Definitions/Theorems/Propositions Flashcards
What is an affine algebraic variety?
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}
What does I mean? where f1,…fm ∈ K where K is a field
I is an ideal of K and is generated by f1,….,fm
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?
V(I) = {(a1,..,an) ∈ K^n | fsubi(a1,…,an) = 0 for all i ∈ [1,m]}
What is a Noetherian Ring?
- 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
What is Hilberts Basis Theorem?
If K is a field, then the polynomial ring K[x1, x2, . . . , xn] is Noetherian for any n ≥ 0.
How can the union of aav’s be written?
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
How can the intersection of aav’s be written?
Wi = V(Ii) be aav’s in K^n. Then ∩ Wi = V(sum (Ii))
This is also an aav
If W1 = V(I1) ⊆ K^m, W2 = V(I2) ⊆ K^n, what does this imply?
W1 x W2 ⊆ K^m x K^n ∼= K^(n+m)
What is an affine subspace?
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 .
If U = u0+W is a non-empty affine subspace in a vector space V, what does this imply?
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
What is the dimension of an affine space?
The dimension of an affine space U = u0 + W is defined to be dim W. (Sometimes it is convenient to define dim ∅ = −1.)
Let K be a field. What is K^n considered to be?
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.
Let V1, V2 be vector spaces. What is an affine map
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.
Let V1, V2 be vector spaces and let Φ : V1 → V2 be an
affine map. What is the image of Φ(U1)
For any affine subspace U1 ⊆ V1, the image Φ(U1) is also an affine subspace
of V2.
Let V1, V2 be vector spaces and let Φ : V1 → V2 be an
affine map. What is the preimage Φ−1
(U2) ?
For any affine subspace U2 ⊆ V2, the preimage Φ−1
(U2) is also an affine
subspace of V1.
Let X, Y be subsets of a vector space V. What is it to say that X and Y are affine equivalent?
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
Let X ⊆ A^n. The ideal of X is defined as..?
I(X)={f ∈ K[x1, x2, . . . , xn] | f(a1, a2, . . . , an) = 0, ∀(a1, a2, . . . ,an) ∈ X}
If J is an ideal of K[x1, x2, . . . , xn], then… ?
J ⊆ I(V(J)), where V(J) is the affine algebraic variety
A field K is algebraically closed iff?
if and only if every polynomial of degree at least 1 in K[x] has a root in K.
Let R be a (commutative) ring. Let I /R be an ideal. The radical of I, denoted by √I or rad I, is… ?
√I = {x ∈ R | ∃n such that x^n ∈ I}, I is called a radical ideal if and only if √I = I
What are the three Hilbert’s Nullstellensatz properties?
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].
An affine algebraic variety V is reducible IFF?
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.
Every affine algebraic variety V can be decomposed into a union such that..?
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.
Let R be a commutative ring. An ideal I of R is called a prime ideal if and only if … ?
if and only if ab ∈ I implies a ∈ I or b ∈ I.
An affine algebraic variety W is irreducible iff?
if and only if I(W) is prime.
Let K be a field and let V ⊂ K^n be an affine algebraic variety. What is a subvariety of V?
W is a subvariety of V if and only if W ⊆ V and W is also an affine algebraicvariety.
Aline ` given in parametric form as {P + tv | t ∈ K}, (v ∈ Kn \ 0), is said to
be tangent to V at P
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.
The tangent space to V at P
is the union of
the tangent lines to V at P and the point P. Tp V
Theorem 3.1
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
dimension of V , denoted by dim V
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
non-singular
The points at which
where dim Tp V = dim V
singular
the points where
dim TP V > dim V
The dimension of V , dim V
is the maximum of the dimensions
of the irreducible components of V
the local dimension of V
the maximum of the
dimension of the irreducible components of V containing P
singular if and only if
dim Tp V is greater than the local dimension
of V at P
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],
dim H = n − 1
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
Then dim Tp V = dim(mp /mp^2
)
Let ϕ : V → W be an isomorphism between affine algebraic varieties. Then dim Tp V = dim T_[ϕ(P)]W for every P ∈ V
therefore dim V =
dim W and ϕ(Sing V ) = Sing W
