- Source: Tangent space to a functor
In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let X be a scheme over a field k.
To give a
k
[
ϵ
]
/
(
ϵ
)
2
{\displaystyle k[\epsilon ]/(\epsilon )^{2}}
-point of X is the same thing as to give a k-rational point p of X (i.e., the residue field of p is k) together with an element of
(
m
X
,
p
/
m
X
,
p
2
)
∗
{\displaystyle ({\mathfrak {m}}_{X,p}/{\mathfrak {m}}_{X,p}^{2})^{*}}
; i.e., a tangent vector at p.
(To see this, use the fact that any local homomorphism
O
p
→
k
[
ϵ
]
/
(
ϵ
)
2
{\displaystyle {\mathcal {O}}_{p}\to k[\epsilon ]/(\epsilon )^{2}}
must be of the form
δ
p
v
:
u
↦
u
(
p
)
+
ϵ
v
(
u
)
,
v
∈
O
p
∗
.
{\displaystyle \delta _{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in {\mathcal {O}}_{p}^{*}.}
)
Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point
p
∈
F
(
k
)
{\displaystyle p\in F(k)}
, the fiber of
π
:
F
(
k
[
ϵ
]
/
(
ϵ
)
2
)
→
F
(
k
)
{\displaystyle \pi :F(k[\epsilon ]/(\epsilon )^{2})\to F(k)}
over p is called the tangent space to F at p.
If the functor F preserves fibered products (e.g. if it is a scheme), the tangent space may be given the structure of a vector space over k. If F is a scheme X over k (i.e.,
F
=
Hom
Spec
k
(
Spec
−
,
X
)
{\displaystyle F=\operatorname {Hom} _{\operatorname {Spec} k}(\operatorname {Spec} -,X)}
), then each v as above may be identified with a derivation at p and this gives the identification of
π
−
1
(
p
)
{\displaystyle \pi ^{-1}(p)}
with the space of derivations at p and we recover the usual construction.
The construction may be thought of as defining an analog of the tangent bundle in the following way. Let
T
X
=
X
(
k
[
ϵ
]
/
(
ϵ
)
2
)
{\displaystyle T_{X}=X(k[\epsilon ]/(\epsilon )^{2})}
. Then, for any morphism
f
:
X
→
Y
{\displaystyle f:X\to Y}
of schemes over k, one sees
f
#
(
δ
p
v
)
=
δ
f
(
p
)
d
f
p
(
v
)
{\displaystyle f^{\#}(\delta _{p}^{v})=\delta _{f(p)}^{df_{p}(v)}}
; this shows that the map
T
X
→
T
Y
{\displaystyle T_{X}\to T_{Y}}
that f induces is precisely the differential of f under the above identification.
References
Borel, Armand (1991) [1969], Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. Zbl 0960.14002.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157