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.
Deep Dive into Injectivity and flatness of semitopological modules.
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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