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This is the Past Paper of Mathematical Tripos which includes Kinetic Theory, Transport Equation, Jacobian Matrix, Positive Constants, Explicit Formula, Characteristics Map, Global Solution, Dispersion Estimate, Expression of Constant etc. Key important points are: Commutative Algebra, Two Nonzero Rings, Normal Domain, Ring of Invariants, Morphism of Affine Schemes, Isomorphism Classes, Algebra Homomorphism, Linear Algebra, Finite Flat Morphism, Hypersurface
Typology: Exams
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Wednesday, 6 June, 2012 1:30 pm to 4:30 pm
Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. All rings are understood to be commutative, unless stated otherwise.
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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(a) Let R be a reduced ring such that Spec(R) is not empty and not connected (as a topological space). Show that R is isomorphic to the product of two nonzero rings.
(b) Let R be a normal domain. Let G be a finite group which acts on R by automorphisms. Show that the ring of invariants RG^ = {f โ R : g(f ) = f for all g โ G} is normal.
Let f : A^1 R โ A^1 R be the morphism of affine schemes over the real numbers R defined by x 7 โ x^4. For each closed point p in A^1 R, compute the number of irreducible components of the closed subscheme f โ^1 (p). How many isomorphism classes of affine schemes over R arise as f โ^1 (p) for closed points p in A^1 R?
Show that any prime ideal in C[x, y, z] of codimension r can be generated by r elements if r = 1 or r = 3. [You may use results from the course.]
On the other hand, show that the kernel of the C-algebra homomorphism C[x, y, z] โ C[t] given by x 7 โ t^3 , y 7 โ t^4 , z 7 โ t^5 is a codimension-2 prime ideal in C[x, y, z] that cannot be generated by 2 elements.
(a) Show that every vector space V over a field k is free as a k-module. [Give a complete proof, without quoting any results from linear algebra. Note that V is not assumed to be finite-dimensional.]
(b) Let f be an irreducible polynomial in k[x 1 ,... , xn], for a field k and a positive integer n. Show that the hypersurface X = {f = 0} โ Ank admits a finite flat morphism to affine (n โ 1)-space over k.
Part III, Paper 4
