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Introducción a la Geometría Algebraica
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Este trabajo es una introducción a la geometría algebraica tratando conceptos básicos y relacionándolo con conceptos del álgebra conmutativa.
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Un poco de geometríaalgebraica
, A
menudo se dice que existe un diccionario entre el Algebra
Commutativa y la Geometria algebraica. Este
diccionario se base en la
idea de asocial subconjuntos de pantos de ke ideals de
k2x., Xn3, de forma las afirmaciones haven
..., que que se en una
se traducen en resultados para otros reciprocaments.
y
1. CONJUNTOS ADEBRAICOS AFiNEs
Seau K un
cuerpo ne/ no nulo.
y
Si pix...., n) ERIX,...., Xn] y act, decimos que a es un cer
de n) Sea ahora S=KEX...., xn], llamaremos
plan, p(a)
5i 0.
SFP,
=
...,
algebraico
conjunto afin de k", definido
por S a:
VISI==1ack"/ pla) 0,
=
xpMeSh / Nota: x (xn,...,Xn1)
=
Se cumplen las propiedades signientes:
(a) K" el coninto vacio son conjuntos algebraicos afines.
y p(X
Afirmo que VIo=R2.
"
Vamos a probarlo
""Trivial por definition (VISI k") =
"Sea atk" =
pla) 0= = atV(o)
Afirmo que VIKEX...., xn]) 0. Vamos =
a probarlo
P.R.A. VIK [Xn,..., xn])+0, dear atV/k2x.,..., xn)).
supongamos que es exist
Esto implicaria que plat 0,
=
Xp(MEKTXn,...,Xn]
Pero, existe q(=1EKIX...., xn] y q(a)=1 to, facK" por
# haber
que
supuesto
-
polinomio cte. 1 V (kEXn, ...,Xn]) 4 +
Concluimos que VIKTX, ...,
Xn]) 0=
(b) Si S, S'=
RIA...., n) S =S', entences VIS' VIS) =
y
Definimos VIS)=1 atk"/p(a) 0
=
, p(eS)
VIS" =
3beK"/q(b) 0,Xq(AcS) =
:VIS' VISI? =
SeS
Sea beVIS' =D q(b) 0,
=
Xq(At S
W
-q(b) 0,rq(tS D
-
=
= beV(S)
, (C) Si SEREX,..., xn] y
<S> es el ideal
generado por S, entences
V(S) v(<S) =
Definimos <S =(rst...+ raSn/SieS, riEKExn,..., Xn], neNY
VISI (
=
ack" / p(a) 0,
=
xp(xeS)
(bek"/q(b) 0,Xq(n<Sx)
V(<Sx) =
=
V(S1 =v(<S)
"-"
V(S)
S
=<S(V(< s)
=
"c"
aeVIS) = D pla) 0, XpIneS=<S
=
Sea q(x-<Sc D
=
q(M ri
=
+...+
ES rnInS q(a) rilalal?...+rnials, FaeUlS)
=
=> atV(S)
Como consecuencia, ECK" es un
conjunto algebraico atin si, y
solo si, E=V(I), donde I es un ideal dil avillo de
policomics
kIA...., Xn] y se complex has
propiedades:
(d) Si Inletes una familia arbitraria de ideates de KIxn,..., n3,
entonces:
SV(In) V(Y 22) V(2n)
= =
Y por to tanto, la intersection arbitraria de
conjuntos algebraicos
afines, as tambien un atin.
conjunto
⑮
-
3 V122) V)kIel? =
Sea (VIIa
at =D
pulal 0,
=
Xprklehn, XR x
=
p(a) 0,
=
XpIeYIr = aEVIV,In
·VIWIn) VIE Ie)? =
Saberos que E12 x, Zn
=
=
VIE,de) VkEdn"V(WInl
=
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