Injectivity and flatness of semitopological modules

Injectivit y and fla t nes s o f semitop ological mo dules Henri Bourl ` es ∗ No v em b er 9, 2018 Abstract The space s D , S and E ′ o ver R n are known to b e flat modules o ver A = C [ ∂ 1 , ..., ∂ n ], whereas their duals D ′ , S ′ and E are known to be injectiv e mod u les o ver the same ring. Let A b e a N oetherian k -algebra ( k = R or C ). The abov e observa tion leads u s to study in this pap er the link existing b et w een the flatness of an A -mo dule E which is a locally conv ex topological k -vector space and the injectivit y of its dual. W e sho w th at, for dual p airs ( E , E ′ ) which are ( K ) ov er A –a notion which is explained in the paper–, injecti vity of E ′ is a stronger condition than flatness of E . A p reprin t of this pap er (dated September 2009) has b een quoted and d iscussed by Sh ank ar [12]. 1 In tro d uction Consider the space s D , S and E ′ ov er R n , as well as their duals D ′ , S ′ and E . Ehrenpreis [5], Malg range [8], [9] and Palamodov [10] prov e d that D , S a nd E ′ are flat mo dules o ver A = C [ ∂ 1 , ..., ∂ n ] whereas D ′ , S ′ and E are injective ov er A . If F is an y of these mo dules, all ma ps F → F : x 7→ a x ( a ∈ A ) ar e contin uous; using Pirkovskii’ s terminolo gy ([11], p. 5), this means that F is semitop olo gic al . This obse r v ation leads to wonder whether there exists a link betw een the injectivity of a semitop olog ical A -module and the flatness of its dual. The existence o f such a link is studied in this pap er. 2 Preliminaries Notation 1 In what fo l lows, A is a No etherian domain ( not ne c essarily c om- mutative) which is a k - algebr a ( k = R or C ). Let E , E ′ be tw o k -vector spaces. Assume tha t E ′ is a left A -mo dule a nd that there exists a nondeg enerate bilinear form h− , −i : E × E ′ → k . Then ∗ SA TIE, ENS Cac han/CNAM, 61 Av en ue Presi den t Wil son, F-94230 Cac han, F rance (Henri.Bourles@satie.ens-cacha n.fr). 1 E and E ′ are lo cally con vex top ological vector spac e s endow ed w ith the weak top ologies σ ( E , E ′ ) and σ ( E ′ , E ) defined by h− , −i ; the pair ( E , E ′ ) is called dual (with r espec t to the bilinear for m h− , −i ). Assume tha t the left A -mo dule E ′ (written A E ′ ) is semitop ologic a l for the top ology σ ( E ′ , E ). Then the k -vector spa ce E b ecomes a right A -mo dule (written E A ), setting h x a, x ′ i = h x, a x ′ i (1) for any x ∈ E , x ′ ∈ E ′ and a ∈ A , and it is obviously semitop ologica l, i.e., all maps E → E : x 7→ x a ( a ∈ A ) are contin uo us. Conv ersely , one can likewise pro ve that if the rig h t A -mo dule E A is semitop ologica l f or the top ology σ ( E , E ′ ), then A E ′ is semitop ological for the top olog y σ ( E ′ , E ). By (1), the transp ose of the left multiplication by a ∈ A , denoted by a • : E ′ → E ′ , is the right multiplication by a , denoted by • a : E → E . Notation 2 In what fol lows, ( E , E ′ ) is a dual p air and E A (or e quivalently A E ′ ) is a semitop olo gic al mo dule. The duality brack et h− , −i is extended to an obvious way to E 1 × k × ( E ′ ) k ; then  E 1 × k , ( E ′ ) k  is again a dual pa ir . Let P ∈ A q × k ; this matrix determines a contin uo us linear map P • : ( E ′ ) k → ( E ′ ) q : x ′ 7→ P x ′ , the transp ose of which is • P : E 1 × q × E 1 × k : x 7→ x P . Example 3 L et E ′ b e the sp ac e of distributions D ′ , S ′ or E ′ over R n and E the asso ciate d sp ac e of test functions. F r om the ab ove, the tr ansp ose of ∂ i • : E ′ → E ′ is • ∂ i : E → E , and for any T ∈ E ′ , ϕ ∈ E , h ϕ ∂ i , T i = h ϕ, ∂ i T i . Sinc e h ϕ, ∂ i T i = − h ∂ i ϕ, T i , one has ϕ ∂ i = − ∂ i ϕ ( ϕ ∈ E ) , i.e., • ∂ i = − ∂ i • . Consider the following sequences where P 1 ∈ A k 1 × k 2 , P 2 ∈ A k 2 × k 3 : A 1 × k 1 • P 1 − → A 1 × k 2 • P 2 − → A 1 × k 3 , (2) E 1 × k 1 • P 1 − → E 1 × k 2 • P 2 − → E 1 × k 3 , (3) ( E ′ ) k 3 P 2 • − → ( E ′ ) k 2 P 1 • − → ( E ′ ) k 1 . (4) The facts reca lled b elow a r e c la ssical: Lemma 4 (i) The mo dule E A is flat if, and only if whenever (2) is exact, ( 3) , de duc e d fr om (2 ) using the functor E N A − , is again exact ([10], Part I, § I.3, Pr op. 5). (ii) The mo dule A E ′ is inje ct ive if, and only if whenever (2) is exact, (4) , de- duc e d fr om (2) u sing the functor Hom A ( − , E ′ ) , is again exact ([10], Part I, § I.3, Pr op. 9). (iii) F or any matrix P 2 ∈ A k 2 × k 3 , ther e exist a natur al inte ger k 1 and a matrix P 1 ∈ A k 1 × k 2 such that (2) is exact. Conversely, given a matrix P 1 ∈ A k 1 × k 2 , ther e exists a matrix P 2 ∈ A k 2 × k 3 such that (2) is exact if, and 2 only if coker A ( • P 1 ) = A 1 × k 2 /  A 1 × k 1 P 2  is torsion-fr e e (se e, e.g., [2], L emma 2.15). (iv) The fol lowing e qualities hold ([1], § IV.6, Cor ol. 2 of Pr op. 6): ker E ′ ( P 1 • ) = ( im E ( • P 1 )) 0 , im E ′ ( P 2 • ) = ( ker E ( • P 2 )) 0 wher e ( . ) 0 is the p olar of ( . ) . Consider the s equence in volving 2 + n maps • P i (1 ≤ i ≤ 2 + n ) A 1 × k 1 • P 1 − → A 1 × k 2 • P 2 − → A 1 × k 3 − → ... • P 2+ n − → A 1 × k 3+ n (5) where n ≥ 0. Definition 5 The mo dule A E ′ is c al le d n -inje ctive if whenever (5) is ex act, (4) is again exact. The following is obvious: Lemma 6 (i) If the mo dule A E ′ is n -inje ct ive ( n ≥ 0) , then it is n ′ -inje ctive for al l inte gers n ′ such that n ′ ≥ n . (ii) The mo dule A E ′ is 0 -inje ctive if, and only if it is inje ctive. Lemma 7 (1) If (3) is exact, then im E ′ ( P 2 • ) = ker E ′ ( P 1 • ) . (2) If (4) is exact, t hen im E ( • P 1 ) = ker E ( • P 2 ) . Pro of. (1) If ( 3) is exact, then ker E ( • P 2 ) = im E ( • P 1 ), there- fore (k er E ( • P 2 )) 0 = (im E ( • P 1 )) 0 with (k e r 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 E ′ ( P 1 • ) = im E ′ ( P 2 • ), therefore (im E • P 1 ) 0 = im E ′ ( P 2 • ), th us ( im E ( • P 1 )) 00 = (im E ′ ( P 2 • )) 0 =  im E ′ ( P 2 • )  0 = (ker E ( • P 2 )) 00 , and im E ( • P 1 ) = ker E ( • P 2 ) by the bip olar theor em s ince ker E ( • P 2 ) is closed. 3 Injectivit y vs. flatness Lemma and Definition 8 (1) L et P ∈ A k × r ; Conditions (i)-(iv) b elow ar e e quivalent: (i) P • : ( E ′ ) r → ( E ′ ) k is a strict morphism and so is also • P : E 1 × k → E 1 × r ; (ii) P • : ( E ′ ) r → ( E ′ ) k is a strict morphism with close d image (in ( E ′ ) k ); (iii) • P : E 1 × k → E 1 × r is a strict morphism with close d image (in E 1 × r ); (iv) b oth maps • P : E 1 × k → E 1 × r and P • : ( E ′ ) r → ( E ′ ) k have a close d image. (2) The dual p air ( E , E ′ ) is said to b e K¨ othe (or ( K ) , for short) over A if for any p ositive inte gers k , r and any matrix P ∈ A k × r , the fol lowing c ondition holds: • P : E 1 × k → E 1 × r has a close d image if, and only if P • : ( E ′ ) r → ( E ′ ) k has a close d imag e. 3 Pro of. (1): see, e.g., ([6], § 3 2 .3). Remark 9 (1) The dual p air ( E , E ′ ) is not ne c essarily ( K ) over A by ([1], § II.6, R emark 2 after Cor ol. 4 of Pr op. 7); se e, also, ([3], Pr op. 2.3). (2) Assum e that E is a F r ´ echet sp ac e (e.g., E = S ), E ′ is its dual and h− , −i is the c anonic al duality br acket. Then for any inte ger k , E 1 × k is again a F r ´ echet sp ac e, and the dual p air ( E , E ′ ) is ( K ) over A by ([1], § IV .4, The or em 1). (3) Likew ise, if E is the dual of a re flexiv e F r´ e chet sp ac e, then the dual p air ( E , E ′ ) is ( K ) over A . Inde e d, let E = F ′ wher e F is a r eflexive F r´ echet sp ac e. If • P : ( F ′ ) 1 × k → ( F ′ ) 1 × r has a close d image, then by the ab ove-quote d the or em P • : F r → F k has a clo se d image and F = F ′′ = E ′ . Conversely, if P • : ( E ′ ) r → ( E ′ ) k has a close d image, then • P : E 1 × k → E 1 × r has a close d image, for E ′ = F and E = F ′ . (4) Whether t he ab ove holds when E is an arbitr ary ( LF ) sp ac e was mentione d in ([4], § 15.1 0) as b eing an op en question; to our know le dge, this question is stil l op en to day. Lemma 10 L et P 1 ∈ A k 1 × k 2 . (i) Assume that A E ′ is inje ctive. Then im E ′ ( P 1 • ) is close d (or e quivalently, • P 1 : E 1 × k 1 → E 1 × k 2 is strict). (ii) Assu me that coker A ( • P 1 ) is torsion-fr e e and E A is flat. Then im E ( • P 1 ) is close d (or e quivalently, P 1 • : ( E ′ ) k 2 → ( E ′ ) k 1 is strict). Pro of. (i): By Lemma 4(iii), there exists a matrix P 0 ∈ A k 0 × k 1 such that the sequence A 1 × k 0 • P 0 − → A 1 × k 1 • P 1 − → A 1 × k 2 is exact, and s ince A E ′ is injective, the s equence ( E ′ ) k 2 P 1 • − → ( E ′ ) k 1 P 0 • − → ( E ′ ) k 0 is exact. Therefor e, im E ′ ( P 1 • ) = ker E ′ ( P 0 • ), thus im E ′ ( P 1 • ) is closed, and • P 1 : E 1 × k 1 → E 1 × k 2 is strict by ([6 ], § 32.3). (ii): Since coker A ( • P 1 ) is torsion-free, b y Lemma 4(iii) there exists P 2 ∈ A k 2 × k 3 such that the sequenc e (2) is exact. Since E A is flat, the s equence (3) is exact. Therefore, im E ( • P 1 ) = ker E ( • P 2 ) is closed, a nd P 1 • : ( E ′ ) k 2 → ( E ′ ) k 1 is strict by ([6 ], § 32.3). Theorem 11 Assume that the dual p air ( E , E ′ ) is ( K ) over A . (1) If A E ′ is inje ctive, then E A is flat. (2) Conversely, if E A is flat, then A E ′ is 1 -inje ctive. Pro of. (1 ) Ass ume that A E ′ is injective and (2) is exac t. Then (4) is exact, which implies that im E ( • P 1 ) = ker E ( • P 2 ) accor ding to Lemma 7(2). By Lemma 10(i), im E ′ ( P 1 • ) is closed. Since ( E , E ′ ) is ( K ) , im E ( • P 1 ) is also closed. Hence im E ( • P 1 ) = ker E ( • P 2 ) , i.e., (3) is e xact. This proves that E A is flat. 4 (2) Assume E A is flat and the s equence (5) is exact with n = 1. Then, the sequence E 1 × k 1 • P 1 − → E 1 × k 2 • P 2 − → E 1 × k 3 • P 3 − → E 1 × k 4 is exact. By Le mma 7(1) we obtain im E ′ ( P 2 • ) = ker E ′ ( P 1 • ) . In additio n, im E ( • P 2 ) = ker E ( • P 3 ), th us im E ( • P 2 ) is closed, and since ( E , E ′ ) is ( K ), im E ′ ( P 2 • ) is closed. This prov es that im E ′ ( P 2 • ) = ker E ′ ( P 1 • ), i.e., the sequence (4) is ex act, a nd A E ′ is 1-injective. 4 Concluding remarks Consider a dual pa ir ( E , E ′ ) which is ( K ) ov er the k -alg ebra A . As shown by Theorem 1 1, injectivity of A E ′ implies flatness o f E A . The conv e r se do es no t hold, since flatness of E A only implies 1 -injectivit y of A E ′ . F or the sequence (5) to b e exa ct with n = 1, coker A ( • P 1 ) must b e tor sion-free, therefore 1 - injectivit y is a w ea k proper t y . T o summar ize, injectivit y of A E ′ is a s tr onger condition than flatnes s of the dual E A . A co n venien t characteriz a tion of dual pairs ( E , E ′ ) which ar e ( K ) over the k -algebra A (b esides the case when E is a F r´ ec het spac e or the dual of a r eflexive F r´ echet space) is an interesting, pro bably difficult, and still op e n pro blem. Ac knowledgemen t 12 The author would like to thank Pr of. Ulrich Ob erst, whose c omments wer e very helpful in impr oving the manuscript. References [1] N. B ourbaki , Esp ac es ve ctoriels top olo giques , Masson, Paris, 1981. (En- glish tra nslation: T op olo gic al V e ctor Sp ac es , Spring er, 1987.) [2] H. Bourl ` es and U. 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