4. Local Study: Tangent Planes, Tangent Cones, Singularities

Back

In this section, we examine the structure of a variety near a point. I begin with

the case of a curve, since the ideas in the general case are the same, but the formulas

are more complicated. Throughout, k is an algebraically closed field.

Tangent spaces to plane curves. Consider the curve

V : F(X, Y ) = 0

in the plane A2 defined by a nonconstant polynomial F(X, Y ). We assume that

F(X, Y ) has no multiple factors, so that (F(X, Y )) is a radical ideal and I(V) =

(F(X, Y )). We can factor F into a product of irreducible _ polynomials, F(X, Y ) =

Fi(X, Y ), and then V = ∪V (Fi) expresses V as a union of its irreducible components.

Each component V (Fi) has dimension 1 (see 1.21) and so V has pure dimension

1. More explicitly, suppose for simplicity that F(X, Y ) itself is irreducible, so that

k[V] = k[X, Y ]/(F(X, Y )) = k[x, y] is an integral domain. If F _= X − c, then x is

transcendental over k and y is algebraic over k(x), and so x is a transcendence basis

for k(V ) over k. Similarly, if F _= Y −c, then y is a transcendence basis for k(V ) over

k.

Let (a, b) be a point on V . In calculus, the equation of the tangent at P = (a, b) is

defined to be

∂F

∂X

(a, b)(X − a) +

∂F

∂Y

(a, b)(Y − b) = 0. (*)

This is the equation of a line unless both ∂F

∂X (a, b) and ∂F

∂Y (a, b) are zero, in which case

it is the equation of a plane.

Definition 4.1. The tangent space TPV to V at P = (a, b) is the space defined

by equation (*).

When ∂F

∂X (a, b) and ∂F

∂Y (a, b) are not both zero, TP (V ) is a line, and we say that P

is a nonsingular or smooth point of V. Otherwise, TP (V ) has dimension 2, and we

say that P is singular or multiple. The curve V is said to be nonsingular or smooth

when all its points are nonsingular.

We regard TP (V ) as a subspace of the two-dimensional vector space TP (A2), which

is the two-dimensional space of vectors with origin P.

Example 4.2. In each case, the reader is invited to sketch the curve. The characteristic

of k is assumed to be _= 2, 3.

(a) Xm+Y m = 1. All points are nonsingular unless the characteristic divides m (in

which case Xm + Y m − 1 has multiple factors).

(b) Y 2 = X3. Here only (0, 0) is singular.

(c) Y 2 = X2(X + 1). Here again only (0, 0) is singular.

(d) Y 2 = X3 +aX +b. In this case, V is singular ⇐⇒ Y 2 −X3 −aX −b, 2Y , and

3X2 + a have a common zero ⇐⇒ X3 + aX + b and 3X2 + a have a common

zero. Since 3X2 + a is the derivative of X3 + aX + b, we see that V is singular

if and only if X3 + aX + b has a multiple root.

(e) (X2 + Y 2)2 + 3X2Y − Y 3 = 0. The origin is (very) singular.

(f) (X2 + Y 2)3 − 4X2Y 2 = 0. The origin is (even more) singular.

60 Algebraic Geometry: 4. Local Study

(g) V = V (FG) where FG has no multiple factors and F and G are relatively

prime. Then V = V (F) ∪ V (G), and a point (a, b) is singular if and only if it is

a singular point of V (F), a singular point of V (G), or a point of V (F) ∩ V (G).

This follows immediately from the equations given by the product rule:

∂(FG)

∂X

= F · ∂G

∂X

+

∂F

∂X

· G,

∂(FG)

∂Y

= F · ∂G

∂Y

+

∂F

∂Y

· G.

Proposition 4.3. Let V be the curve defined by a nonconstant polynomial F without

multiple factors. The set of nonsingular points13 is an open dense subset V .

Proof. We can assume that F is irreducible. We have to show that the set of

singular points is a proper closed subset. Since it is defined by the equations

F = 0,

∂F

∂X

= 0,

∂F

∂Y

= 0,

it is obviously closed. It will be proper unless ∂F/∂X and ∂F/∂Y are identically

zero on V , and are therefore both multiples of F, but, since they have lower degree,

this is impossible unless they are both zero. Clearly ∂F/∂X = 0 if and only if F is

a polynomial in Y (k of characteristic zero) or is a polynomial in Xp and Y (k of

characteristic p). A similar remark applies to ∂F/∂Y. Thus if ∂F/∂X and ∂F/∂Y

are both zero, then F is constant (characteristic zero) or a polynomial in Xp, Y p, and

hence a pth power (characteristic p). These are contrary to our assumptions.

The set of singular points of a variety is often called the singular locus of the variety.

Tangent cones to plane curves. Note that if P = (0, 0), then the equation defining

the tangent space is the linear term of F:si nce (0, 0) is on V ,

F = aX + bY + terms of higher degree,

and the equation of the tangent space is F-(X, Y ) df = aX + bY = 0.

In general a polynomial F(X, Y ) can be written (uniquely) as a finite sum

F = F0 + F1 + F2 + · · · + Fm + · · ·

where Fm is a homogeneous polynomial of degree m. The first nonzero term on the

right (the homogeneous summand of F of least degree) will be written F∗ and called

the leading form of F.

Definition 4.4. Let F(X, Y ) be a polynomial without square factors, and let V

be the curve defined by F. If (0, 0) ∈ V , then the geometric tangent cone to V at

(0, 0) is the zero set of F∗. The tangent cone is the pair (V (F∗), F∗). To obtain the

tangent cone at any other point, translate to the origin, and then translate back.

Example 4.5. (a) Y 2 = X3:the geometric tangent cone at (0, 0) is given by

Y 2 = 0 — it is the X-axis (doubled).

(b) Y 2 = X2(X + 1):the geometric tangent cone at (0,0) is given by Y 2 = X2 —

it is the pair of lines Y = ±X.

13In common usage, “singular” means uncommon or extraordinary as in, for example, he spoke

with singular shrewdness. Thus the proposition says that singular points (mathematical sense) are

singular (usual sense).

Algebraic Geometry: 4. Local Study 61

(c) (X2 + Y 2)2 + 3X2Y − Y 3 = 0:the geometric tangent cone at (0, 0) is given by

3X2Y − Y 3 = 0 — it is the union of the lines Y = 0, Y = ±

√

3X.

(d) (X2 + Y 2)3 − 4X2Y 2 = 0:the geometric tangent cone at (0, 0) is given by

4X2Y 2 = 0 — it is the union of the x and y axes (each doubled).

In general we can factor F∗ as

F∗(X, Y ) =

 

Xr0(Y − aiX)ri .

Then deg F∗ =

_

ri is called the multiplicity of the singularity, multP (V ). A multiple

point is ordinary if its tangents are nonmultiple, i.e., ri = 1 all i. An ordinary double

point is called a node, and a nonordinary double point is called a cusp. (There are

many names for special types of singularities — see any book, especially an old book,

on curves.)

The local ring at a point on a curve.

Proposition 4.6. Let P be a point on a curve V , and let m be the corresponding

maximal ideal in k[V ]. If P is nonsingular, then dimk m/m2 = 1, and otherwise

dimk m/m2 = 2.

Proof. Assume first that P = (0, 0). Then m = (x, y) in k[V] =

k[X, Y ]/(F(X, Y )) = k[x, y]. Note that m2 = (x2, xy, y2), and

m/m2 = (X, Y )/(m2 + F(X, Y )) = (X, Y )/(X2,XY,Y 2, F(X, Y )).

In this quotient, every element is represented by a linear polynomial cx + dy, and

the only relation is F-(x, y) = 0. Clearly dimm/m2 = 1 if F- _= 0, and dimm/m2 =

2 otherwise. Since F- = 0 is the equation of the tangent space, this proves the

proposition in this case.

The same argument works for an arbitrary point (a, b) except that one uses the

variables X_ = X − a and Y _ = Y − b — in essence, one translates the point to the

origin.

We explain what the condition dimk(m/m2) = 1 means for the local ring OP =

k[V ]m — see later for more details. Let n be the maximal ideal mk[V ]m of this local

ring. The map m → n induces an isomorphism m/m2 → n/n2, and so we have

P nonsingular ⇐⇒ dimk m/m2 = 1 ⇐⇒ dimk n/n2 = 1.

Nakayama’s lemma shows that the last condition is equivalent to n being a principal

ideal. Since OP is of dimension 1, n being principal means OP is a regular local ring

of dimension 1, and hence a discrete valuation ring, i.e., a principal ideal domain with

exactly one prime element (up to associates). Thus, for a curve,

P nonsingular ⇐⇒ OP regular ⇐⇒ OP is a discrete valuation ring.

Tangent spaces of subvarieties of Am. Before defining tangent spaces at points

of closed subvarietes of Am we review some terminology from linear algebra.

62 Algebraic Geometry: 4. Local Study

Linear algebra. For a vector space km, let Xi be the ith coordinate function a _→ ai.

Thus X1, . . . ,Xm is the dual basis to the standard basis for km. A linear form

_

aiXi

can be regarded as an element of the dual vector space (km)∨ = Hom(km, k).

Let A = (aij) be an n × m matrix. It defines a linear map α: km → kn, by





a1

...

am



 _→ A





a1

...

am



.

Thus, if α(a) = b, then

bi =

_m

j=1

aijaj.

Write X1, . . . ,Xm for the coordinate functions on km and Y1, . . . , Yn for the coordinate

functions on kn. Then the last equation can be rewritten as:

Yi ◦ α =

_m

j=1

aijXj .

T_his says that, when we apply α to a, then the ith coordinate of the result is m

j=1 aij(Xja) =

_m

j=1 aijaj .

Tangent spaces. Consider an affine variety V ⊂ km, and let a = I(V ). The tangent

space Ta(V ) to V at a = (a1, . . . , am) is the subspace of the vector space with origin

a cut out by the linear equations

_m

i=1

∂F

∂Xi

____

a

(Xi − ai) = 0, F∈ a. (*).

Thus Ta(Am) is the vector space of dimension m with origin a, and Ta(V ) is the

subspace of Ta(Am) defined by the equations (*).

Write (dXi)a for (Xi − ai); then the (dXi)a form a basis for the dual vector space

Ta(Am)∨ to Ta(Am)—in fact, they are the coordinate functions on Ta(Am). As in

advanced calculus, for a function F ∈ k[X1, . . . ,Xm], we define the differential of F

at a by the equation:

(dF )a =

_ ∂F

∂Xi

____

a

(dXi)a.

It is again a linear form on Ta(Am). In terms of differentials, Ta(V ) is the subspace

of Ta(Am) defined by the equations:

(dF )a = 0, F ∈ a (**).

I claim that, in (*) and (**), it suffices to take the F in a generating subset for a.

The product rule for differentiation shows that if G =

_

j HjFj, then

(dG)a =

_

j

Hj(a) · (dFj)a + Fj(a) · (dGj)a.

Algebraic Geometry: 4. Local Study 63

If F1, . . . , Fr generate a and a ∈ V (a), so that Fj(a) = 0 for all j, then this equation

becomes

(dG)a =

_

j

Hj(a) · (dFj)a.

Thus (dG)a(t) = 0 if (dFj)a(t) = 0 for all j.

When V is irreducible, a point a on V is said to be nonsingular ( or smooth) if the

dimension of the tangent space at a is equal to the dimension of V ; otherwise it is

singular ( or multiple). When V is reducible, we say a is nonsingular if dimTa(V ) is

equal to the maximum dimension of an irreducible component of V passing through a.

It turns out then that a is singular precisely when it lies on more than one irreducible

component, or when it lies on only one but is a singular point of that component.

Let a = (F1, . . . , Fr), and let

J = Jac(F1, . . . , Fr) =

_

∂Fi

∂Xj

_

=





∂F1

∂X1

, . . . , ∂F1

∂Xm ...

...

∂Fr

∂X1

, . . . , ∂Fr

∂Xm





.

Then the equations defining Ta(V ) as a subspace of Ta(Am) have matrix J(a). Therefore,

from linear algebra,

dimk Ta(V) = m − rankJ(a),

and so a is nonsingular if and only if the rank of Jac(F1, . . . , Fr)(a) is equal to

m−dim(V ). For example, if V is a hypersurface, say I(V) = (F(X1, . . . ,Xm)), then

Jac(F)(a) =

_

∂F

∂X1

(a), . . . ,

∂F

∂Xm

(a)

_

,

and a is nonsingular if and only if not all of the partial derivatives ∂F

∂Xi

vanish at a.

We can regard J as a matrix of regular functions on V. For each r,

{a ∈ B | rankJ(a) ≤ r}

is closed in V , because it the set where certain determinants vanish. Therefore, there

is an open subset U of V on which rankJ(a) attains its maximum value, and the rank

jumps on closed subsets. Later we shall show that the maximum value of rankJ(a) is

m − dimV , and so the nonsingular points of V form a nonempty open subset of V .

The differential of a map. Consider a regular map

α: Am → An, a _→ (P1(a1, . . . , am), . . . , Pn(a1, . . . , am)).

We think of α as being given by the equations

Yi = Pi(X1, . . . ,Xm), i = 1, . . . n.

It corresponds to the map of rings α∗ : k[Y1, . . . , Yn] → k[X1, . . . ,Xm] sending Yi to

Pi(X1, . . . ,Xm), i = 1, . . . n.

Define (dα)a : Ta(Am) → Tb(An) to be the map such that

(dYi)b ◦ (dα)a =

_ ∂Pi

∂Xj

____

a

(dXj)a,

64 Algebraic Geometry: 4. Local Study

i.e., relative to the standard bases, (dα)a is the map with matrix

Jac(P1, . . . , Pn)(a) =





∂P1

∂X1

(a), . . . , ∂P1

∂Xm

(a)

...

...

∂Pn

∂X1

(a), . . . , ∂Pn

∂Xm

(a)





.

For example, suppose a = (0, . . . , 0) and b = (0, . . . , 0), so that Ta(Am) = km and

Tb(An) = kn, and

Pi =

_m

j=1

cijXj + (higher terms), i = 1, . . . , n.

Then Yi ◦ (dα)a =

_

j cijXj , and the map on tangent spaces is given by the matrix

(cij), i.e., it is simply t _→ (cij)t.

Let F ∈ k[X1, . . . ,Xm]. We can regard F as a regular map Am → A1, whose

differential will be a linear map

(dF )a : Ta(Am) → Tb(A1), b = F(a).

When we identify Tb(A1) with k, we obtain an identification of the differential of F

(F regarded as a regular map) with the differential of F (F regarded as a regular

function).

Lemma 4.7. Let α: Am → An be as at the start of this subsection. If α maps

V = V (a) ⊂ km into W = V (b) ⊂ kn, then (dα)a maps Ta(V ) into Tb(W), b = α(a).

Proof. We are given that

f ∈ b ⇒ f ◦ α ∈ a,

and have to prove that

f ∈ b ⇒ (df )b ◦ (dα)a is zero on Ta(V ).

The chain rule holds in our situation:

∂f

∂Xi

=

_n

i=1

∂f

∂Yj

∂Yj

∂Xi

, Yj = Pj(X1, . . . ,Xm), f = f(Y1, . . . , Yn).

If α is the map given by the equations

Yj = Pj(X1, . . . ,Xm), j= 1, . . . ,m,

then the chain rule implies

d(f ◦ α)a = (df )b ◦ (dα)a, b = α(a).

Let t ∈ Ta(V ); then

(df )b ◦ (dα)a(t) = d(f ◦ α)a(t),

which is zero if f ∈ b because then f ◦ α ∈ a. Thus (dα)a(t) ∈ Tb(W).

We therefore get a map (dα)a : Ta(V ) → Tb(W). The usual rules from advanced

calculus (alias differential geometry) hold. For example,

(dβ)b ◦ (dα)a = d(β ◦ α)a, b = α(a).

Algebraic Geometry: 4. Local Study 65

Example 4.8. Let V be the union of the coordinate axes in A3, and let W be

V (XY (X −Y )) ⊂ A2 (union of three lines). Then V is not isomorphic to W because

To(V ) has dimension 3, but To(W) has dimension 2. (Note that V = V (XY, Y Z,XZ),

from which it is clear that the origin o is the only singular point on V , and that the

tangent space there has dimension 3. An isomorphism V → W would have to send

the singular point to the singular point, i.e., o _→ o, and map To(V ) isomorphically

onto To(W).)

Etale maps. Let V and W be smooth varieties. A regular map α: V → W is ґetale

at a if (dα)a : Ta(V ) → Tb(W) is an isomorphism; α is ґetale if it is ´etale at all points

of V .

Example 4.9. (a) A regular map α = (P1, . . . , Pn) : An → An is ´etale at a if

and only if rank Jac(P1, . . . , Pn)(a) = n, because the map on the tangent spaces has

matrix Jac(P1, . . . , Pn)(a)). Equivalent condition:de t

 

∂Pi

∂Xj

(a)

_

_= 0

(b) Let V = Specm(A) be an affine variety, and let f =

_

ciXi ∈ A[X]. Let

W = Specm(A[X]/(f(X)) (assuming this is an affine k-algebra), and consider the

map W → V corresponding to the inclusion A 8→ A[X]/(f). The points of W lying

over a point a ∈ V correspond to the roots of

_

ci(a)Xi. I claim that the map

W → V is ´etale at a point (a, b) if and only if b is a simple root of

_

ci(a)Xi.

To see this, write A = Specm k[X1, . . . ,Xn]/a, a = (f1, . . . , fr), so that A[X]/(f) =

k[X1, . . . ,Xn]/(f1, . . . , fr, f). The tangent spaces to W and V at (a, b) and a respectively

are the null spaces of the matrices





∂f1

∂X1

(a) . . . ∂f1

∂Xm

(a) 0

...

...

∂fn

∂X1

(a) . . . ∂fn

∂Xm

(a) 0

∂f

∂X1

(a) . . . ∂f

∂Xm

(a) ∂f

∂X (a, b)









∂f1

∂X1

(a) . . . ∂f1

∂Xm

(a)

...

...

∂fn

∂X1

(a) . . . ∂fn

∂Xm

(a)





and the map T(a,b)(W) → Ta(V ) is induced by the projection map kn+1 → kn that

omits the last coordinate. This map is an isomorphism if and only if ∂f

∂X (a, b)_= 0,

because then any solution to the smaller set of equations extends uniquely to a solution

of the larger set. But ∂f

∂X (a, b) = d(Pi ci(a)Xi)

dX (b), which is zero if and only if b is a

multiple root of

_

i ci(a)Xi.

(c) Consider a dominating map α: W → V of smooth affine varieties, corresponding

to a map A → B of rings. Suppose B can be written B = A[Y1, . . . , Yn]/(P1, . . . , Pn)

(same number of polynomials as variables). A similar argument to the above shows

that α is ´etale if and only if det

 

∂Pi

∂Xj

(a)

_

_= 0.

(d) The example in (b) is typical; in fact every ´etale map is locally of this form,

provided V is normal (in the sense defined below). More precisely, let α: W → V be

´etale at P ∈ W, and assume V to normal; then there exist a map α_ : W_ → V _ with

k[W_] = k[V _][X]/(f(X)), and a commutative diagram

W ⊃ U1 ≈ U_

1

⊂ W_

↓ ↓ ↓ ↓

V ⊃ U2 ≈ U_

2

⊂ V _

with the U’s all open subvarieties and P ∈ U1.

66 Algebraic Geometry: 4. Local Study

Warning! In advanced calculus (or differential geometry, or the theory of complex

manifolds), the inverse function theorem says that a map α that is ´etale at a point

a is a local isomorphism there, i.e., there exist open neighbourhoods U and U_ of

a and α(a) such that α induces an isomorphism U → U_. This is not true in

algebraic geometry, at least not for the Zariski topology:a map can be ´etale at a

point without being a local isomorphism. Consider for example the map

α: A1 \ {0} → A1 \ {0}, a_→ a2.

This is ´etale if the characteristic is _= 2, because the Jacobian matrix is (2X), which

has rank one for all X _= 0 (alternatively, it is of the form (4.9b) with f(X) = X2−T ,

where T is the coordinate function on A1, and X2 − c has distinct roots for c _= 0).

Nevertheless, I claim that there do not exist nonempty open subsets U and U_ of

A1−{0} such that α defines an isomorphism U → U_. If there did, then α would define

an isomorphism k[U_] → k[U] and hence an isomorphism on the fields of fractions

k(A1) → k(A1). But on the fields of fractions, α defines the map k(X) → k(X),

X _→ X2, which is not an isomorphism.

Aside 4.10. There is a conjecture that any ´etale map α: An → An is an isomorphism.

If we write α = (P1, . . . , Pn), then this becomes the statement

det

_

∂Pi

∂Xj

(a)

_

_= 0 all a ⇒ α has a inverse.

The condition, det

 

∂Pi

∂Xj

(a)

_

_= 0 all a, implies that det

 

∂Pi

∂Xj

_

is a nonzero constant.

This conjecture, which is known as the Jacobian problem, has not been solved in

general as far as I know. It has caused many mathematicians a good deal of grief.

It is probably harder than it is interesting. See Bass et al., Bull. AMS 7 (1982),

287-330.

Intrinsic definition of the tangent space. The definition we have given of the

tangent space at a point requires the variety to be embedded in affine space. In this

subsection, we give a more intrinsic definition.

By a linear form in X1, . . . ,Xn we mean an expression

_

ciXi, ci ∈ k. The linear

forms form a vector space of dimension n, which is naturally dual to kn.

Lemma 4.11. Let c be an ideal in k[X1, . . . ,Xn] generated by linear forms,

B1, . . . , Br, which we may assume to be linearly independent. Let Xi1, . . . ,Xin−r be

such that {B1, . . . , Br,Xi1, . . . ,Xin−r

} is a basis for the linear forms in X1, . . . ,Xn.

Then k[X1, . . . ,Xn]/c

∼=

k[Xi1, . . . ,Xin−r ].

Proof. This is obvious if the linear forms B1, . . . , Br are X1, . . . ,Xr . In the general

case, because {X1, . . . ,Xn} and {B1, . . . , Br,Xi1, . . . ,Xin−r

} are both bases for the

linear forms, each element of one set can be expressed as a linear combination of the

elements of the second set. Therefore

k[X1, . . . ,Xn] = k[B1, . . . , Br,Xi1, . . . ,Xin−r ]

and so

k[X1, . . . ,Xn]/c = k[B1, . . . , Br,Xi1, . . . ,Xin−r ]/(B1, . . . , Br)

∼=

k[Xi1, . . . ,Xin−r ].

Algebraic Geometry: 4. Local Study 67

Let V = V (a) ⊂ kn, and assume the origin P ∈ V. Let a- be the ideal generated

by the linear terms f- of the f ∈ a. By definition, TP (V) = V (a-). Let A- =

k[X1, . . . ,Xn]/a-, and let m be the maximal ideal in k[V ] corresponding to the origin;

thus m = (x1, . . . , xn).

Proposition 4.12. There are canonical isomorphisms

Homk-linear(m/m2, k)

∼=

→ Homk-alg(A-, k)

∼=

→ TP (V ).

Proof. First isomorphism. Let n = (X1, . . . ,Xn) be the maximal ideal at the

origin in k[X1, . . . ,Xn]. Then m/m2 = n/(n2 + a), and as f − f- ∈ n2 for every

f ∈ a, we have m/m2 = n/(n2 + a-). Let f1,-, . . . , fr,- be a basis for the vector space

a-; there are n − r indeterminates Xi1 . . . ,Xin−r forming with the fi,- a basis for the

linear forms on kn. Then Xi1 + m2, . . . ,Xin−r + m2 form a basis for m/m2 as a kvector

space, and the lemma shows that A- = k[Xi1 . . . ,Xin−r ]. Any homomorphism

α: A- → k of k-algebras is determined by its values α(Xi1 ), . . . , α(Xin−r ), and they

can be arbitrarily given. Since the k-linear maps m/m2 → k have a similar description,

the first isomorphism is now obvious.

Second isomorphism. To give a k-algebra homomorphism A- → k is the same as to

give an element (a1, . . . , an) ∈ kn such that f(a1, . . . , an) = 0 for all f ∈ A-, which

is the same as to give an element of TP (V ).

Lemma 4.13. Let m be a maximal ideal of a ring A, and let n = mAm. For all n,

the map

a + mn _→ a + nn : A/mn → Am/nn

is an isomorphism. Moreover, it induces isomorphisms

mr/mn → nr/nn

for all r < n.

Proof. The second statement follows from the first, because of the exact commutative

diagram:

0 −−−→ mr/mn −−−→ A/mn −−−→ A/mr −−−→ 0 _

 _

≈

 _

≈

0 −−−→ nr/nn −−−→ Am/nn −−−→ Am/nr −−−→ 0.

To simplify the exposition, in proving that the first map is an isomorphism, I’ll assume

A ⊂ S−1A. In order to show that the map A/mn → An/nn is injective, we have to

show that nm ∩A = mm. But nm = S−1mm, S = A−m, and so we have to show that

mm = (S−1mm)∩A. An element of (S−1mm)∩A can be written a = b/s with b ∈ mm,

s ∈ S, and a ∈ A. Then sa ∈ mm, and so sa = 0 in A/mm. The only maximal ideal

containing mm is m (because m_ ⊃ mm ⇒ m_ ⊃ m), and so the only maximal ideal in

A/mm is m/mm; in particular, A/mm is a local ring. As s is not in m/mm, it is a unit

in A/mm, and so sa = 0 in A/mm implies a = 0 in A/mm, i.e., a ∈ mm.

We now prove that the map is surjective. Let a

s

∈ Am. Because s /∈ m and m is

maximal, we have that (s) + m = A, i.e., (s) and m are relatively prime. Therefore

(s) and mm are relatively prime (no maximal ideal contains both of them), and so

there exist b ∈ A and q ∈ mm such that bs+q = 1. Then b maps to s−1 in Am/nm and

68 Algebraic Geometry: 4. Local Study

so ba maps to a

s . More precisely:b ecause s is invertible in Am/nm, a

s is the unique

element of this ring such that sa

s = a; since s(ba) = a(1 − q), the image of ba in Am

also has this property and therefore equals a

s .

Therefore, we also have a canonical isomorphism

TP (V )

≈→

Homk-lin(nP /n2

P , k),

where nP is now the maximal ideal in OP (= Am).

Definition 4.14. The tangent space TP (V) at a point P of a variety V is

Homk-lin(nP /n2

P , k), where nP the maximal ideal in OP .

When V is embedded in affine space, the above remarks show that this definition

agrees with the more explicit definition on p68. The advantage of the present definition

is that it depends only on a (small) neighbourhood of P. In particular, it doesn’t

depend on an affine embedding of V .

A regularmap α: V → W sending P to Q defines a local homomorphism OQ →OP ,

which induces maps mQ → mP , mQ/m2

Q

→ mP /m2

P, and TP (V ) → TQ(W). The last

map is written (dα)P . When some open neighbourhoods of P and Q are realized

as closed subvarieties of affine space, then (dα)P becomes identified with the map

defined earlier.

In particular, if f ∈ mP, then f is represented by a regular map U → A1, P _→ 0,

and hence defines a linear map (df )P : TP (V ) → k. This is just the map sending a

tangent vector (element of Homk-lin(mP /m2

P , k)) to its value at f mod m2

P . Again, in

the concrete situation V ⊂ Am this agrees with the previous definition. In general,

for f ∈ OP , i.e., for f a germ of a function at P, we define

(df )P = f − f(P) modm2.

The tangent space at P and the space of differentials at P are dual vector spaces—in

contrast to the situation in advanced calculus, for us it is easier to define first the

space of differentials, and then define the tangent space to be its dual.

Consider for example, a ∈ V (a) ⊂ An, with a a radical ideal. For f ∈ k[An] =

k[X1, . . . ,Xn], we have (trivial Taylor expansion)

f = f(P) +

_

ci(Xi − ai) + terms of degree ≥ 2 in the Xi − ai,

that is,

f − f(P) ≡

_

ci(Xi − ai) modm2

P .

Therefore (df )P can be identified with

_

ci(Xi − ai) =

_ ∂f

∂Xi

____

a

(Xi − ai),

which is how we originally defined the differential.14 The tangent space Ta(V (a)) is

the zero set of the equations

(df )P = 0, f∈ a,

14The same discussion applies to any f ∈ OP . Such an f is of the form g

h with h(a) _= 0, and has

a (not quite so trivial) Taylor expansion of the same form, but with an infinite number of terms,

i.e., it lies in the power series ring k[[X1 − a1, . . . ,Xn − an]].

Algebraic Geometry: 4. Local Study 69

and the set {(df )P |Ta(V ) | f ∈ k[X1, . . . ,Xn]} is the dual space to Ta(V ).

The dimension of the tangent space. In this subsection we show that the dimension

of the tangent space is at least that of the variety. First we review some

commutative algebra.

Some commutative algebra. Let S be a multiplicative subset of a ring A, and let S−1A

be the corresponding ring of fractions. Any ideal a in A, generates an ideal S−1a in

S−1A. If a contains an element of S, then S−1a contains a unit, and so is the whole

ring. Thus some of the ideal structure of A is lost in the passage to S−1A, but, as

the next lemma shows, some is retained.

Proposition 4.15. Let S be a multiplicative subset of the ring A. The map p _→

S−1p = p(S−1A) is a bijection from the set of prime ideals of A disjoint from S to

the set of prime ideals of S−1A.

Proof. It is straightforward to verify that

q _→ (inverse image of q in A)

provides an inverse to p _→ S−1p. (See Atiyah and MacDonald 1969, p41–42.)

For example, let V be an affine variety and P a point on V . The proposition shows

that there is a one-to-one correspondence between the prime ideals of k[V ] contained

in mP and the prime ideals of OP . In geometric terms, this says that there is a

one-to-one correspondence between the prime ideals in OP and the irreducible closed

subvarieties of V passing through P.

Now let A be a local Noetherian ring with maximal ideal m. Then m is an Amodule,

and the action of A on m/m2 factors through k df = A/m.

Proposition 4.16. The elements a1, . . . , an of m generate m as an ideal if and

only if their residues modulo m2 generate m/m2 as a vector space over k. In particular,

the minimum number of generators for the maximal ideal is equal to the dimension

of the vector space m/m2.

Proof. If a1, . . . , an generate m, it is obvious that their residues generate m/m2.

Conversely, suppose that their residues generate m/m2, so that m = (a1, . . . , an)+m2.

Since A is Noetherian and (hence) m is finitely generated, Nakayama’s lemma, applied

with M = m and N = (a1, . . . , an), shows that m = (a1, . . . , an).

Lemma 4.17 (Nakayama’s Lemma). Let A be a local Noetherian ring, and let M

be a finitely generated A-module. If N is a submodule of M such that M = N +mM,

then M = N.

Proof. After replacing M with the quotient module M/N, we can assume that

N = 0. Thus we have to show that if M = mM, then M = 0. Let x1, . . . , xn generate

M, and write

xi =

_

j

aijxj

70 Algebraic Geometry: 4. Local Study

for some aij ∈ m. We see that x1, . . . , xn can be considered to be solutions to the

system of n equations in n variables

_

j

(δij − aij)xj = 0, δij = Kronecker delta,

and so Cramer’s rule tells us that det(δij − aij) · xi = 0 for all i. But on expanding it

out, we find that det(δij −aij) = 1+m with m ∈ m. In particular, det(δij −aij) /∈

m,

and so it is a unit. We deduce that all the xi are zero, and that M = 0.

A Noetherian local ring A of Krull dimension d is said to be regular if its maximal

ideal can be generated by d elements. Thus A is regular if and only if its Krull

dimension is equal to the dimension of m/m2.

Two results from Section 7. We shall need to use two results that won’t be proved

until §7.

4.18. For any irreducible variety V and regular functions f1, . . . , fr on V , the

irreducible components of V (f1, . . . , fr) have codimension ≤ r.

Note that for polynomials of degree 1 on kn, this is familiar from linear algebra:A

system of r linear equations in n variables either has no solutions (the equations are

inconsistent) or has a family of solutions of dimension at least n − r.

Recall that the Krull dimension of a Noetherian local ring A is the maximum length

of a chain of prime ideals:

m = p0  p1  · · ·  pd.

In §7, we shall prove:

4.19. If V is an irreducible variety of dimension d, then the local ring at each

point P of V has dimension d.

The height of a prime ideal p in a Noetherian ring A, is the maximum length of a

chain of prime ideals:

p = p0  p1  · · ·  pd.

Because of (4.15), the height of p is the Krull dimension of Ap. Thus the above result

can be restated as:If V is an irreducible affine variety of dimension d, then every

maximal ideal in k[V ] has height d.

Sketch of proof of (4.19):I f V = Ad, then A = k[X1, . . . ,Xd], and all maximal

ideals in this ring have height d, for example,

(X1 − a1, . . . ,Xd − ad) ⊃ (X1 − a1, . . . ,Xd−1 − ad−1) ⊃ . . . ⊃ (X1 − a1) ⊃ 0

is a chain of prime ideals of length d that can’t be refined. In the general case,

the Noether normalization theorem says that k[V ] is integral over a polynomial ring

k[x1, . . . , xd], xi ∈ k[V ]; then clearly x1, . . . , xd is a transcendence basis for k(V ), and

the going up and down theorems (see Atiyah and MacDonald 1969, Chapt 5) show

that the local rings of k[V ] and k[x1, . . . , xd] have the same dimension.

Algebraic Geometry: 4. Local Study 71

The dimension of the tangent space. Note that (4.16) implies that the dimension of

TP (V ) is the minimum number of elements needed to generate nP ⊂ OP .

Theorem 4.20. Let V be irreducible; then dimTP (V ) ≥ dim(V ), and equality

holds if and only if OP is regular.

Proof. Suppose f1, . . . , fr generate the maximal ideal nP in OP. Then f1, . . . , fr

are all defined on some open affine neighbourhood U of P, and I claim that P is an

irreducible component of the zero-set V (f1, . . . , fr) of f1, . . . , fr in U. If not, there

will be some irreducible component Z _= P of V (f1, . . . , fr) passing through P. Write

Z = V (p) with p a prime ideal in k[U]. Because V (p) ⊂ V (f1, . . . , fr) and because

Z contains P and is not equal to it, we have

(f1, . . . , fr) ⊂ p _ mP (ideals in k[U]).

On passing to the local ring OP = k[U]mP , we find (using 4.15) that

(f1, . . . , fr) ⊂ pOP _ nP (ideals in OP ).

This contradicts the assumption that the fi generate mP . Hence P is an irreducible

component of V (f1, . . . , fr), and (4.18) implies that

r ≥ codimP = dimV.

Since the dimension of TP (V ) is the minimum value of r, this implies that

dimTP (V ) ≥ dimV . If equality holds, then mP can be generated by dimV elements,

which (because of 4.19) implies that OP is regular. Conversely, if OP is regular, then

the minimum value of r is dimV , and so equality holds.

As in the affine case, we define a point P to be nonsingular if dimTP (V ) = dimV .

Thus a point P is nonsingular if and only if OP is a regular local ring. In more geometric

terms, we can say that a point P on a variety V of dimension d is nonsingular

if and only if it can be defined by d equations in some neighbourhood of the point;

more precisely, P is nonsingular if there exists an open neighbourhood U of P and d

regular functions f1, . . . , fd on U that generate the ideal mP .

According to (Atiyah and MacDonald 1969, 11.23), a regular local ring is an integral

domain. This provides another explanation of why a point on the intersection of two

irreducible components of a variety can’t be nonsingular:t he local ring at such a

point in not an integral domain. (Suppose P ∈ Z1 ∩ Z2, with Z1 ∩Z2 _= Z1, Z2. Since

Z1∩Z2 _= Z1, there is a nonzero regular function f1 defined on an open neighbourhood

U of P in Z1 that is zero on U ∩Z1∩Z2. Extend f1 to a neighbourhood of P in Z1∪Z2

by setting f1(Q) = 0 for all Q ∈ Z2. Then f1 defines a germ of regular function at P.

Similarly construct a function f2 that is zero on Z1. Then f1 and f2 define nonzero

germs of functions at P, but their product is zero.)

An integral domain that is integrally closed in its field of fractions is also called a

normal ring.

An algebraic variety is normal if OP is normal for all P ∈ V . Equivalent condition

(Atiyah and MacDonald 1969, 5.13):f or all open affines U ⊂ V , k[U] is a finite

product of normal rings. Since, as we just noted, the local ring at a point lying on

two irreducible components can’t be an integral domain, a normal variety is a disjoint

union of irreducible varieties.

72 Algebraic Geometry: 4. Local Study

A regular local Noetherian ring is always normal (cf. Atiyah and MacDonald 1969,

p123); conversely, a normal local integral domain of dimension one is regular (ibid.).

Thus nonsingular varieties are normal, and normal curves are nonsingular. However,

a normal surface need not be nonsingular:the cone

X2 + Y 2 − Z2 = 0

is normal, but is singular at the origin — the tangent space at the origin is k3.

However, it is true that the singular locus of a normal variety must have codimension

≥ 2. For example, a normal surface can only have isolated singularities— the singular

locus can’t contain a curve.

Singular points are singular. The set of singular points on a variety is called the

singular locus of the variety.

Theorem 4.21. The nonsingular points of a variety V form a dense open subset.

Proof. We have to show that the singular points form a proper closed subset of

every irreducible component of V .

Closed: We can assume that V is affine, say V = V (a) ⊂ An. Let P1, . . . , Pr

generate a. Then the set of singular points is the zero set of the ideal generated by

the (n − d) × (n − d) minors of the matrix

Jac(P1, . . . , Pr)(a) =





∂P1

∂X1

(a) . . . ∂P1

∂Xm

(a)

...

...

∂Pr

∂X1

(a) . . . ∂Pr

∂Xm

(a)





Proper: Suppose first that V is an irreducible hypersurface in Ad+1, i.e., that it is the

zero set of a single nonconstant irreducible polynomial F(X1, . . . ,Xd+1). By (1.21),

dimV = d. In this case, the proof is the same as that of (4.3):i f ∂F

∂X1

is identically

zero on V (F), then ∂F

∂X1

must be divisible by F, and hence be zero. Thus F must be a

polynomial in X2, . . . Xd+1 (characteristic zero) or in Xp

1,X2, . . . ,Xd+1 (characteristic

p). Therefore, if all the points of V are singular, then F is constant (characteristic 0)

or a pth power (characteristic p) which contradict the hypothesis.

We shall complete the proof by showing (Lemma 4.21) that there is a nonempty

open subset of V that is isomorphic to a nonempty open subset of an irreducible

hypersurface in Ad+1.

Two irreducible varieties V and W are said to be birationally equivalent if k(V ) ≈

k(W).

Lemma 4.22. Two irreducible varieties V and W are birationally equivalent if and

only if there are open subsets U and U_ of V and W respectively such that U ≈ U_.

Proof. Assume that V and W are birationally equivalent. We may suppose that

V and W are affine, corresponding to the rings A and B say, and that A and B have

a common field of fractions K. Write B = k[x1, . . . , xn]. Then xi = ai/bi, ai, bi ∈ A,

and B ⊂ Ab1...br . Since Specm(Ab1...br) is a basic open subvariety of V , wemay replace

Algebraic Geometry: 4. Local Study 73

A with Ab1...br , and suppose that B ⊂ A. The same argument shows that there exists

a d ∈ B ⊂ A such A ⊂ Bd. Now

B ⊂ A ⊂ Bd ⇒ Bd ⊂ Ad ⊂ (Bd)d = Bd,

and so Ad = Bd. This shows that the open subvarieties D(b) ⊂ V and D(b) ⊂ W are

isomorphic. This proves the “only if” part, and the “if” part is obvious.

Lemma 4.23. Let V be an irreducible algebraic variety of dimension d; then there

is a hypersurface H in Ad+1 birationally equivalent to V .

Proof. Let K = k(x1, . . . , xn), and assume n > d + 1. After renumbering, we

may suppose that x1, . . . , xd are algebraically independent. Then f(x1, . . . , xd+1) = 0

for some nonzero irreducible polynomial f(X1, . . . ,Xd+1) with coefficients in k.

Not all ∂f/∂Xi are zero, for otherwise k will have characteristic p _= 0 and f

will be a pth power. After renumbering, we may suppose that ∂f/∂Xd+1 _= 0.

Then k(x1, . . . , xd+1, xd+2) is algebraic over k(x1, . . . , xd) and xd+1 is separable over

k(x1, . . . , xd), and so, by the Primitive Element Theorem (my notes on Fields and Galois

Theory 5.1), there is an element y such that k(x1, . . . , xd+2) = k(x1, . . . , xd, y).

Thus K is generated by n − 1 elements (as a field containing k). After repeating

the process, possibly several times, we will have K = k(z1, . . . , zd+1) with zd+1 separable

over k(z1, . . . , zd). Now take f to be an irreducible polynomial satisfied by

z1, . . . , zd+1 and H to be the hypersurface f = 0.

Corollary 4.24. Any algebraic group G is nonsingular.

Proof. From the theorem we know that there is an open dense subset U of G of

nonsingular points. For any g ∈ G, a _→ ga is an isomorphism G → G, and so gU

consists of nonsingular points. Clearly G = ∪gU.

In fact, any variety on which a group acts transitively by regular maps will be

nonsingular.

Aside 4.25. If V has pure codimension 1 in Ad+1, then I(V) = (f) for some

polynomial f.

Proof. We know I(V) = ∩I(Vi) where the Vi are the irreducible components of

V , and so if we can prove I(Vi) = (fi) then I(V ) = (f1 · · · fr). Thus we may suppose

that V is irreducible. Let p = I(V ); it is a prime ideal, and it is nonzero because

otherwise dim(V) = d + 1. Therefore it contains an irreducible polynomial f. From

(0.3) we know (f) is prime. If (f) _= p , then we have

V = V (p) _ V ((f)) _ Ad+1,

and dim(V ) < dim(V (f)) < d + 1 (see 1.22), which contradicts the fact that V has

dimension d.

Aside 4.26. Lemma 4.22 can be improved as follows:if V and W are irreducible

varieties, then every inclusion k(W) ⊂ k(V ) is defined by a regular surjective map

α: U → U_ from an open subset U of W onto an open subset U_ of V .

Aside 4.27. An irreducible variety V of dimension d is said to rational if it is birationally

equivalent to Ad. It is said tobe unirational if k(V ) can be embedded in k(Ad)

— according to the last aside, this means that there is a regular surjective map from

74 Algebraic Geometry: 4. Local Study

an open subset of AdimV onto an open subset of V. L¨uroth’s theorem (which sometimes

used to be included in basic graduate algebra courses) says that a unirational

curve is rational, that is, a subfield of k(X) not equal to k is a pure transcendental

extension of k. It was proved by Castelnuovo that when k has characteristic zero every

unirational surface is rational. Only in the seventies was it shown that this is not

true for three dimensional varieties (Artin, Mumford, Clemens, Griffiths, Manin,...).

When k has characteristic p _= 0, Zariski showed that there exist nonrational unirational

surfaces, and P. Blass (UM thesis 1977) showed that there exist infinitely many

surfaces V , no two birationally equivalent, such that k(Xp, Y p) ⊂ k(V ) ⊂ k(X, Y ).

Aside 4.28. Note that, if V is irreducible, then

dimV = min

P

dimTP (V )

This formula can be useful in computing the dimension of a variety.

Etale neighbourhoods. Recall that a regular map α: W → V is said to be ´etale at

a nonsingular point P of W if the map (dα)P : TP (W) → Tα(P)(V ) is an isomorphism.

Let P be a nonsingular point on a variety V of dimension d. A local system of

parameters at P is a family {f1, . . . , fd} of germs of regular functions at P generating

the maximal ideal nP ⊂ OP . Equivalent conditions:t he images of f1, . . . , fd in nP /n2

P

generate it as a k-vector space (see 4.16); or (df1)P , . . . , (dfd)P is a basis for dual space

to TP (V ).

Proposition 4.29. Let {f1, . . . , fd} be a local system of parameters at a nonsingular

point P of V . Then there is a nonsingular open neighbourhood U of P

such that f1, f2, . . . , fd are represented by pairs ( ˜ f1, U), . . . , ( ˜ fd, U) and the map

( ˜ f1, . . . , ˜ fd) : U → Ad is ґetale.

Proof. Obviously, the fi are represented by regular functions ˜ fi defined on a

single open neighbourhood U_ of P, which, because of (4.21), we can choose to be

nonsingular. The map α = ( ˜ f1, . . . , ˜ fd) : U_ → Ad is ´etale at P, because the dual map

to (dα)a is (dXi)0 _→ (d ˜ fi)a. The next lemma then shows that α is ´etale on an open

neighbourhood U of P.

Lemma 4.30. Let W and V be nonsingular varieties. If α: W → V is ґetale at P,

then it is ґetale at all points in an open neighbourhood of P.

Proof. The hypotheses imply that W and V have the same dimension d, and

that their tangent spaces all have dimension d. We may assume W and V to

be affine, say W ⊂ Am and V ⊂ An, and that α is given by polynomials

P1(X1, . . . ,Xm), . . . , Pn(X1, . . . ,Xm). Then (dα)a : Ta(Am) → Tα(a)(An) is a linear

map with matrix

 

∂Pi

∂Xj

(a)

_

, and α is not ´etale at a if and only if the kernel of

this map contains a nonzero vector in the subspace Ta(V ) of Ta(An). Let f1, . . . , fr

generate I(W). Then α is not ´etale at a if and only if the matrix

_

∂fi

∂Xj

(a)

∂Pi

∂Xj

(a)

_

has rank less than m. This is a polynomial condition on a, and so it fails on a closed

subset of W, which doesn’t contain P.

Algebraic Geometry: 4. Local Study 75

Let V be a nonsingular variety, and let P ∈ V. An ґetale neighbourhood of a point

P of V is pair (Q, π : U → V ) with π an ´etale map from a nonsingular variety U to

V and Q a point of U such that π(Q) = P.

Corollary 4.31. Let V be a nonsingular variety of dimension d, and let P ∈ V .

There is an open Zariski neighbourhood U of P and a map π : U → Ad realizing (P, U)

as an ґetale neighbourhood of (0, . . . , 0) ∈ Ad.

Proof. This is a restatement of the Proposition.

Aside 4.32. Note the analogy with the definition of a differentiable manifold:

every point P on nonsingular variety of dimension d has an open neighbourhood

that is also a “neighbourhood” of the origin in Ad. There is a “topology” on algebraic

varieties for which the “open neighbourhoods” of a point are the ´etale neighbourhoods.

Relative to this “topology”, any two nonsingular varieties are locally isomorphic (this

is not true for the Zariski topology). The “topology” is called the ґetale topology —

see my notes Lectures on Etale Cohomology.

Dual numbers and derivations. In general, if A is a k-algebra and M is an Amodule,

then a k-derivation is a map D: A → M such that

(a) D(c) = 0 for all c ∈ k;

(b) D(a + b) = D(a) + D(b);

(c) D(a · b) = a · Db + b · Da (Leibniz rule).

Note that the conditions imply that D is k-linear (but not A-linear). We write

Derk(A,M) for the space of all k-derivations A → M.

For example, the map f _→ (df )P

df = f−f(P) mod n2

P is a k-derivation OP → nP /n2

P .

Proposition 4.33. There are canonical isomorphisms

Derk(OP, k)

≈→

Homk-lin(nP /n2

P , k)

≈→

TP (V ).

Proof. Note that, as a k-vector space,

OP = k ⊕ nP, f↔ (f(P), f − f(P)).

A derivation D: OP → k is zero on k and on n2

P (Leibniz’s rule). It therefore defines

a linear map nP /n2

P

→ k, and all such linear maps arise in this way, by composition

OP

f_→→(df)P nP /n2

P

→ k.

The ring of dual numbers is k[ε] = k[X]/(X2), ε = X mod X2. As a k-vector

space it has a basis {1, ε}.

Proposition 4.34. The tangent space

TP (V) = Hom(OP, k[ε]) (local homomorphisms of local k-algebras).

76 Algebraic Geometry: 4. Local Study

Proof. Let α: OP → k[ε] be a local homomorphism of k-algebras, and write

α(a) = a0 + Dα(a)ε. Because α is a homomorphism of k-algebras, a _→ a0 is the

quotient map OP →OP /m = k. We have

α(ab) = (ab)0 + Dα(ab)ε, and

α(a)α(b) = (a0 + Dα(a)ε)(b0 + Dα(b)ε) = a0b0 + (a0Dα(b) + b0Dα(a))ε.

On comparing these expressions, we see that Dα satisfies Leibniz’s rule, and therefore

is a k-derivation OP → k. All such derivations arise in this way.

For an affine variety V and a k-algebra A (not necessarily an affine k-algebra), we

define V (A), the set of points of V with coordinates in A, to be Homk-alg(k[V ], A).

For example, if V = V (a) ⊂ An, then

V (A) = {(a1, . . . , an) ∈ An | f(a1, . . . , an) = 0 all f ∈ a}.

Consider an α ∈ V (k[ε]), i.e., a k-algebra homomorphism α: k[V ] → k[ε]. The

composite k[V ] → k[ε] → k is a point P of V , and

mP = Ker(k[V ] → k[ε] → k) = α−1((ε)).

Therefore elements of k[V] not in mP map to units in k[ε], and so α extends to

a homomorphism α_ : OP → k[ε]. By construction, this is a local homomorphism

of local k-algebras, and every such homomorphism arises in this way. In this way

we get a one-to-one correspondence between the local homomorphisms of k-algebras

OP → k[ε] and the set

{P_ ∈ V (k[ε]) | P_ _→ P under the map V (k[ε]) → V (k)}.

This gives us a new interpretation of the tangent space at P.

Consider, for example, V = V (a) ⊂ An, a a radical ideal in k[X1, . . . ,Xn], and let

a ∈ V . In this case, it is possible to show directly that

Ta(V ) = {a

_ ∈ V (k[ε]) | a

_

maps to a under V (k[ε]) → V (k)}

Note that when we write a polynomial F(X1, . . . ,Xn) in terms of the variables Xi−ai,

we obtain a formula (trivial Taylor formula)

F(X1, . . . ,Xn) = F(a1, . . . , an) +

_ ∂F

∂Xi

____

a

(Xi − ai) + R

with R a finite sum of products of at least two terms (Xi − ai). Now let a ∈ kn be a

point on V , and consider the condition for a + εb ∈ k[ε]n to be a point on V. When

we substitute ai + εbi for Xi in the above formula and take F ∈ a, we obtain:

F(a1 + εb1, . . . , an + εbn) = ε(

_ ∂F

∂Xi

____

a

bi).

Consequently, (a1 + εb1, . . . , an + εbn) lies on V if and only if (b1, . . . , bn) ∈ Ta(V )

(original definition p68).

Geometrically, we can think of a point of V with coordinates in k[ε] as being a

point of V with coordinates in k (the image of the point under V (k[ε]) → V (k))

together with a “direction”

Algebraic Geometry: 4. Local Study 77

Remark 4.35. The description of the tangent space in terms of dual numbers is

particularly convenient when our variety is given to us in terms of its points functor.

For example, let Mn be the set of n × n matrices, and let I be the identity matrix.

Write e for I when it is to be regarded as the identity element of GLn. Thenwe have

Te(GLn) = {I + εA | A ∈ Mn} ≈ Mn;

Te(SLn) = {I + εA | det(I + εA) = I} = {I + εA | trace(A) = 0}.

Assume the characteristic _= 2, and let On be orthogonal group:

On = {A ∈ GLn | AAtr = I}.

(tr=transpose). This is the group of matrices preserving the quadratic form X2

1 +

· · · + X2

n. Then det: On → {±1} is a homomorphism, and the special orthogonal

group SOn is defined to be the kernel of this map. We have

Te(On) = Te(SOn)

= {I + εA ∈ Mn | (I + εA)(I + εA)tr = I}

= {I + εA ∈ Mn | A is skew-symmetric}.

Note that, because an algebraic group is nonsingular, dimTe(G) = dimG — this

gives a very convenient way of computing the dimension of an algebraic group.

On the tangent space Te(GLn) = Mn of GLn, there is a bracket operation

[M,N] df = MN −NM

which makes Te(GLn) into a Lie algebra. For any closed algbraic subgroup G of GLn,

Te(G) is stable under the bracket operation on Te(GLn) and is a sub-Lie-algebra of

Mn, which we denote Lie(G). The Lie algebra structure on Lie(G) is independent of

the embedding of G into GLn (in fact, it has an intrinsic definition), and G _→ Lie(G)

is a functor from the category of linear algebraic groups to that of Lie algebras.

This functor is not fully faithful, for example, any ´etale homomorphism G → G_

will define an isomorphism Lie(G) → Lie(G_), but is nevertheless very useful.

Assume k has characteristic zero. A connected algebraic group G is said to be

semisimple if it has no closed connected solvable normal subgroup (except {e}). Such

a group G may have a finite nontrivial centre Z(G), and we call two semisimple

groups G and G_ locally isomorphic if G/Z(G) ≈ G_/Z(G_). For example, SLn is

semisimple, with centre μn, the set of diagonal matrices diag(ζ, . . . , ζ), ζn = 1, and

SLn /μn = PSLn. A Lie algebra is semisimple if it has no commutative ideal (except

{0}). One can prove that

G is semisimple ⇐⇒ Lie(G) is semisimple,

and the map G _→ Lie(G) defines a one-to-one correspondence between the set of local

isomorphism classes of semisimple algebraic groups and the set of isomorphism classes

of Lie algebras. The classification of semisimple algebraic groups can be deduced

from that of semisimple Lie algebras and a study of the finite coverings of semisimple

algebraic groups — this is quite similar to the relation between Lie groups and Lie

algebras.

78 Algebraic Geometry: 4. Local Study

Tangent cones. In this subsection, I assume familiarity with parts of Atiyah and

MacDonald 1969, Chapters 11, 12.

Let V = V (a) ⊂ km, a = rad(a), and let P = (0, . . . , 0) ∈ V . Define a∗ to be the

ideal generated by the polynomials F∗ for F ∈ a, where F∗ is the leading form of F

(see p66). The geometric tangent cone at P, CP (V ) is V (a∗), and the tangent cone

is the pair (V (a∗), k[X1, . . . ,Xn]/a∗). Obviously, CP (V ) ⊂ TP (V ).

Computing the tangent cone. If a is principal, say a = (F), then a∗ = (F∗), but if

a = (F1, . . . , Fr), then it need not be true that a∗ = (F1∗, . . . , Fr∗). Consider for

example a = (XY,XZ +Z(Y 2−Z2)). One can show that this is a radical ideal either

by asking Macaulay (assuming you believe Macaulay), or by following the method

suggested in Cox et al. 1992, p474, prob 3 to show that it is an intersection of prime

ideals. Since

Y Z(Y 2 − Z2) = Y · (XZ + Z(Y 2 − Z2)) − Z · (XY ) ∈ a

and is homogeneous, it is in a∗, but it is not in the ideal generated by XY , XZ. In

fact, a∗ is the ideal generated by

XY, XZ, Y Z(Y 2 − Z2).

This raises the following question:gi ven a set of generators for an ideal a, how do

you find a set of generators for a∗? There is an algorithm for this in Cox et al. 1992,

p467. Let a be an ideal (not necessarily radical) such that V = V (a), and assume

the origin is in V . Introduce an extra variable T such that T “>” the remaining

variables. Make each generator of a homogeneous by multiplying its monomials by

appropriate (small) powers of T , and find a Gr¨obner basis for the ideal generated by

these homogeneous polynomials. Remove T from the elements of the basis, and then

the polynomials you get generate a∗.

Intrinsic definition of the tangent cone. Let A be a local ring with maximal ideal n.

The associated graded ring is

gr(A) = ⊕ni/ni+1.

Note that if A = Bm and n = mA, then gr(A) = ⊕mi/mi+1 (because of (4.13)).

Proposition 4.36. The map k[X1, . . . ,Xm]/a∗ → gr(OP ) sending the class of Xi

in k[X1, . . . ,Xm]/a∗ to the class of Xi in gr(OP ) is an isomorphism.

Proof. Let m be the maximal ideal in k[X1, . . . ,Xm]/a corresponding to P. Then

gr(OP) =

_

mi/mi+1

=

_

(X1, . . . ,Xm)i/(X1, . . . ,Xm)i+1 + a ∩ (X1, . . . ,Xm)i

=

_

(X1, . . . ,Xm)i/(X1, . . . ,Xm)i+1 + ai

where ai is the homogeneous piece of a∗ of degree i (that is, the subspace of a∗

consisting of homogeneous polynomials of degree i). But

(X1, . . . ,Xm)i/(X1, . . . ,Xm)i+1 + ai = ith homogeneous piece of k[X1, . . . ,Xm]/a∗.

79

For a general variety V and P ∈ V , we define the geometric tangent cone CP (V )

of V at P to be Specm(gr(OP )red), where gr(OP )red is the quotient of gr(OP) by its

nilradical.

Recall (Atiyah and MacDonald 1969, 11.21) that dim(A) = dim(gr(A)). Therefore

the dimension of the geometric tangent cone at P is the same as the dimension of V

(in contrast to the dimension of the tangent space).

Recall (ibid., 11.22) that gr(OP ) is a polynomial ring in d variables (d = dimV )

if and only if OP is regular. Therefore, P is nonsingular if and only if gr(OP) is a

polynomial ring in d variables, in which case CP (V ) = TP (V ).

Using tangent cones, we can extend the notion of an ´etale morphism to singular

varieties. Obviously, a regular map α: V → W induces a homomorphism gr(Oα(P)) →

gr(OP ). We say that α is ґetale at P if this is an isomorphism. Note that then there

is an isomorphism of the geometric tangent cones CP (V ) → Cα(P)(W), but this map

may be an isomorphism without α being ´etale at P. Roughly speaking, to be ´etale

at P, we need the map on geometric tangent cones to be an isomorphism and to

preserve the “multiplicities” of the components.

It is a fairly elementary result that a local homomorphism of local rings α: A → B

induces an isomorphism on the graded rings if and only if it induces an isomorphism on

the completions. Thus α: V → W is ´etale at P if and only if the map is Oˆα(P) → OˆP

an isomorphism. Hence (4.29) shows that the choice of a local system of parameters

f1, . . . , fd at a nonsingular point P determines an isomorphism OˆP → k[[X1, . . . ,Xd]].

We can rewrite this as follows:let t1, . . . , td be a local system of parameters at a

nonsingular point P; then there is a canonical isomorphism OˆP → k[[t1, . . . , td]]. For

f ∈ OˆP , the image of f ∈ k[[t1, . . . , td]] can be regarded as the Taylor series of f.

For example, let V = A1, and let P be the point a. Then t = X − a is a local

parameter at a, OP consists of quotients f(X) = g(X)/h(X) with h(a) _= 0, and the

coefficients of the Taylor expansion

_

n≥0 an(X −a)n of f(X) can be computed as in

elementary calculus courses: an = f(n)(a)/n!.

80