in [C.sup.n], R [subset] Hol(G) be a Noetherian ring containing C[[z.sub.1],...,[z.sub.n]], and X be a reproducing R-module on G. For each x in G let [M.sub.x] = {f [element of] R [where] f(x) = 0} be the corresponding maximal ideal and let [Mathematical Expression Omitted] be the localization of R at x. We call R algebraically admissible if for every x in G the Noetherian local ring [Mathematical Expression Omitted] is regular and [dim.sub.C] [Mathematical Expression Omitted] is finite. It is easily checked that the polynomial ring C[[z.sub.1],...,[z.sub.n]] and the ring of bounded rational function

[Mathematical Expression Omitted] for all [Mathematical Expression Omitted]

are admissible.

We call R admissible if it is both analytically and algebraically admissible. Thus, the two preceding examples are both admissible.

THEOREM 3.3. Let G be a nonempty bounded open subset of [C.sup.n]. Let X be a reproducing R-module on G with R admissible, and let I, J be ideals of R of height at least 2 such that every algebraic component of their zero sets meets G. If there exist bounded R-module maps A: [I] [approaches] [J] and B : [J] [approaches] [I] with dense ranges, then I = J.

Proof. Fix x in G and fix an integer k [greater than or equal to] 0. Since A is an R-module homomorphism, we have [Mathematical Expression Omitted] so that we have an induced map

[Mathematical Expression Omitted].

But [dim.sub.C] [Mathematical Expression Omitted] and [Mathematical Expression Omitted] is bounded with dense range so that [Mathematical Expression Omitted] must be surjective. Similarly, [Mathematical Expression Omitted] is surjective so that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are finite dimensional (over C) isomorphic R-modules.

Consider the quotient map [Mathematical Expression Omitted]. By finite dimensionality over C and denseness, g is surjective and has kernel [Mathematical Expression Omitted]. But by Theorem 2.7 and Proposition 2.9, I and hence [Mathematical Expression Omitted] are contracted. Thus, [Mathematical Expression Omitted], and arguing analogously for J we see that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are isomorphic as R-modules. Passing to the local ring, this implies that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are isomorphic as [Mathematical Expression Omitted]-modules for all k [greater than or equal to] 0. Moreover, for each k, this isomorphism is induced by A; thus, the hypotheses of Lemma 3.2 are met and [Mathematical Expression Omitted]. Hence, for every x in G, [Mathematical Expression Omitted] and arguing as in Theorem 2.7, we have

[Mathematical Expression Omitted].

This completes the proof of the theorem.

COROLLARY 3.4. Let G be a nonempty bounded open subset of [C.sup.n], n [greater than or equal to] 2, X be a reproducing R-module with R admissible and assume that vp(X) = G and X = [R]. If Y and W are closed R-submodules of X of finite codimension, and A : Y [approaches] W, and B : W [approaches] Y are bounded R-submodule maps with dense ranges, then Y = W.

Proof. Apply Corollary 2.8.

We now turn our attention to generalizations of some of the above results. We start with some algebraic results.

PROPOSITION 3.5. Let R be a Noetherian local ring with maximal ideal M. Assume that R contains C and that dime (M/[M.sup.2]) [less than] [infinity]. If E and F are R-modules of finite type such that E/[M.sup.j+1] E [approximately equal to] F/[M.sup.j+1] F as R-modules for each j [greater than or equal to] 0, then E [approximately equal to] F as R-module.

Sketch of the proof. There are two steps in the proof.

Step 1. We first show that [Mathematical Expression Omitted]. Let [T.sub.j] denote the set of all R-module isomorphisms from E/[M.sup.j+1] E to F/[M.sup.j+1] F. Evidently, the set [T.sub.j] is mapped into [T.sub.j-1] by using the associativity of tensor product. In view of the assumptions, each [T.sub.j] is a nonempty constructible set (i.e., an intersection of a Zariski open with a Zariski closed set) in a suitable affine space of finite dimension (depending on j) over C. We therefore have an inverse sequence of nonempty affine constructible sets and morphisms. Our first step amounts to showing that the inverse limit of [T.sub.j], j [greater than or equal to] 0, is not empty. Let [Mathematical Expression Omitted] denote the Zariski closure of [T.sub.j] so that each [Mathematical Expression Omitted] is a finite union of irreducible Zariski closed subsets. If we consider the inverse sequence of finite sets made up from the components of [Mathematical Expression Omitted] (and the maps induced by composing the morphisms), we may conclude that the inverse limit of these nonempty finite sets is not empty (this is just the dual version of the Theorem of Konig stating that a locally finite tree containing arbitrarily long paths must contain an infinite path. Its proof is easily obtained by considering the case of a rooted tree through the selection of a base point.). In other words, we can assume at the beginning that each [Mathematical Expression Omitted] is irreducible and nonempty. The image of [Mathematical Expression Omitted] in [Mathematical Expression Omitted] is therefore nonempty, irreducible, and has a well defined dimension over C. This decreasing sequence of irreducible subsets of [Mathematical Expression Omitted] therefore achieves a common minimum dimension. Without loss of generality, we may assume that the images for j [greater than or equal to] [j.sub.0] are equal to the nonempty irreducible Zariski closed subset T of [Mathematical Expression Omitted]. If [dim.sub.C] T = 0, then T is a single point and coincides with the nonempty image of [T.sub.j] for j [greater than or equal to] [j.sub.0]. Arguing as before, we can conclude that the inverse limit of [T.sub.j] is not empty. If [dim.sub.C] T [greater than] 0, then the image of [T.sub.j] in T is obtained by deleting a finite union of proper Zariski closed subsets of T for j [greater than or equal to] [j.sub.0]. Since C is uncountable, the irreducible set T is not a countable union of proper Zariski closed subsets. It follows that the image of [T.sub.j] in T is not empty for j [greater than or equal to] [j.sub.0] and the inverse limit of [T.sub.j] is not empty by the same argument. We note that the present argument is an algebraic analogue of the Baire category argument.

Step 2. [Mathematical Expression Omitted] implies E [approximately equal to] F. This is a special case of Proposition 2.5.8 in Grothendieck's EGA IV [15; p. 25]. We translate it for convenience.

Let A be a commutative semilocal ring (only a finite number of maximal ideals). Let B be an A-algebra (not necessarily commutative) with V and W denoting two B-modules. Let A[prime] be a commutative A-algebra which is a faithfully flat A-module. Let B[prime], V[prime] and W[prime] be defined by tensoring with A[prime] over A. Assume one of the following two conditions:

(a) A and A[prime] are both Noetherian and V, W are A-modules of finite type.

(b) B is an A-module of finite type, V is B-projective of finite type and V is an A-module of finite presentation.

Then V[prime] [approximately equal to] W[prime] as B[prime]-modules if and only if V [approximately equal to] W as B-modules.

For the case at hand, A = B = R, V = E, [Mathematical Expression Omitted]. By assumption, condition (a) holds because A is Noetherian so that the completion A[prime] with respect to any M-adic topology is again Noetherian. We still have to check that A[prime] is a faithfully flat A-module. However, since A = R is Noetherian, the completion [Mathematical Expression Omitted] is a faithfully flat A-module if and only if the ideal M is contained in the Jacobson radical of A, see Bourbaki [8, Proposition 9, p. 206]. The Jacobson radical of A is the intersection of all maximal ideals of A. Since A is local with M as its unique maximal ideal, M is precisely the Jacobson radical of A. We have checked all the hypotheses.

Remark. It should be noted that the complicated hypotheses in the result quoted above are used in its proof. Roughly speaking, one performs various reduction steps. In some of the reduction steps, the more complicated situation arises. It is thus necessary to pose general hypotheses that are "stable" with respect to the reduction steps. In both Step 1 and Step 2, the problem we face is NOT linear. Namely, the problem involves GL(n) rather than M(n), n [greater than] 1.

The techniques used in the proof of our main theorem, Theorem 3.3, can actually be exploited to prove a more general rigidity result. Although Theorem 3.3 is a special case of this more general result, Theorem 3.3 is more readily applicable in the context of operator theory. For this reason we have stated Theorem 3.3 as our main result. We will now sketch the more general result in the subsequent paragraph for the interested reader.

Let R = C[[z.sub.1], ..., [z.sub.n]/P be a Noetherian domain, i.e., P is a prime ideal of C[[z.sub.l],...,[z.sub.n]]. Thus, the maximal ideal space V = Z(P) can be identified with a subset of [C.sup.n] by Hilbert's Nullstellen Satz. Let [Omega] be a nonempty open subset of V in the classical topology of V induced by [C.sup.n]. A locally convex linear Hausdorff topology T on R is said to have property (A) if the following condition holds:

(A) for each point x [element of] [Omega] and each integer k [greater than or equal to] 0,

[Mathematical Expression Omitted] is closed in (R, T).

As usual, [M.sub.x] is the maximal ideal of R associated to x and we identify x with [M.sub.x] when there is no chance of confusion.

If I is an ideal of R such that each of its associated prime ideals is contained in some M of [Omega], then the argument in our discussion about contracted ideals shows that such an I must be contracted in (R, T).

The completion of R with respect to T is denoted by [R] and the closure of I in [R] is denoted by [I]. We assume that [R] is an R-module under the natural action. Let M = [M.sub.1] ... [M.sub.s] with [M.sub.1] in [Omega] and I be a contracted ideal as before. Then [I] [[cross product].sub.R] R/[M.sup.k] is of finite dimension over C and is R-module isomorphic to I/[M.sup.k]. Now, if I and J are contracted ideals of height at least 2 and if there exist continuous R-module maps, A: [I] [approaches] [J] and B: [J] [approaches] [I] with dense ranges, then the same argument as in the proof of Theorem 3.3 yields,

[Mathematical Expression Omitted]

where "[caret]" denotes the M-adic completion. Thus, .[Mathematical Expression Omitted]. Since the argument is valid for all [M.sub.i] in [Omega], we can select the [M.sub.i]'s from the components of the zero sets of I and J and use the contractedness of I and J with respect to the M-adic topology to conclude that I = J.

Let G be a nonempty bounded open subset of an algebraic submanifold in [C.sup.n]. Let R be a Noetherian subring of Hol(G) containing C. Let X be a reproducing R-submodule on G. R is said to be analytically admissible, if to each point x in G with local coordinates ([z.sub.1],...,[z.sub.n]) the following is true for each j [greater than or equal to] 0,

[Mathematical Expression Omitted],

where [M.sub.x] = {f [element of] R [where]f(x) = 0} and [Mathematical Expression Omitted]. We can define R to be algebraically admissible as before, and R is called admissible if it is both algebraically and analytically admissible.

A straightforward application of the above discussion yields.

THEOREM 3.6. Let G be a nonempty bounded open subset of an algebraic manifold. Let R be a Noetherian subring of Hol(G) containing C, which is R admissible, X be a reproducing R-module on G, and I, J be ideals of R of height at least 2 such that each algebraic component of their zero sets meets G. If there exist bounded R-module maps A: [I] [approaches] [J] and B: [J] [approaches] [I] with dense ranges, then I = J.

4. Tensor products. In this section we examine tensor products of Banach modules and interpret many of the results of section 3 in terms of tensor products. Let X be a Banach space and let R. be an algebra over the complex numbers with unit 1. We call X a Banach R-module provided that X is an R-module with 1 [center dot] x = x for all x in X, and that for each fixed f in R the map x [approaches] f [center dot] x is a bounded map on X. If X and Y are Banach spaces, we let X [cross product] Y denote their tensor product as complex vector spaces and we let [Mathematical Expression Omitted] denote the completion of the tensor product in the projective (i.e., greatest) cross-norm. Recall that for u in X [cross product] Y the projective cross-norm of u is defined by,

[Mathematical Expression Omitted]

If X and Y are both Banach R-modules, then let N denote the smallest closed subspace of [Mathematical Expression Omitted] which contains the set

{(f [center dot] x) [cross product] y - x [cross product] (f [center dot] y) [where] f [element of] R, x [element of] X, y [element of] Y}.

We let [Mathematical Expression Omitted] denote the quotient Banach space [Mathematical Expression Omitted]. For x in X and y in Y, we let [Mathematical Expression Omitted] denote the image of x[cross product]y in [Mathematical Expression Omitted]. Note that (f [center dot]x)[[cross product].sub.R]y = x[[cross product].sub.R] (f[center dot]y) and if we define f [center dot](x[[cross product].sub.R]y) = (f[center dot]x)[[cross product].sub.R]y, then this extends linearly to a bounded linear map on [Mathematical Expression Omitted], and makes this space into a Banach R-module. If [X.sub.i] and [Y.sub.i] are Banach R-modules and [[Phi].sub.i]: [X.sub.i] [approaches] [Y.sub.i], i = 1, 2, are bounded R-module maps, then it is easily checked that there is a bounded R-module map [[Phi].sub.1][[cross product].sub.R][[Phi].sub.2]: [X.sub.i] [[cross product].sub.R][X.sub.2] [approaches] [Y.sub.1] [[cross product].sub.R][Y.sub.2] satisfying ([[Phi].sub.1] [[cross product].sub.R][[Phi].sub.2]([x.sub.1][[cross product].sub.R][x.sub.2]) = [[Phi].sub.1]([x.sub.1])[[cross product].sub.R][[Phi].sub.2]([x.sub.2]). See [18] and [12] for further properties.

We shall call a Banach R-module X finitely generated if there exists a finite set of vectors [x.sub.l], ..., [x.sub.n] in X such that R[x.sub.l] + ... + R[x.sub.n] is dense in X. The minimum cardinality of such a set is called the rank of X as an R-module and is denoted by [rank.sub.R](X).

PROPOSITION 4.1. Let X and Y be Banach R-modules with X finitely generated and Y finite dimensional. Then

[Mathematical Expression Omitted] is finite dimensional and dim [Mathematical Expression Omitted] [iless than or equal to] [rank.sub.R](X) [center dot] dim (Y).

Proof. If R[x.sub.l] + ... + R[x.sub.n] is dense in X and [y.sub.l] ..., [y.sub.m] is a basis for Y, then it is easily seen that [Mathematical Expression Omitted], 1 [less than or equal to] i [less than or equal to] n, 1 [less than or equal to] j [less than or equal to] m, spans [Mathematical Expression Omitted].

Let G be a bounded open subset of [C.sup.n] and let X be a reproducing R-module on G. If J is an ideal of finite codimension in R with Z(J) [subset] G, then J is contracted by Theorem 2.6. Consequently [R]/[J] and R/J are isomorphic as R-modules.

PROPOSITION 4.2. Let G be a bounded open subset of [C.sup.n] and let X be a reproducing R-module on G with R analytically admissible. If I is a contracted ideal in R and if J is an ideal of finite codimension in R with Z(J) [subset] G, then

[Mathematical Expression Omitted] and I/J [center dot] I are isomorphic as R-modules.

Proof. Since R is Noetherian, [I] is finitely generated and hence [Mathematical Expression Omitted] and [I]/[J [center dot] I] are finite dimensional.

Set e = 1 + [J] in [R]/[J] and define [Mathematical Expression Omitted] via [Mathematical Expression Omitted]. Note that [Psi] is bounded, onto, and an R-module map with [J [center dot] I] [subset] ker [Psi]. Thus, we have an induced map [Mathematical Expression Omitted], which is onto.

Now choose [f.sub.1], ..., [f.sub.m] in R so that [f.sub.l] + [J], ..., [f.sub.m] + [J] span [R]/[J] and [[[f.sub.i] + [J]]] = 1. Using the Banach space isomorphism of [Mathematical Expression Omitted] with m copies of [I] equipped with the 1-norm, we see that there exists a constant c so that

[Mathematical Expression Omitted].

Since each map x [approaches] [f.sub.i] [center dot] x is bounded on X, there exists a constant K with [[[f.sub.i] [center dot] x]] [less than or equal to] K [center dot] [[x]] for 1 [less than or equal to] i [less than or equal to] m. It is now easily seen that the map,

[Mathematical Expression Omitted]

given by [Phi]([summation over i] [x.sub.i] [cross product]([f.sub.i] + [J])) = [summation over i][f.sub.i] [center dot] [x.sub.i] + [J [center dot] I], 1 [less than or equal to]i [less than or equal to]m, is a well-defined, bounded map with [[[Phi]]] [less than or equal to] K/c, and induces a bounded, R-module map,

[Mathematical Expression Omitted]

It is easily checked that [Mathematical Expression Omitted] is an inverse to [Mathematical Expression Omitted]. Thus, [Mathematical Expression Omitted] and [I]/[J [center dot] I] are isomorphic as R-modules. Finally, if we consider the map I [approaches] [I]/[J [center dot] I], then this map is onto because the target space is finite dimensional and has kernel I [intersection] [J [center dot] I] = J [center dot] I (by virtue of the fact that J [center dot] I is contracted in accordance to Proposition 2.9).

Remark 4.3. The above results remain true if the projective cross-norm is replaced by any reasonable cross-norm in the definition of the module tensor product.

Our earlier versions of the rigidity theorems from section 3 were based on a tensor product formulation, and we wish to make the connection between those proofs and the proofs given in section 3. To this end assume that we have the assumptions of Theorem 3.3. Let L be a finite dimensional R-module. Since R is Noetherian, [I] and [J] are finitely generated. Hence, by Proposition 4.1, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are finite dimensional. From this it follows that the maps [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are mutual inverses and thus [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are module isomorphic. Using Proposition 4.3 we see that if L = R/H with Z(H) [subset] G, then [Mathematical Expression Omitted]. Our earlier arguments using the isomorphism between J [[cross product].sub.R] L and I [[cross product].sub.R] L for L of the above form, yield J = I.

5. Examples. In the last part of this paper, we give some examples to illustrate our main result. This result is very general. It extends many known results and exhausts the applicability of most techniques used in the study of this topic. We examine some aspects to understand the generality of our results.

1) In our result, the underlying domain G of the module can be any nonempty bounded domain, even a domain in a complex manifold. This generalizes many previous works which assume G to be a classical domain, such as the polydisk or the unit ball in [C.sup.n] or some pseudo-convex domain in [C.sup.n].

2) We can allow the norm on the module X to define a Banach space or, more generally, we can allow even an admissible locally convex Hausdorff linear topological space. It is not necessary to assume that X is a Banach space. This enables us to claim results for more general spaces such as the space of [L.sup.p]-integrable analytic functions over certain domains.

3) The condition we put on the ring R is coordinate free. This allows us to apply our results to situations where the ring in question is not a polynomial ring.

EXAMPLE 1. Let V be an algebraic manifold in [C.sup.n], [Omega] be a nonempty bounded open subset of V and dv be a volume form on V. Let R be the coordinate ring of V and [B.sup.p]([Omega]) be the closure of R in [L.sup.p](dv) where 0 [less than] p [less than or equal to] [infinity]. Then [B.sup.p]([Omega]) is an R-module. Theorem 3.6 applies to this example.

[B.sup.p]([Omega]) is a reproducing [Omega]-space because for each point x in [Omega], there is a neighborhood U of x in [Omega] and a constant C [less than] [infinity] such that

[Mathematical Expression Omitted].

Since every point x in [Omega] is a smooth point, R, is locally at x a subring of analytic functions in local coordinates of [Omega] at x. Thus, R is both algebraically and analytically admissible at every point of [Omega].

If V = [C.sup.n], p = 2, and [Omega] is the unit ball of [C.sup.n], then Theorem 3.6 implies the result of Agrawal and Salinas [2].

Another class of examples that fit into this category is obtained by taking quotient modules. Let [Omega] be a bounded domain in [C.sup.n]. Let [B.sup.p]([Omega]) be defined as above. If I is a prime ideal of C[[z.sub.l],...,[z.sub.n]] with smooth zero set V intersecting [Omega], then we can form the quotient module:

H = [B.sup.p]([Omega])/[I], where [I] is the closure of I in [B.sup.p]([Omega]).

Thus, we get a class of quotient modules that satisfy the conditions in Theorem 3.6, and we can deduce many rigidity results for submodules in these quotient modules.

EXAMPLE 2. Let [Mu] be a compactly supported probability measure on [C.sup.n] and assume that [Mu] is invariant under the action of the n-torus [T.sup.n] on [C.sup.n]:

[Mu]([e.sup.[i[Theta].sub.1]][z.sub.1],...,[e.sup.[i[Theta].sub.n]][z.sub.n]) = [Mu]([z.sub.1],...,[z.sub.n]).

Let R be the ring C[[z.sub.l],...,[z.sub.n]] and X be the closure of R in [L.sup.2](d[Mu]). Then X is an R-module and G = vp(X) is the interior of the polynomial hull of the support of [Mu] (cf. [10], [11]). By a recent result of Curto-Yan [11], the triple X, R and G satisfies the conditions of Theorem 3.3. Hence we obtain a rigidity result for the invariant subspaces of X.

When d[Mu] = [(d[Theta]).sup.n], where d[Theta] denotes Lebesgue measure on the unit circle T, then the space X defined above is the Hardy space [H.sup.2]([D.sup.n]), and Theorem 3.3 recovers some of the results in [14] and the result in [1].

REFERENCES

[1] O. P. Agrawal, D. N. Clark, and R. G. Douglas, Invariant subspaces in the polydisk, Pacific J. Math. 121 (1986), 1-11.

[2] O. P. Agrawal and N. Salinas, Sharp kernels and canonical subspaces, Amer. J. Math. 109 (1987), 23-40.

[3] P. Ahern and D. N. Clark, Invariant subspaces and analytic continuations in several variables, J. Math. Mech. 19 (1969/1970), 963-969.

[4] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.

[5] S. Axler and P. Bourdon, Finite codimensional invariant subspaces of Bergman spaces, Trans. Amer. Math. Soc. 305 (1988), 1-13.

[6] C. A. Berger, L. A. Coburn, and A. Lebow, Representation and index theory for C*-algebras generated by commuting isometries, J. Funct. Anal. 27 (1978), 51-99.

[7] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239-255.

[8] N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading, 1972.

[9] M. J. Cowen and R. G. Douglas, On moduli of invariant subspaces, Operator Theory: Advances and Applications, vol. 6, Birkhauser, Boston, 1982, pp. 35-73.

[10] R. Curto and P. S. Muhly, C*-algebras of multiplication operators on Bergman spaces, J. Funct. Anal. 64 (1985), 315-329.

[11] R. Curto and K. Yan, Spectral theory of Reinhardt measures, Bull. Amer. Math. Soc. 24 (1991), 379-385.

[12] R. G. Douglas and V. I. Paulsen, Hilbert modules for function algebras, Research Notes in Math., vol. 217, Longman, 1989.

[13] R. G. Douglas, V. I. Paulsen, and K. Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc. 20 (1990), 67-71.

[14] R. G. Douglas and K. Yan, Rigidity of Hardy submodules, J. Integral Equations Appl. 13 (1990), 350-363.

[15] A. Grothendieck, Elements de Geometrie Algebrique, IV, Inst. Hautes Etudes Sci. Publ. Math. 20, 1961.

[16] M. Putinar and N. Salinas, Analytic transversality and Nullstellensatz in Bergman spaces, Contemp. Math. (to appear).

[17] A. Shields, Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys Monographs, vol. 13, American Mathematical Society, Providence, RI, 1974, pp. 49-128.

[18] J. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172-191.

[19] O. Zariski and P. Samuel, Commutative Algebra, I, II, van Nostrand, 1958, 1960.
COPYRIGHT 1995 Johns Hopkins University Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 1995 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Douglas, R.G.; Paulsen, V.I.; Sah, C.H.; Yan, K.
Publication:American Journal of Mathematics
Date:Feb 1, 1995
Words:9435
Previous Article:Moduli spaces.
Next Article:Non-Euclidean elastica.
Topics:

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters