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This is the Final Exam of Mathematical Tripos which includes Lie Groups, Lie Algebras, and Their Representations, Homomorphism, Inner Derivations, Irreducible Root System, Real Vector Space, Weyl Group, Irreducible Summands, Space of Diagonal Matrices etc. Key important points are: Commutative Algebra, Artinian Ring, Injective Endomorphism, Zariski Topology, Closed Subsets, Subspace Topology, Correspondance Explicitly, Integral Domain, Krull Dimension, Group of Cartier Divisors
Typology: Exams
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Monday, 31 May, 2010 9:00 am to 12:00 pm
Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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Suppose that A is a ring. Define an Artinian A-module. What does it mean to say that A is Artinian?
Prove carefully that if A is an Artinian ring and M is a finitely generated A-module then M is an Artinian A-module.
Give an example of an Artinian ring A and a non-Artinian A-module M.
Show that if M is an Artinian A-module and f is an injective endomorphism of the A-module M , then f is surjective.
Suppose that A is a ring. Define the Zariski topology on Spec(A). Show that if A is a finitely generated C-algebra then there is a 1-1 correspondance between radical ideals of A and closed subsets of maxSpec(A) equipped with the subspace topology.
Describe this correspondance explicitly for A = C[x].
Suppose that A is a subring of an integral domain B. What does it mean to say that B is integral over A?
Define the Krull dimension of a ring. Show that if B is integral over A then the two rings have the same Krull dimension.
Suppose that K is an algebraic field extension of Q and O is its ring of integers. What is the Krull dimension of O? Justify your answer.
Suppose that A is an integral domain. Define Pic(A) the Picard group of A and Cart(A) the group of Cartier divisors of A. Carefully show that there is a natural exact sequence of abelian groups
1 → A×^ → Q(A)×^ → Cart(A) → Pic(A) → 0.
Part III, Paper 3
