Compléments S schemes pointsand

theYoneda lemma

TheYonedalemma This is a fundamental result in Category theory which is used a lot

Let l be a category Then everyobject X of C defines acontravariantfunctor

x COP SetsA 1 Home A X CCAX

Let F COP Sets be a contravariantfunctor If 0 h X F is a natural

transformation then we can consider themapOx h X Home X X F X

TheYonedalemma is thefollowingresult

YonedalemmaThemap

set of HornHQ F FIXLanaturaltransformations 0 Ox idx

is a bijection

ProofConsider a natural transformation 0 Then bydefinition the followingsquare

is commutativeHome X X f X

ou Fluv r

Home A X 0A F A

foreveryobject AEC andevery morphism u EHomCAX In particular 0 is entirely

determined by Ox Idx since

u Idiou F u Dx Idx

To prouethestatement it is thereforeenough tosee that if x C f X the collection

of maps Os Home A X F A defined by 0A u f u x for uCHomeAXis a natural transformation If v A B is a morphismin E thenthesquare

HomeCBX OB FCBou Fcov q v

HomeCAX A F A

iscommutatice since for UE HomeCB X we have

f v 00ps u f v of u x Fluov x

and Gaofav u nov Fluov x la

Recall that a functor F C D is said tobe fullyfaithful iff foreveryobjectX Yin C themap Home X x Hom MX FLY is a bijection

CoroLet E be a category then the functor

h C Fun COPSetsh X

is fullyfaithful

ProofNotethat if ce X Y is amorphismin C then theassociated natural

transformationHu b X h Y is given forevery object A in C byHuta Home A X Home A Y

v 1 nov

TheYoneda lemmatells us thatthemapHomChX h Y h Y X Home X Y

O Ox Idx

is a bijection Now the image ofthebythis bijection is trucidx u Id ce This

concludes theproof

In particular C can beseen as a full subcategory of Fun Sets

DefinitionA functorCOP Sets is said tobe representable iff it is isomorphic

to h X forsome Xobject in C

Rink such an X is unique up to isomorphism

S schemesand points

DefinitionLet s be a scheme A S scheme is apair XT where X is a

scheme and T X S is a morphismofschemes A morphism of 5 schemes

is a morphism ofschemes f X Y such that Tyof Tx where Tx is thestructural

morphismof X and Ty thestructuralmorphismof Y

Let A be a ring We haie seenthat given a scheme X to gce a morphism

SpecA of schemes amounts to specifying a morphismofrings A TEX

that is a A algebra structure on T It turnsOx into a sheafof A algebras

Instead of specA schemes we simplysay A schemes

Rink Schemes are exactlythe Z schemes

DefinitionLet s be a schemeand X be a Scheme AI point where Tisa

5 scheme is a morphism of of S schemes T X The set of T points

is written asXT Hansens

T X

Example Let ALe Spec2ft be theattirelineover Te thenforeveryscheme Xwe have a canonical isomorphism

THÉ X Hansen X Ale n Homprings Elt TEXQD T X Gx

Let us now relate scheme points and solutionsofpolynomial equations

Yonedaand solutionsofpolynomialequations Consider a field k and a familyofpolynomials Pi _Pr C KIK XD Theydefine an affine k scheme

X Spec k Xi XIIwhere I L Pi Pr

The Yoneda lemma applied with C Affschlk tells us that the category ofattirek schemes is a full subcategoryof thecategory

Fun AffsetypiP Sets Fun Rings Setswe usethefundamentalequivalence

HereX ismapped to its functor of points its restriction toattire k schemes So notall

functor comes fromschemes but a scheme is determined byits functorof pointsThis is fundamental in moduliproblems a functor ofsomesignificance is given and

the problem is to showthat it is representable

Now take a k algebra A then theA_points of X are

X A Hansenµ SpecA X Hank T XOx A

HonkAlg kf4 XD PrA

and therefore

X A ai an CAn Pilai an P ai an 0

In particular therationalpoints X K is thesetofsolutions in k of Pi Pr