The spaces D, S and E' over \mathbb{R}^(n) are known to be flat modules over A=\mathbb{C}[\partial_{1},...,\partial_{n}], whereas their duals D', S' and E are known to be injective modules over the same ring. Let A be a Noetherian k-algebra (k=\mathbb{R} or \mathbb{C}). The above observation leads us to study in this paper the link existing between the flatness of an A-module E which is a locally convex topological k-vector space and the injectivity of its dual. We show that, for dual pairs (E,E') which are (K) over A--a notion which is explained in the paper--, injectivity of E' is a stronger condition than flatness of E. A preprint of this paper (dated September 2009) has been quoted and discussed by Shankar.
Consider the spaces D, S and E ′ over R n , as well as their duals D ′ , S ′ and E. Ehrenpreis [5], Malgrange [8], [9] and Palamodov [10] proved that D, S and E ′ are flat modules over A = C [∂ 1 , ..., ∂ n ] whereas D ′ , S ′ and E are injective over A. If F is any of these modules, all maps F → F : x → a x (a ∈ A) are continuous; using Pirkovskii' s terminology ( [11], p. 5), this means that F is semitopological. This observation leads to wonder whether there exists a link between the injectivity of a semitopological A-module and the flatness of its dual. The existence of such a link is studied in this paper.
Notation 1 In what follows, A is a Noetherian domain (not necessarily commutative) which is a k-algebra (k = R or C).
Let E, E ′ be two k-vector spaces. Assume that E ′ is a left A-module and that there exists a nondegenerate bilinear form -, -: E × E ′ → k. Then E and E ′ are locally convex topological vector spaces endowed with the weak topologies σ (E, E ′ ) and σ (E ′ , E) defined by -, -; the pair (E, E ′ ) is called dual (with respect to the bilinear form -, -).
Assume that the left A-module E ′ (written A E ′ ) is semitopological for the topology σ (E ′ , E). Then the k-vector space E becomes a right A-module
for any x ∈ E, x
The duality bracket -, -is extended to an obvious way to
k is again a dual pair. Let P ∈ A q×k ; this matrix determines a continuous linear map P
Example 3 Let E ′ be the space of distributions D ′ , S ′ or E ′ over R n and E the associated space of test functions. From the above, the transpose of
Consider the following sequences where P 1 ∈ A k1×k2 , P 2 ∈ A k2×k3 :
The facts recalled below are classical:
where (.) 0 is the polar of (.).
Consider the sequence involving 2 + n maps
where n ≥ 0.
The module A E ′ is called n-injective if whenever (5) is exact, (4) is again exact.
The following is obvious:
Proof.
(1) If ( 3) is exact, then ker E (•P 2 ) = im E (•P 1 ), therefore (ker E (•P 2 )) 0 = (im E (•P 1 )) 0 with (ker E (•P 2 )) 0 = im E ′ (P 2 •) and (im E (•P 1 )) 0 = ker E ′ (P 1 •).
(2) If ( 4) is exact, then ker
) 00 , and im E (•P 1 ) = ker E (•P 2 ) by the bipolar theorem since ker E (•P 2 ) is closed.
Lemma and Definition 8 (1) Let P ∈ A k×r ; Conditions (i)-(iv) below are equivalent: (i) P • : (E ′ ) r → (E ′ ) k is a strict morphism and so is also
The dual pair (E, E ′ ) is said to be Köthe (or (K), for short) over A if for any positive integers k, r and any matrix P ∈ A k×r , the following condition holds: (2) Assume that E is a Fréchet space (e.g., E = S), E ′ is its dual and -, -is the canonical duality bracket. Then for any integer k, E 1×k is again a Fréchet space, and the dual pair (E, E ′ ) is (K) over A by ( [1], §IV.4, Theorem 1).
(3) Likewise, if E is the dual of a reflexive Fréchet space, then the dual pair
1×r has a closed image, then by the above-quoted theorem P • : F r → F k has a closed image and
(4) Whether the above holds when E is an arbitrary (LF ) space was mentioned in ( [4], §15.10) as being an open question; to our knowledge, this question is still open today.
Proof. (i): By Lemma 4(iii), there exists a matrix P 0 ∈ A k0×k1 such that the sequence
is exact, and since A E ′ is injective, the sequence
(ii): Since coker A (•P 1 ) is torsion-free, by Lemma 4(iii) there exists P 2 ∈ A k2×k3 such that the sequence (2) is exact. Since E A is flat, the sequence (3) is exact. Therefore, im E (•P 1 ) = ker E (•P 2 ) is closed, and
Theorem 11 Assume that the dual pair
Proof.
(1) Assume that A E ′ is injective and (2) is exact. Then (4) is exact, which implies that im E (•P 1 ) = ker E (•P 2 ) according to Lemma 7(2). By Lemma 10(i), im
is exact. This proves that E A is flat.
(2) Assume E A is flat and the sequence ( 5) is exact with n = 1. Then, the sequence
, the sequence ( 4) is exact, and A E ′ is 1-injective.
Consider a dual pair (E, E ′ ) which is (K) over the k-algebra A. As shown by Theorem 11, injectivity of A E ′ implies flatness of E A . The converse does not hold, since flatness of E A only implies 1-injectivity of A E ′ . For the sequence (5) to be exact with n = 1, coker A (•P 1 ) must be torsion-free, therefore 1injectivity is a weak property. To summarize, injectivity of A E ′ is a stronger condition than flatness of the dual E A . A convenient characterization of dual pairs (E, E ′ ) which are (K) over the k-algebra A (besides the case when E is a Fréchet space or the dual of a reflexive Fréchet space) is an interesting, probably difficult, and still open problem.
and only if whenever (2) is exact, (3), deduced from (2) using the functor E A -, is again exact ([10], Part I, §I.3, Prop. 5). (ii) The module A E ′ is injective if, and only if whenever (2) is exact, (4), deduced from (2) using the functor Hom A (-, E ′ ), is again exact ([10], Part I,
and only if whenever (2) is exact, (3), deduced from (2) using the functor E A -, is again exact ([10], Part I, §I.3, Prop. 5). (ii) The module A E ′ is
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