Morphism of Affine Varieties Quiz

Question 1

Which algebraic structure plays a key role in defining morphisms between affine varieties?

Accepted Answer

Ring homomorphisms

Answer

Topological mappings

Answer

Group homomorphisms

Answer

Linear transformations

Question 2

If φ: X → Y is a morphism between affine varieties, what is the direction of the induced ring homomorphism k[Y] → k[X]?

Accepted Answer

From k[Y] to k[X]

Answer

No specific direction

Answer

From k[X] to k[Z]

Answer

From k[X] to k[Y]

Question 3

In the context of affine varieties, what is a regular function?

Accepted Answer

A polynomial function

Answer

An analytic function

Answer

Any continuous function

Answer

A holomorphic function

Question 4

If two affine varieties X and Y are isomorphic, what is the relationship between their coordinate rings k[X] and k[Y]?

Accepted Answer

They are ring isomorphic

Answer

They are only similar

Answer

They have no relation

Answer

They may be homomorphic

Question 5

In the category of affine varieties, what is called an automorphism?

Accepted Answer

An isomorphism with itself

Answer

Any homomorphism

Answer

An exomorphism

Answer

An endomorphism

Question 6

What type of ring is the coordinate ring of an affine variety?

Accepted Answer

A finitely generated k-algebra

Answer

A principal ideal domain

Answer

A field

Answer

A unique factorization domain

Question 7

If f: X → Y is a closed immersion, what can be said about the induced ring homomorphism k[Y] → k[X]?

Accepted Answer

It is surjective

Answer

It is bijective

Answer

It is injective

Answer

It is not surjective

Question 8

What is true about the induced ring homomorphism for an open immersion?

Accepted Answer

It is injective

Answer

It is never a homomorphism

Answer

It is always zero

Answer

It is surjective

Question 9

According to Nullstellensatz, for a prime ideal I ⊆ k[X], what does I(V(I)) equal?

Accepted Answer

I itself

Answer

Zero ideal

Answer

The radical of I

Answer

The whole ring

Question 10

How are regular functions defined in the structure sheaf of an affine variety?

Accepted Answer

Locally polynomial functions

Answer

Undefined functions

Answer

All continuous functions

Answer

Only global polynomial functions

Morphism of Affine Varieties

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