SCHEMES 17

Our goal in this subsection is to show that 9Jtot(X, Y7) is closed in 9Hot(X, Y) if

Yf

is closed in Y and if some additional condition holds. We need some preparation:

Consider an open covering (Xj)jej of X and a closed subfunctor Y' of Y. Let

Pj : dJlot(X, Y) —* DJlox(Xj,Y) denote the obvious restriction map. We claim:

(2) mox(X, Yf) = p | pjifOtotiXj, Y').

jeJ

Of course, one inclusion ("c") is trivial. Consider on the other hand / G 9Jtot(X,

Y)(A) =

MOT(XA,YA)

for some /c-algebra A with pj(A)f G Mor(XJA,YA) for all

j G J, i.e., with XjA C

f~1(YA)

for all j . Now the (XjA)jej are an open covering

of XA and /

_ 1

(1^ ) is a closed subfunctor of XA, SO 1.12(5) yields

f~1(YA)

= XA,

hence / G OTot(X, F'X^)-

(3) Le£ X and Y be k-functors and

Yf

C Y a closed subfunctor. If X admits

an open covering (Xj)jej with affine schemes such that each k[Xj] is free as a

k-module, then 97tot(X, Y') is closed in 0Jlor(X, Y).

(If X is a scheme, then X is called locally free if and only if there is an open covering

as above.)

One sees using (2) that it suffices to prove (3) in the case where X = SpkR

for some ^-algebra R that is free as a /c-module. We now have to show for each

fc-algebra A and each morphism / : SpkA — • DJlot(X, Y) that

f~19Jlox(X,

Y') is

closed.

We have for each /c-algebra B natural bijections

Mor(SpkB,Wlox{SpkR,Y)) ~ Wlox(SpkR,Y)(B) ~ Mor((SpkR)B,YB)

~ Moi(SpB(R 0 B), YB) ^ Y(R 0 B)

arising from Yoneda's lemma, the definitions, or the universal property of the tensor

product.

Taking B = A, we see that any / as above corresponds to some y G Y(R®A).

One checks then for all B that

f(B) : (SpkA)(B) = Homfe_alg(A, B) - Wlox{SpkR, Y)(B) ~ Y(# 0 B)

maps any /3 G Homfe_aig(A, S) to y(id/j ®/?)(y) G V(ii (g) £) .

On the other hand y G Y(R®A) defines also a morphism f : Spk{R®A) — * F

that maps for all B any 7 G Homfc_aig(i? 0 A, B) to F(7)(y). So we get above

f(B)((3) = f'(R®B)(idR®f3).

So

f~lmox{SpkR, Yr){B)

consists of all f3 with /'(i j 0 B)(idfl 0/3) G y ' ( # 0 5) .

Since Y' is closed in Y, there is an ideal

If

in R®A with ( / ' J "

1

^ ' ) = V(J').

We get now for all B

f^MoxiSpkRXXB) = {/3e Homfc_alg(A,£) | V c ker(idR®(3) }.

Now we use for the first time that R is free as a fc-module. It implies that

ker(id# 0/3) = R 0 ker(/?), hence

f^VJloxiSpkR,

Yf)(B)

= { p G Hom^_alg(A, B ) | J ' c f l ® ker(/3) }.