Last Updated：2026/01/01
Sentence
If
𝒲
is
a
locally
closed
subset
(i.e.,
the
intersection
of
open
and
closed
sets)
of
an
algebraic
variety,
it
becomes
an
algebraic
variety
in
a
natural
manner
since
the
germs
of
regular
functions
at
x∈𝒲
axe
taken
to
be
the
germs
of
functions
on
𝒲
induced
by
functions
in
the
stalk
Rₓ.
The
definitions
of
irreducibility
and
local
rings
of
subvarieties
for
algebraic
varieties
are
given
in
the
same
manner
as
before.
From
now
on,
by
a
variety
is
meant
an
algebraic
variety.
Any
variety
(X,R),
by
definition,
is
a
local-ringed
space.
Quizzes for review
If 𝒲 is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x∈𝒲 axe taken to be the germs of functions on 𝒲 induced by functions in the stalk Rₓ. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,R), by definition, is a local-ringed space.
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