Tutorial on Differential Galois Theory II

T. Dyckerhoff

Description

TutorialonDifferential Galois Theory IIT. DyckerhoffDepartment of MathematicsUniversity of Pennsylvania02/13/08 / OberflockenbachOutlineToday’s planPicard Vessiot ringsThe ∂ Galois group schemeThe Torsor theorem and applicationsDescent theory for Picard Vessiot extensionsn th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... .. .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 i⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... . . .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1Systems of ∂ equationsYesterday we considered:a field F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ...

Informations

• Publié par: Uproi
• Nombre de lectures: 20
• Langue: English

Extrait

Tutorial on Differential Galois Theory

T. Dyckerhoff

Department of Mathematics University of Pennsylvania

02/13/08 / Oberflockenbach

II

Outline

Today’s plan

Picard-Vessiot rings

The∂-Galois group scheme

The Torsor theorem and applications

Descent theory for Picard-Vessiot extensions

    )(y y ∂   y( 1−n ∂ . W⇒    ) 1... .0 00 . . . − 0 . − 12 a 0 aa− n a 1− ..−. erensyalemstf 1 so ots-redrauqenoit edevelopPicard-Vseistohtoeyrofgr

a 0

y

an

equation

∂

∂

n

a fieldFwith derivation

Systems of∂oitauqe-sn Yesterday we considered:

.]A[yb

with

a i

=

0

)

(y

∙

+

∙

∙

a 1

+

(y

∂

+

)

AhF∈wyti)yA= :s (∂ notewedehich n×n w

F

∈

uation⇒ahordereqnt-   : ∂ nsioatqureedrots-1fometsys 10..   0  =   ( 1 )y∂ . −n )y y  (∂

elopedev   ⇒W  tohtseisdrV-iPacsyalerenrgforyeo redrots-1 fosmets qeauitno:s (∂)yA=ywithA∈F n×n whichedewetonA[yb

n-th order equation⇒a system of 1-st order equations: 0 0 ∂  ∂( y y)   0 0   ∂( y y) =   ∂ n− . 1 (y)     −a 0 −a n−1     ∂ n− . 1 (y )

.]

1 0 . −a 1

Systems of∂nsuqe-oita Yesterday we considered: a fieldFwith derivation∂ an equation∂ n (y) +∙ ∙ ∙+a 1 ∂(y) +a 0 y=0 witha i ∈F

. . . . . . . . . . . .

0 1 −a 2

Systems of∂ionsquat-e Yesterday we considered: a fieldFwith derivation∂ an equation∂ n (y) +∙ ∙ ∙+a 1 ∂(y) +a 0 y=0 witha i ∈F n-th order equation⇒a system of 1-st order equations: ∂  ∂(y y )   100 010 . ... .. 0 0   ∂( y y)  =   ∂ n− . 1 (y)     −a 0 −a 1 . −a 2 . . . . . . −a n−1     .   ∂ n−1 (y) ⇒We develop Picard-Vessiot theory for general systems of 1-st order equations: ∂(y) =AywithA∈F n×n which we denote by[A].

)R( n Y(∂:∃ :xi LG∈Y[Y=F,d ij AY)=dRan R 2 sinani teY( − )] 1 main 3 R/F tegraldo .e.i,cirtemoegsiwcnenoas)h(RotQu -∂isR 4 si nastnots on-t.non,i.emple Definition APcira-deVssoirtni]i[Aorgfngri∂-sa 1 htiwF/R negsiF/Redbyeratdameafunosultnlaamrtitno draciPfotoisseV-ffdoel=finsioctraV-seacdrefidlisto al∂-rivils Pi idea

a

system

of

∂-equations

ioit

Picard-Vessiot rings

Given

n

n×n

a

∂-fieldFwith field of constants

K

A

[A],

F

∈

yhseetdryas’ednfi ring coincideswit

Definition APicard-Vessiot ringfor[A]is a∂-ringR/Fwith 1 R/Fgenerated by a fundamental solution matrix:is ∃Y∈GL n (R) :∂(Y) =AYandR=F[Y ij ,det(Y) −1 ] Ris an integral domain R/Fis geometric, i.e. Quot(R)has no new constants Ris∂-simple, i.e. no non-trivial∂-ideals

Given a∂-fieldFwith field of constantsK a system of∂-equations[A],A∈F n×n

2 3 4

no

Picard-Vessiot rings

initdsfieday’sterthyeeswi dlforfcafitle=defid-VessioPicar ioc g dicn oiss nir taricVed-ontifPso

Picard-Vessiot rings

Given a∂-fieldFwith field of constantsK a system of∂-equations[A],A∈F n×n

Definition APicard-Vessiot ringfor[A]is a∂-ringR/Fwith 1 R/Fis generated by a fundamental solution matrix: ∃Y∈GL n (R) :∂(Y) =AYandR=F[Y ij ,det(Y) −1 ] 2 Ris an integral domain 3 R/Fis geometric, i.e. Quot(R)has no new constants 4 Ris∂-simple, i.e. no non-trivial∂-ideals

Picard-Vessiot field= field of fractions of Picard-Vessiot ring coincides with yesterday’s definition

y 00

∂

evogt(Rr

+y=0

Example overR(t)

)

2

order equation translates into the system  y y 2 1  =  − 01 01   y y 1 2 

the

2-nd

s(cosit) nbve =  y Y c)t(t(so )t(n nis− mentalsoafundartxisiigulitnoam aPis =1) 2 (tin+s) 2 nirtoisseV-draci (t),[cosR(t) )  R= sot( ti c h)tw]is(n

=Rc[so(R)tsin((t),ith c t)]w s+ 2 )t(so1= 2 )t(ni aricaPisiossVed-rtniogevRrt()

+y=0

Example overR(t)

the 2-nd order equation translates into the system ∂  y y 1 2  =  − 10 10   y y 2 1 

y 00

a fundamental solution matrix is given by  − cos sni( ( t t ) ) cso nis ( ( t t ) ) 

Y=

Example overR(t)

y 00

+y=0 the 2-nd order equation translates into the system ∂  y y 2  =  −01 01   y y 1 2  1

a fundamental solution matrix is given by Y=  cos(t ) ) cso nsi ( ( t t ) )  −sin(t

R=R(t)[cos(t),sin(t)]with

cos(t) 2 +sin(t) 2 =1

is a Picard-Vessiot ring overR(t)