The category of Hausdorff spectra over a semiabelian categor Текст научной статьи по специальности «Математика»

УДК 512.7

E. I. Smirnov

The Category of Hausdorff Spectra over a Semiabelian Category

In this article the category H of Hausdorff spectra is introduced into the discussion by means of an appropriate factorization of the category of Hausdorff spectra Spect G over the category G. If G is a semiabelian complete subcategory of the category TG, then H is a semiabelian category (in the sense of V. P. Palamodov [1]).

Key words: Hausdorff spectra, semiabelian category, commutative diagram.

Let X = {Xs,F, hg's] and Y = {Yp, F1,hp'p] be Hausdorff spectra over some category G. We will call any set of morphisms ups : Xs ^ Yp of the category G which satisfies the following conditions a mapping of spectra uyX : X ^ Y •'

(1) there exist mappings p : F ^ F1, 0 : F1 ^ F (VF e F), XT' : T1 ^ T (VT1 G F

XT k1

T i

Tq1 с T1 such that (w denotes mapping of elements)

X

P

G T G F G F

X

ф ф

G t 1 g f1 g F1

(2) for each pair (p,x(p)) a morphism upx(p) : Xxp ^ Yp of the category G is defined in such a way that if hp*p : Yp ^ Yp*, up*x(p*) : Xx(p*) ^ Yp*, then there exists a morphism hx(p*)x(p) : Xx(p) ^ Xx(p*), and the following diagram is commutative:

YP

шрх(Р)

h

■p'p

Yp*

шр*х(р*)

(Ф)

hx(p*)x(p) Xx(p) -Xx(p*)

s

(3) if hx(p*)x(p) : Xx(p) W Xx(p*) , шрх(р) : Xx(p) W YP , шр*х(р*) : Xx(p*)

W Yp*

then there

exists a morphism hp*p : Yp

Yp* and the diagram (Ф) is commutative.

It follows from condition (3) and the definition of a Hausdorff spectrum that, for example, every diagram

Yp

X

x(p)

Yp'

Yp*

xx(p')

(Ф' )

© Smirnov E. I., 2010

is commutative.

In particular, if |f| = |F1| = z (z is the set of whole numbers), F = {|F|}, F1 = {IF11}, then we obtain a mapping of inverse spectra. Moreover, V. P. Palamodov's version of mapping of spectra [1] is a mapping of Hausdorff spectra. In fact, for each fi G z there exists the largest a = ) G z such that (a, ,5) G A^, i.e. the inverse function of ) from Condition II of Definition 2 in [1] defines a set of morphisms «a : X«^) ^ Y^ of the category K (^ G z) which satisfies Condition I - this corresponds to fulfilling (1) and (2).

Suppose that uyX : X ^ Y and uZy : Y ^ Z are mappings of Hausdorff spectra so that Uyx = xX Uzy = x;). Let us put p* = p' O 0* = 0 O x* = x O ^ so

that p* : F ^ F2, 0* : F2

Urp ◦ Ups

F (VF G F), X* : T2 ^ T (VT2 G F2), setting ^

are defined. It is easy to verify that the set of morphisms Zr of the category G satisfies conditions (1) and (2) for a mapping of Hausdorff

{Xs

whenever morphisms ups and urp

wrs : Xs _ _ . , . ,

spectra. We will call the mapping i : X ^ X, where X = {Xs, F, hs/s}, the identity mapping if it is formed by means of all the identity morphisms uss : Xs ^ Xs (s G |F|) of the category G ; it is clear that i is a left and right identity under composition.

Thus, the set of Hausdorff spectra over G and their mappings form a category, which (by analogy with [1]) we will denote by Spect G. We may consider the category G as a subcategory in Spect G - namely, to each object A G G we assign the Hausdorff spectrum A = {A, {A}, 0}. Let X = {Xs, F, hs/s} G Spect G. Then every mapping : X ^ B is given by a set of morphisms wbs : Xs ^ B, where p : F ^ {B}, 0F : {B} ^ F (VF G F), Xf : {B} ^ 0f({B}) and s = XF(0F({B})) (F G F). Correspondingly, every mapping uXA : A ^ X is given by a set of morphisms WsA : A ^ Xs, where p' : {A} ^ F, 0 : F ^ {A} (F = p'({A})), x : T ^ {A} (VT G F), s G IFI, F = p'({A}).

Let X = {Xs, F,hs/s}, Y = {Yp, F1,hp/p} be objects from Spect G. We will say that two mappings of Hausdorff spectra uyX : X ^ Y and Jyx : X ^ Y are equivalent if for any F G F there exists F* G F1 such that the diagram

и

ps

Yp

h

-p'p

Xs

Yp*

U

p's

Yp,

h

,p*p/

is commutative for any p* G |F*| (s G |F|, p G |p(F)|, p' G |p'(F)|). The relation introduced is reflexive, i.e. in this case p,p',p* G |Fs = x(p), s = x(p'), s* = x(p*) and the following diagram is commutative because of ($):

Xs

hs

и

ps

Xs

Yp

Yp,

h

и

p* s*

p* p

hp*p'

yp*

*

Specifically, the existence of a morphism hs*s of the Hausdorff spectrum X follows from ($).

We now establish the transitivity of the relation. Let uyX ~ u'yx and u'yx ~ uyx. Then transitivity follows from the directedness of the class F1 and the commutativity of the following diagram (V p e |F |) :

Here, p e |^(F)|, p' e |^'(F)|, p'' e |^''(F)|, p* e |F*|, p** e |F**| and F* x F, F** x F. It is clear that the equivalence relation is preserved under composition.

Thus the set Hom(X, Y) is decomposed into equivalence classes; let us now consider a new category H whose objects are the objects of the category Spect G, while the set HomH(X, Y) is formed by the equivalence classes of mappings uyX : X ^ Y. We will denote these classes by ||uyx11 .

Let G be a semiabelian complete subcategory of the category TG, in which it is possible to construct direct sums and direct products. Then for each Hausdorff spectrum X = {Xs, F, hs/s} over G there exists (as already shown) a unique (up to isomorphism) object of the category

G, called the H-limit of the Hausdorff spectrum X and denoted by lim hs/sXs. Moreover, if

F

(yX : X ^ Y, then there exists a unique morphism

uyx : lim hs/sXs ^ lim hp/pYp

F Fi

of the category G. In fact, let x e lim hs/sXs, i.e. x e UFeF Pitef ^Vf , where ^ : S ^ S is

F

the canonical mapping and

Vp = {a e JJ Xs : Xs = hssXs, s,s e T} . F

Then there exists F e F such that x e ^V/ (T e F), and, consequently, x = ^aT, where aT = (xT)F , aT e Vp , T e F. Therefore by the definition of a mapping of Hausdorff spectra there exist F1 e F1, F1 = p(F), 0 : F1 ^ F and x : T1 ^ T (VT1 e F1) which allow us to define a morphism of the category

gfif : II Xs ^ II Yp ,

F F1

where gFiF = {upx(p)}p€|Fi|. For each T1 e F1 we define an element 1 e V^i C nFi Yp such that pTi = {upx(p)xX(p)}Pe|Fi|, where T = 0(T1). Here, given hpp : Y ^ Yp , there exists by ($)

hx(p)x(p) : Xx(j5) ^ Xx(p), and moreover /¿pp(ipx(P)xX(P)} = i^px(p)xX(p) ' where G T1- Now if is the canonical mapping for the Hausdorff spectrum Y, then by ($) we obtain -0'ßTi = ^'ßTi

for arbitrary Tf,Tl G F1. It remains to put y = i (T1 G F1), where y G f|TieFi ,

and, consequently, y G lim hp>pYp and UyXx = y. Additivity and continuity of UyX are obvious

7 _

and come directly from the definition of the H-limit of a Hausdorff spectrum, therefore uyX is a morphism of the category TG. We will employ the notation H(uyX) = uyX .

It is clear that H translates the identity mapping into the identity and a composition of mappings into a composition. Therefore H is a covariant functor from the category Spect G into the category G. Moreover, we have the following result:

Proposition 1. Let H : Spect G — G- Then H can be extended to the category H and is additive on it-

Proof. We show first of all that HomH (X, Y) is an abelian group. Let uyX : X — Y,

JyX : X — Y, where X, Y G Spect G, ^yX = <, x), u'yX = u>'(ip', <', x'). For each F G F we can find F* G F1 such that p(F) F* and p'(F) F*. Let us construct mappings of Hausdorff spectra w^X : X — Y so that uyX ~ UyX and JyX ~ UyX, |JF* x1(p) = UF* X2(p). In fact, forp G |F 1| there exists sp G |F| such that hx(p)Sp : Xsp — XxW, hx/(p)Sp : Xsp — Xx>p and moreover, if p G T*, uFF* : F* — F, uFF* : F* — F', then sp G T, where T = <(T*), T d <[uFF*(T*)], T d <'[uFF*(T*)]. Putting x1(p) = sp and x2(p) = sp in this case, we obtain the necessary identity. Now if we put p(F) = F*, then the mappings of Hausdorff spectra UyX = <,x1), 4X = <,x2) are equivalent to uyX and JyX respectively by ($). Therefore we define ||^yX|| + ||^yX|| to be the element of HomH(X, Y) containing

{^pxi(p) + upx2(p)}pe|F*| (F G F, F* = P(F)).

Clearly, this class does not depend on the choice of representatives uyX, JyX in their equivalence classes. The operation of addition which has been introduced converts HomH(X, Y) into an abelian group. Now the extension of the functor H to the category H and its additivity there are obvious. The proposition is proved.

We will reserve the notation H = Haus for the case G = TLC.

We introduce a semiabelian structure on the category H. For any objects X, Y, Z G H the law of composition defines a bilinear mapping

Homw(X, Y) x Homw(Y, Z) — Homw(X, Z).

Thus H is an additive category.

Proposition 2. (See [1].) The category H is semiabelian.

Proof. Let ||wyX || : X —► Y, where X, Y are Hausdorff spectra over G. We will construct for a morphism ||^yX || of the category H its kernel and cokernel. We choose in the class ||wyX || some element uyX G Spect G so that uyX = <, x). Now for each s G |F|, where F G F, let us consider an object Ns G G, Ns c Xs, provided with the topology induced from Xs, and such that Ns = ker ups for s = x(p) (p G |p(F)|)). By ($) the restriction ns/s of the morphism hs/s translates Ns into Ns/, therefore the family N = {Ns, F, ns/s} is a Hausdorff subspectrum of the Hausdorff spectrum X = {Xs, F, hs/s}. We will show that the identity embedding iXN : N — X

is the kernel of uyX . For this it is enough to establish that for any morphism mxz : Z ^ X of the category Spect G such that uyX ◦ mxz = 0,

Ze Spect G , Z = {Zt, F°,hft}

there exists a morphism n^z : Z ^ N of the category Spect G such that the following diagram is commutative:

Я

nNZ

iXN

Uyx

Y

(N)

mxz

Z

(Here the zero mapping of spectra signifies that for uyX = 0 = ^yx ◦ mXZ its component morphisms upt = ups ◦ u>st are such that u>pt(Zt) = 0.)

At the same time it is clear that, if for Jyx ~ uyX , m'XZ ~ mXZ such that Jyx o m'XZ = 0', where 0' ~ 0, there also exist n'z ^ Spect G and i'X' ~ iXN such that the diagram

is commutative, then nNZ ~ n^z . Therefore, if diagram (N) applies, each morphism ||uyX || of the category H such that ||uyX|| o ||mxz|| = 0, where ||mxz|| : Z ^ X, and iXN e ||iXN||, has kernel ||iXN|| such that there exists ||n_NZ|| with commutative diagram

||nNZ11

X lkyxl1 Y

z

Thus, for the existence of the kernel of the morphism ||wyX || it is enough to establish the existence of unz : Z ^ N and the commutativity of diagram (N).

If the mapping of spectra is mXZ = m(ip° ,0° ,x°), then taking into account the fact that Im(¿sX°(s) C Ns (s G |0°(F°)|, F° G F°) by assumption, we can construct a mapping of Hausdorff spectra unZ : Z ^ N, where unZ = n(^°, 0°, x°), so that its constituent morphisms ^ Ns are restrictions of the morphisms (¿sx°(s). Commutativity of the diagram is

obvious.

Now we will construct the cokernel of the morphism ||wyX|| ; let uyX G ||wyX||. For each p G |^(F)| (F G F) let us consider the factor group Rp = Yp/Im&px(p) with the topology

induced from Yp. It is clear that because of ($) the subgroups Im wpx(p) form a Hausdorff spectrum, therefore the factor groups Rp (p e |p(F)|) also form a Hausdorff spectrum; let

R = (RP,F0,hp'p] , Yo = {Yp,F0,hp'p] ,

where F, = F1!^) (without loss of generality we may assume that p(F) = F1)- Let us denote by : Y ^ R the canonical mapping of Hausdorff spectra; we will show that || uRy || is the cokernal of the morphism ||uyX||- For this it follows that we have to establish that, for any morphism mZy : Y ^ Z of the category Spect G such that mZy ◦ ^yx = 0, there exists a morphism nZR : R ^ Z of the category Spect G such that the following diagram is commutative:

If mZy = m(f, f, f), then nZR = n(f, f, f), and since wz£(z) (Yx(z)) = 0 for all z e |f(F 1)| (F1 e F), then Im&x(z)x(x(z)) c Nx(z), and, consequently, because the category G is semiabelian there exists a morphism ofz£(z) : Rx(z) ^ Zt such that the following diagram is commutative:

Thus, as is not difficult to see, the set of morphisms u)zx(z) defines a mapping of Hausdorff spectra in such a way that diagram (K) is commutative. The proposition is proved.

Библиографический список

1. Palamodov V.P.: Functor of projective limit in the category of topological linear spaces. Math. Collect., V.75. N4 (1968), P.567-603.

2. Palamodov V.P.: Homological methods in the theory of locally convex spaces. UMN., V.26. N1(1971), P.3-65.

3. Smirnov E.I. Hausdorff spectra in functional analysis. Springer-Verlag, London, 2002. 209p.

4. Zabreiko P.P., Smirnov E.I.: On the closed graph theorem. Siberian Math. J., V.18. N.2 (1977), P.305-316.

5. Rajkov D.A.: On the closed graph theorem for topological linear spaces . Siberian Math. J., V.7. N2 (1966), P.353-372.

6. Smirnov E.I.: On continuity of semiadditive functional. Math. Notes., 1976. V.19. N4 (1976), P.541-548.

7. Wilde M. : Reseaus dans les espaces lineaires a seminormes. Mem. Soc. Roi. Sci. Liege., V.19. N4 (1969), P.1-104.

8. Smirnov E.I.: The theory of Hausdorff spectra in the category of locally convex spaces. Functiones et Aproximatio, XXIV. UAM, 1996, P.17-33.

9. Retakh V.S.: On dual homomorphism of locally convex spaces. Funct. Anal. and its Appl., V.3. N4 (1969), P.63-71.

10. Nobeling C. : Uber die Derivierten des Inversen und des Directen Limes einer Modulfamilie. Topology I., 1962, P.47-61.

11. Cartan A., Heilenberg C. Homological algebra. M.: Mir. 1960, 510p.

12. Smirnov E.I. Hausdorff spectra and the closed graph theorem. In: Pitman Research Notes in Mathematics Series, Longman, England, 1994. P.37-50.