1. 1, 2, 3, 4 ...

...bend down and touch the floor. I have collected a bunch of what I think are instructive exercises.

1.1. Finite Rings. Give a structure theorem for finite rings of order p3. Is it possible to “easily” generalise this for pn for all n?

1.2. Hensel’s lemma. Use Hensel’s lemma to show that any matrix A over a field of characteristic 0 can be written as the sum A = S + N with S and N commuting with each other, N nilpotent and the minimal polynomial of S is separable (“has distinct roots”). Can you generalise this to characteristic p?

1.3. Affine schemes. Show that there cannot be an isomorphism between Spec(k[X,Y ]/(X2 - Y 3)) and Spec(k[T]) by considering these as functors on the category of finite dimensional k algebras.

1.4. Finite morphisms. Find a finite morphism Spec(k[X,Y ]/(XY (X +Y )-1)) --> Spec(k[T]). Your proof should work over any field k!

1.5. Closed morphisms. Show that the morphism Spec(k[X,Y ]) --> Spec(k[X]) is not closed.

1.6. Hilbert basis theorem. Consider the ideals defining circles with centres along points of the y-axis. What is a finite basis for the ideal defining the intersection of the corresponding closed subschemes?

1.7. Primary decomposition. Let Spec(R/I) be a closed subscheme of a scheme Spec(R). If I is contained in a component of Spec(R) then show that every element of I is a zero=divisor. What about the converse?

1.8. Segre embedding. Consider the natural transformations of functors Pn Pm --> Pnm+n+nm which, for each finite ring A, has the form

((a0,...,an);(b0,...,bm)) '--> (aibj)i=0,j=0

Show that this is a bijection onto the projective variety in Pnm+n+m defined by the equations ZijZkl = ZilZkj as i, k (respectively j, l) vary from 0 to n (respectively from 0 to m).

1.9. Empty projective schemes. Let R be a finitel generated graded ring which is a finite over the subring R0 of elements of degree 0. Show that Proj(R)(A) is empty for every finite ring A. What about the converse?

1.10. Dimension theory. Let R be a graded k-algebra with R0 = k such that the graded dimension of R is r. Let k[Z0,...,Zr] --> R be a graded homomorphism such that it induces a morphism Proj(R) --> Pr. Show that the homomorpihsm is finite.