Definitions/Theorems/Propositions

Question 1

What is an affine algebraic variety?

Answer 1

Let K be a field and I be an ideal of K.

V(I) = {(a1,a2,…an) ∈ K^n | f(a1,a2,….,an) = 0, for all f ∈ I}

Question 2

What does I mean? where f1,…fm ∈ K where K is a field

Answer 2

I is an ideal of K and is generated by f1,….,fm

Question 3

K is a field and let f1,…fm ∈ K[x1,…,xn] and I is an ideal of K generated by f1,….,fm. This implies what?

Answer 3

V(I) = {(a1,..,an) ∈ K^n | fsubi(a1,…,an) = 0 for all i ∈ [1,m]}

Question 4

What is a Noetherian Ring?

Answer 4

• All ideals are finitely generated
• There exists N such that In = IN for an increasing set of ideals I1 ⊆ I2 ⊆ ….. ⊆ In ⊆ I(n+1) ⊆ …. where n is greater than or equal to N

Question 5

What is Hilberts Basis Theorem?

Answer 5

If K is a field, then the polynomial ring K[x1, x2, . . . , xn] is Noetherian for any n ≥ 0.

Question 6

How can the union of aav’s be written?

Answer 6

Let W1 = (I1), W2 = V(I2), . . . , Wk = V(Ik) be affine
algebraic varieties in K^n. Then
W1 ∪ W2 ∪ . . . ∪ Wk = V(I1 ∩ I2 ∩ . . . ∩ Ik) = V(I1I2 . . . Ik)
This is also an aav

Question 7

How can the intersection of aav’s be written?

Answer 7

Wi = V(Ii) be aav’s in K^n. Then ∩ Wi = V(sum (Ii))

This is also an aav

Question 8

If W1 = V(I1) ⊆ K^m, W2 = V(I2) ⊆ K^n, what does this imply?

Answer 8

W1 x W2 ⊆ K^m x K^n ∼= K^(n+m)

Question 9

What is an affine subspace?

Answer 9

Let K be a field, and let V be a vector space over K. U ⊆ V is an affine subspace of V iff U = ∅ or U = u0 + W = {u0 + w | w ∈ W}, where u0 ∈ U and W is a linear subspace of V .

Question 10

If U = u0+W is a non-empty affine subspace in a vector space V, what does this imply?

Answer 10

U = u + W for any u ∈ U, in other words, u0 can be chosen to be an arbitrary element of U, but W is uniquely determined by U

Question 11

What is the dimension of an affine space?

Answer 11

The dimension of an affine space U = u0 + W is defined to be dim W. (Sometimes it is convenient to define dim ∅ = −1.)

Question 12

Let K be a field. What is K^n considered to be?

Answer 12

K^n, considered as an affine subspace of itself is called an n-dimensional affine space over K, and is denoted by A^n(K) or A^n if the field is understood.

Question 13

Let V1, V2 be vector spaces. What is an affine map

Answer 13

A function Φ : V1 → V2 which can be written in the form

Φ(x) = T(x) + b, where T : V1 → V2 is a linear transformation and b ∈ V2.

Question 14

Let V1, V2 be vector spaces and let Φ : V1 → V2 be an

affine map. What is the image of Φ(U1)

Answer 14

For any affine subspace U1 ⊆ V1, the image Φ(U1) is also an affine subspace
of V2.

Question 15

Let V1, V2 be vector spaces and let Φ : V1 → V2 be an
affine map. What is the preimage Φ−1
(U2) ?

Answer 15

For any affine subspace U2 ⊆ V2, the preimage Φ−1
(U2) is also an affine
subspace of V1.

Question 16

Let X, Y be subsets of a vector space V. What is it to say that X and Y are affine equivalent?

Answer 16

X and Y are affine equivalent if and only if there exist mutually inverse affine maps Φ, Ψ : V → V such that Φ(X) = Y and Ψ(Y ) = X

Question 17

Let X ⊆ A^n. The ideal of X is defined as..?

Answer 17

I(X)={f ∈ K[x1, x2, . . . , xn] | f(a1, a2, . . . , an) = 0, ∀(a1, a2, . . . ,an) ∈ X}

Question 18

If J is an ideal of K[x1, x2, . . . , xn], then… ?

Answer 18

J ⊆ I(V(J)), where V(J) is the affine algebraic variety

Question 19

A field K is algebraically closed iff?

Answer 19

if and only if every polynomial of degree at least 1 in K[x] has a root in K.

Question 20

Let R be a (commutative) ring. Let I /R be an ideal. The radical of I, denoted by √I or rad I, is… ?

Answer 20

√I = {x ∈ R | ∃n such that x^n ∈ I}, I is called a radical ideal if and only if √I = I

Question 21

What are the three Hilbert’s Nullstellensatz properties?

Answer 21

Let K be an algebraically closed
field.
(i) Every maximal ideal of K[x1, x2, . . . , xn] is of the form , where ai ∈ K, 1 ≤ i ≤ n.
(ii) If J is an ideal of K[x1, x2, . . . , xn and, J is not equal to K[x1, x2, . . . , xn], then V(J) is not equal to ∅
(iii) I(V(J)) = √J for any J ideal of K[x1, x2, . . . , xn].

Question 22

An affine algebraic variety V is reducible IFF?

Answer 22

If and only if it can be written as V = V1 ∪ V2 where V1 not equal to V which is not equal to V2. V1 and V2 are aav’s. If V is not reducible it is irreducible.

Question 23

Every affine algebraic variety V can be decomposed into a union such that..?

Answer 23

V = V1 ∪ V2 ∪ . . . ∪ Vk such that every Vi, 1 ≤ i ≤ k, is an irreducible affine algebraic variety and Vi is not ⊆ Vj
for i is not equal to j. The decomposition is unique
up to the ordering of the components.

Question 24

Let R be a commutative ring. An ideal I of R is called a prime ideal if and only if … ?

Answer 24

if and only if ab ∈ I implies a ∈ I or b ∈ I.

Question 25

An affine algebraic variety W is irreducible iff?

Answer 25

if and only if I(W) is prime.

Question 26

Let K be a field and let V ⊂ K^n be an affine algebraic variety. What is a subvariety of V?

Answer 26

W is a subvariety of V if and only if W ⊆ V and W is also an affine algebraicvariety.

Question 27

Aline ` given in parametric form as {P + tv | t ∈ K}, (v ∈ Kn \ 0), is said to be tangent to V at P

Answer 27

if and only if for every f ∈ I(V ), f(P + tv) has a zero | of multiplicity at least 2 at t = 0, or is identically 0.

Question 28

The tangent space to V at P

Answer 28

is the union of | the tangent lines to V at P and the point P. Tp V

Question 29

Theorem 3.1

Answer 29

Let V ⊆ A n be an affine algebraic variety. Let f1, f2, . . . , fr be a set of generators for I(V ). Let P ∈ V and let v = (v1, v2, . . . , vn) ∈ Kn \ {0}. The line ` = {P + tv | t ∈ K} is tangent to V at P if and only if JP v = 0, where J is the Jacobian matrix

Question 30

dimension of V , denoted by dim V

Answer 30

Let V be an irreducible affine algebraic variety. If V 6= ∅, there exists a proper subvariety of V (possibly ∅) such that dim TP V is constant outside this subvariety

Question 31

non-singular

Answer 31

The points at which | where dim Tp V = dim V

Question 32

singular

Answer 32

the points where | dim TP V > dim V

Question 33

The dimension of V , dim V

Answer 33

is the maximum of the dimensions | of the irreducible components of V

Question 34

the local dimension of V

Answer 34

the maximum of the | dimension of the irreducible components of V containing P

Question 35

singular if and only if

Answer 35

dim Tp V is greater than the local dimension | of V at P

Question 36

Let K be an algebraically closed field. Let H ⊂ A^n(K) be a hypersurface, that is, H = V(hfi) for some non-constant polynomial f ∈ K[x1, x2, . . . , xn],

Answer 36

dim H = n − 1

Question 37

4 Let V ⊆ A n be an affine algebraic variety and let P ∈ V . Let mP = {F ∈ K[V ] | F(P) = 0} be the maximal ideal of K[V ] corresponding to P

Answer 37

Then dim Tp V = dim(mp /mp^2 | )

Question 38

Let ϕ : V → W be an isomorphism between affine algebraic varieties. Then dim Tp V = dim T_[ϕ(P)]W for every P ∈ V

Answer 38

therefore dim V = | dim W and ϕ(Sing V ) = Sing W