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 ⁿ × ^{m nm+n+nm} which, for each finite ring A, has the form

Show that this is a bijection onto the projective variety in ^nm+n+m defined by the equations Z_ijZ_kl = Z_ilZ_kj 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,...,Z_r] R be a graded homomorphism such that it induces a morphism Proj(R) ^r. Show that the homomorpihsm is finite.