Some Kan extensions of non-additive functors

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMAT Y CZNE X X (1978)

STANisbAw B alcerzyk (Torun)

Some Kan extensions of non-additive functors

1. Let В , В' denote rings and let J t, resp. J i', be the category of all left B-, resp. E'-modules. We define an additive category ê = ${Jt) as follows. Objects of S’ are pairs < Л , n}, where M is an object of J t and n is a non-negative integer, while H o r n n}, <JV, fc» = Ext^"fe( J f , N) and compositions are given by th e Yoneda product (we pu t E x tm = 0 for m < 0). We have a full and faithful functor K : J t ^ S , K ( M) = < M , 0>, К (a) = a for M, a in J t. I t is known (see [2], p. 239) th a t we can identify a connected sequence of left derived functors L+T of an additive right exact functor T: J t- ^ J t' with a right additive Kan extension E( T) of T along К in the following way: E ( T ) ( ( M, n}) = L nT( M) for all M in Jt, n = 0 ,1 , . . . and E( T) maps an element of H o r n ny, <Ж', n — 1 »

= E x t^ ( Л " , M') corresponding to an exact sequence 0-^M' ->0 onto a connecting map L nT ( M ' ' ) ^ L n_xT(M' ) induced by this sequence.

In this note we study right K an extensions E( T) of not necessarily additive functors T: J t- ^ J t' along K : Jt-+S. We prove a reduction theorem and compute E(T)((_M, n}) in terms of a projective resolution of M. We present some explicit computations in case when В is the ring of rational integers or a ring of polynomials and T is of the form )G, Am(-)®#. Results of parts 3 and 4 remain valid when categories of modules are replaced by appropriate abelian categories. For . homo- logical notions used in the paper see [3].

2. Let J t, resp. J t ', denote the category of all left B-, resp. E'-modules and let T : Jt-> Jt' be a (covariant) functor such th a t T(0) = 0. In [1]

cross-effect functors T x = T, T 2, T3, ... of T are defined. We remind the

г1 *2

definition of T 2. Let M x, M 2 be objects of J t and let M x M x® M 2^ M 2

P l P2

be a representation of a direct sum ( = product) M x ® M 2. The map 1 t ( m 1® m 2) - ( T ( 4 P i )+T(4P2)) may be represented as Ц М Х, M 2) q {Mx, M 2), where q ( Mx, M 2) is epimorphic and Д( Mx, M 2) is monomorphic. Since T ( i xp x), T ( i 2p 2) are orthogonal idempotents, then

Ц М Х, М 2)

T { M X@M2) ъ T ( M X) @T ( M2)@T2( MX, M 2) *~" , T 2( MX, M 2)

e (M v M 2)

where T 2( Ml , M 2) = 1ш(Л(Ж1? 3I2)o(3f 1, М 2)). I t is easy to see th a t maps f x: M x->M'l f f 2: M 2-^M'2 induce such a natural map T 2( f x, f 2) : T t ( Mu ж 2 )->т 2 (ж ;, M 2) th a t T ( f 1@f2)X{M1, M 2) = Ц31[, M 2) T 2(f x, f 2) and thus T 2 is a functor of two variables.

We prove th a t for any maps f, g: M->M' we have (1) 24 f + g ) = T ( } ) + T ( g ) + r { M' ) T, ( f , g) o{ M) , where

o{M): T ( M ) — I t ( M @ 3 1 ) - ^ ^ I t 2( M , M ) ,

t (M'): T 2{31', Ж ' ) - ^ ^ Р ( Ж ' © Ж ' ) - ^ Р ( Ж ' )

and A, V' denote diagonal and codiagonal maps. In fact, we have fA-g

= r ' ( f ® g ) d , tben T ( f + g ) = T( V' ) T( f ®g ) T( A) , T( f ) = T( V ) T { } ®

© 0)T(A) and from F'% = 1, p 2zI = 1, T 2IJ, 0) = 0 it follows T ( f + g ) - T ( f ) = T ( P " ) ( T ( f ® g ) - T ( f ® 0 ) ) T ( A )

= T( V' )\T(f ®g) - T ( f ® 0)) {T( h )T(Pi) + T( h )1’(p2) + + X(M, M) g ( M, M))T(A)

= T ( V ’)( X{ i [ ) T( f ) ~T( i [) T( f ) ) T( p1) T( A) +

+X(P' )X(i ' )X(g)X(p2)X(A) + +X ( P ' ) { X ( f ®g ) - X ( f ®0 ) ) X ( M, M) g ( M, M) T( A)

= T( g) +T( V' ) MM' , M ’)(X2( f , g ) - .

- T 2( / , 0 )] q ( M , M ) T ( A )

= Х(д) + т(М')Хг(/, g)a(M) and ( 1 ) is proved.

3. I t is known (see [2] p. 233) th a t right K an extension E( T) of a functor T : Л -> Л ' along К : Л -> ё is given on objects by a formula.

E{ T ) { ( M, n y ) = lim To

<M,tA j. к

where (31, n ) \ K denotes a comma category with objects (X , a: (31, n ) -+K(X)), X in Л , a in ê, while a m ap£: X ^ X ! in Л maps {X, a) in ( X \ a) if K{!-)a — a and

, is such a functor th a t

T heorem 1. Let T : Л -> Л ' be such a functor that T (0) = 0 , n > 1 and let

0 ->P-->Pn_2->... ->Р0->Ж ->Ю

be an exact sequence in Л with projectiveP 0, ..., P w_2 ; then there exist natural isomorphisms

E{ T) { ( M, n}) * E { T ) { ( L , 1 » , Щ Т ) « М , 0 » » T(M).

P ro o f. Since К is full and faithful, then the second formula follows by [2] p. 235. We have a natural equivalence /*: E x t” ( i f , *)->ЕхР(1/, •) which induces a functor /t: < i f , n y \K -+ (L , i y \ K defined as follows:

Ц ( Х , а : <M, п >- *ЩХ) ) = ( X , ц(Х)(а): < i , 1 > ^ ( X ) ) , fï(() = £ for f: X - + X ’.

ji is an isomorphism of categories and —

then it induces an isomorphism

E { T ) ( ( M, n } ) = lim T ог<М)П> = lim T o i<Ltl>oji ъ lim T o i<Ltl>

<•M , n y \ K i M , n y \ K <l,i> |x

~ E ( T ) « L , 1 » and the theorem follows.

Let us remark th a t objects < i f , n}, <P, 1> are not isomorphic if n > 1 and i f Ф 0 because Hom^(<L, 1>, < i f , n » = E x t 1 - n (P, if ) = 0 and Hom)f(< if, w>, < if, n'y) — HomM( i f , if ) Ф 0.

4. We prove

T heorem 2. Let T : jii-^ J l' be such a functor that T (0) = 0 , let

o ^ l X p -+ m ^ o

be an exact sequence in J i with projecitve P; then

P ( P ) « i f , l » = Ker T (j) nK er (T2( l , j ) a (L)j.

P ro o f. Let A be an object in Ji' and let for any object (X, a:

( M, 1} - >K( X) ) of < i f , 1 >\K a map \ x>a): A - ^ T o i <M>1>(X, a) = T( X) in M' be given, and suppose th a t for any map (X, a)->(X', a) we have T(£)h{Xa) = \ х >^. Since a e E x t 1( M , X ) , then a corresponds to an exact sequence a: 0 - > X Y-> if-» 0 .

We denote by m the exact sequence O^X -^>P->if^O . By projectivity of P it follows th a t there exists a map (g0, f 0, 1M): n ^ a . I t is easy to see th a t any other map ( g, f , 1 дД: тг->й is of a form (go + s j , f 0 + i s , 1 M), where s: P->X is an arbitrary map. Consequently a pair of maps g0:

L->X, s : P-+X determines a map g0 + sj: (L, 1)->(X, a) in < i f , 1)4_йГ and any map of (L, n) in (X , a) is of this form for a certain s: P->X.

Silice T (g q A sj) \ L>„) = h{x>a) for a lls , then T(g0 + sj)h(L>n} = Т(д9) \ ь>я).

By (1) it follows

(2) 8 (T(sj) + t ( X ) T z(g0, sj)e {L )}\Ltn) = 0.

For any map g0: L->X there exists such a e E x t 1 ( i f , X ) th a t (# 0 , / 0 ,1 M):

n-+a for some / 0, then ( 2 ) holds for all s and all g0. If we p u t X = P ,

s = 1P , g0 = 0 , then we get T ( j ) \ Ltn) = 0. If we put X = P ® P , g0 — i x:

L*— —>L ^^P, s = î > 2 * 0 Pj then

0 = t{L@P) T z{ix1i 2j)a{L)h{Ltn) = t { L ® P ) T z(ix1i 2) Tz( l , j ) a { L ) \ Ltn) and

r ( L ® P ) T 2(il1 i 2) = T ( V ) À ( L ® P , L @ P ) T 2(iXJ i 2)

= T ( V ) T ( b ® i z) l ( L , P ) = X ( L , P ) . because VLeP(ix @г2) = 1 L@P. Since X(L, P) is monomorphic, then we get a t last

(3) T ( j ) \ L,n) 3)а{^)\ь,п) — О

thus all \x,a) factorize through a map h: А -*-KerT (j )nK er (T2( 1 , j ) a (Z)j.

I t is easy to see th a t (3) implies ( 2 ) and the correspondence {à(Xta)}i-^h is one-to-one, thus lim T o

1

^ K e rT (j)n K er(T a(l,j)<x(L)).

<M,V>\k

Let us rem ark ‘th a t if T is additive, then T 2 = 0 and E ( T ) ( ( M , 1 >)

= K erT (j) is a usual formula for the first left satellite of T.

5. In this p art we present some computations of E(T) in case T( X)

= X 0 B ... <S> r X <8>r G = X®m 0 д(г, where В is a commutative ring, X and G are JS-modules. We have

т е х 1 e x 2) t ( x 1) @ t { x 2) ® © 6 x 8^ 0 ... 0 X e(m) 0 (?,

where e runs over all non-constant functions e: { 1 , ..., 2 }, then we identify T 2{ Xl, X 2) with ©eX £(1) 0 ... 0 X e(m) 0 (x.

Let 0-> хЛ Р -> Ж -> 0 be an exact sequence of .R-modules with pro­

jective P. For x Xl ..., xm e l , g s G we get

o{L)(xx® ... 0 #m<8>g) — Xeco x0 ... ®xm®g.

We p u t j° = 1L, j 1 = j, L° = L, L 1 — P ; then T 2 ( l, j) a{L){xx® ...

... < 8 >жт 0 0 ) - ^ J 8 ( 1 )W 0 ... ®js{m)(xm) ®g and the term (xx) 0 ...

... 0 gr is in L'W 0 ... 0 Le(m) ®G<=->T2{L \ L l) = T 2(L, P). Thus K er(IT 2 ( l , j)or(L)) is equal to the intersection of kernels of all maps

m

j e(1) 0 ... 0 j e(m) 0 l G and this intersection coincides with П Ker j k, where

k = l

Зк = 1 ь ® . . . 0 j 0 . . . 0 l£ 0 l( ? (j on fc-th place). Since T(j) = j 0 ...

... 0 J 0 1 , then by Theorem 2 we get

m

(4) E { T ) { ( M , 1 » = П K erjfc.

fc=i

If J f is a flat JB-module, then B](T)((M, n » = 0 for w > 1. In fact,

if n > 1 then, in notation of Theorem 1, К is flat, then by Theorem 1

we reduce to the case n — 1. In this case we have Kerj k ъ K e r(£ ® i® (w- 1 )®G£i4- 1 P(8)i:®(w- 1 )0 ^ )

^ Torf (M, L®(m~v'®G) = 0 and the formula is proved.

Now we assume R = Z = the ring of rational integers. Let M be a ^-module; then Ж = (J Ma, where {Ma}aeA is the family of all finitely

aeA

generated submodules of M, partially ordered by inclusion. We show th a t (5) Щ Т Ж М , 1 » = И тР(Т )(< Ж а, 1 » ,

where T = -®m ®G, thus computations of Р(Т)(<Ж , 1 » are reduced to the case of finitely generated ^-modules. To prove ( 6 ) let us denote by F, resp. F a, a free ^-module on the set of elements of M, resp. M a, a e A . Then we have natural epimorphisms F~>M, F a->Ma and commutative diagrams

0 ^{— >F} i* * ■*“ a - > M a- — >0

( 6 ) 0 ^{—>L -—} г ^{A n F J, n —> M n —>0}

F a<^~>F is a splitting monomorphism then L acz-*L is also a splitting mono­

morphism because Co ker (La <=_>!,) is contained in a free ^-module Coker(Pac:-^_P). I t is clear th a t (J L a — L. ( 6 ) induces a commutative diagram for к = 1 , ..., m a

T ( L a) = L®m <g>G^>L®{k~V ® P a ® L f w- fc) ® 0 T(L) = L ® m®G£- ^ > L 0 (* -I)® P ® L 0 (w- A:)® 61

and vertical maps are monomorphisms. Since T(L) = lim T (L a) and

m m *"

Kerj k = liin Ker j aA., then pj K erjfc = lim f] K erja fc and formula (5) follows

“*■ ’ fc=l bfc=l

from (4).

From (4) we get immediately th a t F ( T ) ( ( Z j n Z , 1 » = {g e G\ ng = 0}

then by ( 6 ) it follows th a t E(T){<fQjZ, 1 » is the torsion subgroup of G (Q denotes the group of rationals). From (5) we see also th a t Р(Т)(<Ж , 1 » depends only on torsion subgroup of M, nevertheless it is not equivalent to T o rf(M, G).

Let us assume now th a t R = 1c[Xx, ..., X n] is a ring of polynomials over a commutative ring k. We p u t I = R X x + ... + R X n and Ж = R /I then 0 ^ Z - > P ^ P /Z ->0 is exact. I is a free fc-module on monomials in X x, . . . , X n (of positive degrees). Z0m = I ®й ... <g>RI is a factor module of I <s>k ... ®ftZ which is a free fc-module on ц х <g> ... <8>/лт, where p x, ..., pm

2 — Roczniki PTM Prace Mat. XX.2

are monomials. One can easily show th a t if /q, • • • ? /4 are also monomials, then

/iX 0 Й ... ®Rf*m = 0 • • • 0 / 4 iff

(a) /*i... t*m = /* !• .. / 4 in case deg/q > m, and (b) /q = / 4 /im = /4 in case 2 1 = w -

Consequently 70m is a free 7-module on free generators / (”1) = /q 0 ...

••• 0 /*m? /* == /q ... /*m corresponding to monomials ц of degree > m and pm) _ X v(l) 0 ... 0 ) X^(TO) corresponding to functions rj: { 1 , . . . , m}-* { 1 , ...

..., n}. We denote n(rj) ~ X n(1)... X^(m) and identify all products Z® ...

. . . 0 7 0 1 2 0 7 0 . . . 0 7 with 7®(m~1); then j fc: maps *q 0 ...

... 0 \ onto % щ 0 ... 0 %._! 0 w fc+10 ... 0 and is independent of 7' = 1 , ..., m.

Moreover, j k(fi?l)) = j k(f[m)) = f% J1} and an element a j™ +

m M

-f-JJ « ,д т) belongs to П Kerj* = K erji iff

4 fc=l

(a) = 0 in case deg /n > m, and

(b) JT 1 av = 0 for each /* of degree m.

M »?)=**

Thus Kerfy is a free 7-module generated by elements

__ 0 ... 0 0 ... 0 -3Tn 0 ... 0 -5ГП 0 • « • 0-3Z»j(m)>

81 «П

where s* is a number of elements in rj~1(k), 7 = 1, ..., n and rj runs over non-monotonic .functions rj: {1 , . . . , . . . , n} . I t is clear th a t Х {Ъ%1) = 0 for all i = 1 ,

Since K e rjx<=-*.7®"* splits then we have a commutative diagram with exact rows and columns

. 0 — > K erj x ®kG — > 7®m ®kG 7®(m~1) 0 * G

(7) 7®w ®BG 7®(w-b ®r G

I I

0 0

where K err is generated by all elements of a form Х {и ®д — и ® Х {д, и e 7®m, g 6 G, i = 1, . . . , n. Using this diagram one can show th a t JS/(T)(<72/7,1 » = K e r(ji 0 Bl) s 7®m0 B6t is generated by elements of a form

П

7 ^ 0 y , Х ц Х) 0 ...

i = 1

where ^ X igi = 0 and £: { 1 ,..., m —!}->-{1,..., n).

A cyclic Л-module R jl admits a Koszid complex

0 ->L = Rex л ...

en — >P = Re[ @ ... @Ren-^- ... ~>R->RIT>i) as a free resolution, where <?'• — ex л ... л et_x л ei+1 л ... л en and j (ех л ...

... л e,rt) = ( — 1 )г_ 1 Х г-е'-. We can identify L®rn 0 G with G, L<g> ... 0 L 0

г=1

0 P 0 i 0 ... 0 i 0 6 with P 0 G ; then respecting this identification we

have /I W

j*<9) =

г = 1 * t = l

. SO

Kerj* = fer Ig - 0 } = 0 :&J and finally E { T ) { ( R j I , n)) = 0 :GI.

6 ^. In this part we present some computations of E( T) in case T( X)

= Д m(X)<S>G, where X and G are modules over a commutative ring R and m ^ 2. We have

TH—l

T(X ’ © X 2) ~ T ( X 1)©2'(X2)© © A k( X l) ® A m- ' ,( X 2) ® e k =1

m— 1

then we identify T 2{ X \ X 2) with © Д ^ Х 1) 0 0 <?. Let fc=i

0 — >P — >АГ— >0 be an exact sequence of P-modules with projective P . For any x x, ..., x.m g L we get

m— 1

*=i where the term

ji(Æ1A ...A æ m0<?) - £ Sgll (ç>,

) ^ 1)

. . .

^

) 0 ^ (

(1))

. . .

j(®v(TO_fc))0flr (<P,y>)

is in Д*(Р) 0 Д ш_А:(Р) 0 6 r for Л = 0, w —1, (ç>, y) runs over all (Jc, m — ft)-shuffles. Since = f \ m(j) 0 1 and it is clear th a t

m— l

K er(P 2 ( 1 , j)a(L)] = (© K erj*, then by Theorem 2

(

8

Р ( Т ) « Ж , 1 » « П K e rj;.

& = o

Now we assume th a t R — Z and we show th a t formula (5) is valid also for the functor T = f\ m( • ) 0 6 r. Diagram ( 6 ) induces for fc = 1, ...

..., m — 1 a commutative diagram

T(i.) = A » ( i . ) ® e - ^ - A * ( i a) ® A —

p / p

Т(Х) = A m(X )® e— —►л*(Х)®Лт “':(-г')®<?

and -we deduce formula (5) from ( 8 ) similarly as in part 5.

Since E ( T ) ( ( Z ( n Z , 1 » = 0 then E( T) ( ( QI Z, 1 » = 0. If M is a direct sum of cyclic groups of finite orders rx, ..., rn, then using ( 8 ) we can show by standard computation th a t E ( T ) ( ( M, 1 » = K erj'^.j and it is iso­

morphic to ф {g e ( t ; d{rj)g = 0 }, where r) runs over all strictly monotonie

V

functions rj: { 1 , . . . , w}-> { 1 , . . . , w} and d(r\) denotes the greatest common divisor of r„( 1 ), . . . , r „ (m).

Let us assume now th a t В = &[X1} X n] and preserve the nota­

tion of part 5. We have T 2( I , B ) = / \ m~1(I) (&G, j f0 = T(j) = 0 and, moreover, for i x < ... < im

m

T 2{1, j) a {I) {Xh л • • • A X im<S>g) = - ±)к~1хч х ч л • • •л Х гк л • • • л X in ®д

к — 1

ш

= У ( ... л Х ,т _1 ® ÿ

fc=l

Xv Хг- л (g )<7 for odd m , *

_ гт г1 *кг—1 £ 7

О for even m.

Using the description of as a й-module one can easily show th a t f \ m(I) as a ^-module is a direct sum $ m>1 ® $W)2, where $ m>1 is a free ^-module on ^ Х ^ л . ' . л Х ф ) for all strictly monotonie functions rj: { 1 , ...

.. . , т}-з*{ 1 , .. . , n} and Sm>2 is a direct sum of cyclic ^-modules ks^n) f* к 12k, where s f l) = X :(m+1)... X f(e)X f( 1 )A... л X C(m) for all strictly monotonie functions £: {1, ..., s }-^{1, ..., n} and s > m. From a diagram similar to (7) one can deduce th a t

E ( T ) « R I I , 1 » f \ m(I) ® 0 for even m , for odd m,

where $ (m) denotes a submodule of f \ m{I)®G generated by elements 2 s *w)® 0 (for all strictly monotonie functions r\\ ( 1 , ..., ..., a}

and all g e G) and elements ^ Х г(1) л ... л X C(m_1) л Хг- ® д{ (for all strictly

г = 1

monotonie functions £ : { 1 , ..., m — 1 }-> { 1 , n} and ail relations

П

JT Х ^ г- = 0). I t is clear th a t I anihilates SSm).

i= 1

References

[1] S. E ile n b e r g and S. M a cL a n e, On the groups Н ( л , п ) , II, Ann. of Math.

60 (1954), p. 49-139.

[2] S. M a cL a n e, Categories for the Working Mathematician, Berlin 1971.

[3] —, Homology, Berlin 1963.

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