• Nie Znaleziono Wyników

Faithfully flat descent of linear actions of affine group schemes

N/A
N/A
Protected

Academic year: 2021

Share "Faithfully flat descent of linear actions of affine group schemes"

Copied!
4
0
0

Pełen tekst

(1)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PR ACE MATEMATYCZNE XXI (1979)

Jo a n n a Sw iç c ic k a (Warszawa)

Faithfully flat descent of linear actions of affine group schemes

In this paper we shall investigate the problem whether modules with some extrastructure (the “linear structure”) which become isomorphic after an ap­

propriate extension of the ring of scalars were isomorphic before. We obtain in this way a theorem on equivalent representations of affine group schemes which is analogous to the theorem for abstract groups ([1], Theorem 2).

This result does not follow from the main result of [1] since we do not assume any finiteness condition on the algebraic group. Besides that, it follows from our result that the faithfully flat descent holds for many other cases of additional structures that can be defined on modules.

We fix the following notation: Я is a local commutative ring, A a finitely generated commutative Я-algebra, S a finitely generated Л-module, T an arbitrary Л-module, M an Л-module finitely presented over R, R' a faithfully flat R-algebra.

De f in it io n. An А-linear structure on M of type ((1, S), (1, T )) or ((0, S), (1, T)) or ((1, S), (0, T )) is an Л-linear homomorphism q>M: M 0 S -*■ M (g) T

A A

(or (p'M : S -* M (g) Г, or (p'lf : M (x) S -* T, respectively).

A A

De f in i t io n. We say that an element r e H o m A(M,iV) is a morphism of А-linear structures (Pm><Pn ° f *УРе ((F S), (1, Г)) if the following diagram is commutative:

M ® T

jrgnd |t®id (T(x)id)spM = (pN(t (x) id) N ® S N ® T

(for Л-linear structures (pM, (pN of other types the homomorphism т must satisfy equalities (t ® id) (pM = q>N or (pM = (pN(т (g) id), respectively).

We shall denote by Mor (<pM, (pN) the set of morphisms of linear struc-

(2)

254 J. Swiçcick a

tures (pM,(pN• It is in fact an Л-submodule of the Л-module HomA{ M , N ) determined by the exact sequence

0 -* Mor (ç>M, <pN) - Н о т., (M, N) * (Hom^ (M, JV ® T ) f , where

Ф (t)* (m) = (pN (t (m) (x) x) - (r (x) id) (pM (m ® x) or

Ф(т)х(т) = <jOJV(x)-(T(g)id)<ipM(x) or Ф(т)х(т) = q>N( z( m)®x )- q> M{m<g)x) for linear structures of type ((1, S), (1, Г)), ((0, S), (1, T)) or ((1, S), (0, T)), respectively, and X is any finite set of generators of the Л-module S.

Ex a m p l e s. Г Let A = R and let S be an Я-algebra. An Я-linear struc­

ture on an Я-module M of type ((1, S), (1, Я)) determines the structure of S-module on М ,фм : M (x) S -> M if cpM satisfies <pM (x) id) = (pM(id (x)/*)

R

where fi: S ® S - + S is the multiplication in S.

R

2° Let Г be a Hopf algebra of an affine group scheme G over Spec (Л), and let End (M) denote the functor from the category of Л-algebras to the category of semigroups:

End (M)(P) = Н о т р (М ® Я ,М ® Я ) ~ Н о т Л(М ,М ® Я ).

A A

Thus any natural transformation of functors from Л-algebras to semigroups ф: G End (M) determines an homomorphism of Л -т о - dules (p £ Hom^ (M, M (x) T ) (because Nat (G, End (M)) = End (M) (T ) = HornA( M , M 0 T ) ) . Such cp is a subject to the following conditions: The diagram :

M — -— > M ® T

A

(1) <P

M ( x ) T ^ M (x )T ® T where q: T

(2)

T (x) T is the comultiplication in T, is commutative ; M -> M(x) T

id

M ~ М(Х)Л where j is the counity in T, is commutative.

So from each representation of G in End (M) we obtain the linear structure on M of type ((1, A), (1, T)).

In the case Л is a field any linear structure of type ((1, A), (1, T))

(3)

Faithfully flat descent of linear actions 2 5 5

which satisfies (1) and (2) determines a morphism of group schemes G -*■ Gl{n, A), where n is the dimension of the vector space M.

3° К-linear structure q>M of type ((0, Я), (1, Я)) determines an element (pM( 1) of M and, vice-versa, every element a e M determines an Я-linear structure (pM of type ((0, Я), (1, Я)) given by q>M(r) = ra for every r e R .

4° Я-linear structure of type ((1, R), (0, R)) is an element of Horn (M, R),

<pM: M -► R.

The morphisms of such structures are:

(i) homomorphisms of S-modules, (ii) G-homomorphisms,

(iii) fixed point preserving homomorphisms, (iv) linear form preserving homomorphisms.

From now on we shall consider only Л-linear structures of type ((1,S),(1, T)), but our result is true for other types too.

Each Л-linear structure induces in a natural way an Л 0 Я'-Нпеаг

R

structure (pM®R’ — <Рм®^ of type ((1, 5 0 Я ' ) , (1, Т 0 Я ')) on the Л 0 К '-

R

module М ® Я ',

R

( M 0 R') 0 (S0R O ~ ( M 0 S ) 0 K ' ^ ( M 0 Г ) 0 Д '

R A Q R ' A R A R

- (M 0 R') 0 (Т 0 Я ').

A ® R '

A morphism t: q>M -> q>N induces a morphism i': (Pm®r> <Pn®r’> ^ — T 0 i d 0 i d .

Th e o r e m 1. (pM ~ q>N iff <Pm®r' <Pn®r’-

A A ® R '

P ro o f. The general idea of the proof is similar to that of Theorem 2 in [1]. It is obvious that the crucial point in this proof is fact (A) ([1] p. 372).

We shall prove the analogue of it in the case of the Л-Iinear struc­

tures on M.

Le m m a. МотА((рм , (pN) ® R ' ^ Mora®r'((Pm®r'’ (Pn®r')-

R

P ro o f. From Proposition 11 in [2], p. 39, we infer that for any Я-module F

(i) Horn R( M , F)(g) R' ~ Horn R>(M ® R', F ® R').

The following sequence is exact (from the definition)

0 ^ HornA(M, F) -► HornR(M, F) % (HornR{ M , F))\

where X is any finite set of generators of the Я-algebra Л and Ф(т)х{ш)

= хт(т) — т(хт).

(4)

256 J. Swiçcicka

If we tensor this sequence with Я' over R and apply (i), we obtain (ii) НоmA( M , F ) ® R ' ^ HornA®R- ( M ® R', F ® R').

R R R

The definition implies the exactness of the following sequence 0 -> Могл (q)M, (pN) -► Н о т л (M, N ) -> (Н о т л (M , N ® T ) f ,

A

where X' is any finite set of generators of the Л-module S and Ф(т)х(т)

= ф^(т(т)®х) — (i® id ) cpM (m®x). If we tensor the above sequence with R' over Я, we obtain the exact sequence

0 -> Mor^(фм , ^ ) ® Я ' -> Н о т л (M, N ) ® R ,<P^+ (Н о т , (M, N ® T))x ’ ®R'

R R A R

if

Н о т ^ я - (M ® R \ N ® R') -»■ ( Н о т ^ г ( M ® R ' , N ( ^ T ® R'))r . So ЫогА(фм , q>N) ® R ' ^ Могa®r' №m®r' ^ n®r') and we have proved our lemma.

The rest of the proof of the theorem is very similar to the proof of Theorem 2 in [1], so we leave it to the reader.

Co r o l l a r y. Let ф,ф': G -*■ EndA (M ) be two linear representations of the affine group scheme G = Spec(T). Then ф and ф' are equivalent iff the induced representations of G' = Spec ( T ® R r) ф,ф': G' -»• Enda®R' (M ®R ') are equi­

valent.

In the case where R = A = к is a field we infer from the corollary that two representations ф , ф G -> Gl(n, к) are equivalent iff the induced representations over an algebraic closure of к are equivalent.

Pr o p o s it io n. If R is an Artinian local ring, then it is enough to assume that A in [1] is countably generated over R.

P ro o f. Let A = lim A„, where A„ are finitely generated over R. We assume, as before, that R' is a faithfully flat Я-algebra, and M , N are A -modules finitely presented as R-modules. For each n we know that if M ® R' ~ N ® R \ then M ® R ' ~ N ® R ' ; so M ~ N.

R A ® R ’ R R A „ ® R ' R A n

But HomAn( M , N) a HomR( M , N) and HomR(M ,N ) is an Artinian Я-module. Hence the decreasing sequence of its submodules Н о т A ( M, N )

=>...=> Н о т Ak( M, N ) =>...=> Н о т A( M, N) stabilizes and we have M ~ N

(because Iso^r(M,iV) Ф 0 for each r). A

References

[1] A. B ia ly n ic k i-B iru la , On the equivalence o f integral representations of groups, Proc.

Amer. Math. Soc. 26 (1970) p. 37b377.

[2] N. B o u rb a k i. Algèbre commutative, Chap. 1, 2, Actualités Sci. Indust. No. 1290, Hermann, Paris 1960.

Cytaty

Powiązane dokumenty

• ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO. Séria I:

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1991) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGOJ. Séria I: PR ACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVII (1987) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE