Solutions to the exercises of Introduction to Commutative Algebra, Atiyah, Chapter 8.
Suppose is an isolated primary component. Then is an Artin local ring, hence if is its maximal ideal we have for all sufficiently large , hence for all large .
If is an embedded primary component, then is not Artinian, hence the powers are all distinct, and so the are all distinct. Hence in the given primary decomposition we can replace by any of the infinite set of -primary ideals where , and so there are infinitely many minimal primary decompositions of which differ only in the -component.
is discrete and finite;
Proof. i) ii) iii) is clear.
is a finite -algebra.
If is Artinian, then is a finite direct product of Artin local rings by (8.7). Clearly each multiplicand is also a finitely generated -algebra. So it reduces to the case that is local. Let be the maximal ideal of , then is a finite extension of by the Hilbert Nullstellensatz.
If , then ; each is a -vector space of finite dimension since is Noetherian. Hence is finite. Therefore is also finite.
Every ideals of is also a -vector subspace of . Since is a finite-dimensional -vector space, it satisfies d.c.c on ideals.
the fibres of are discrete subspaces of ;
for each prime ideal of , the ring is a finite -algebra ( is the residue field of );
the fibres of are finite.
Prove that i) ii) iii) iv).
If is finite, then also is . Therefore is a finite -algebra.
Let denote . The fibres of are . Since is of finite type, there is a surjective homomorphism . Tensoring with gives a surjective homomorphism . Therefore is a finitely generated -algebra, so Noetherian.
By Exercise 2, discrete if and only if is Artinian. By Exercise 3 it’s equivalent to that is a finite -algebra.
Under the same notation above, iii) implies that is Artinian. Hence is finite.
If is integral and the fibres of are finite, is necessarily finite?
Proof. Let and denote the coordinate ring of and respectively. Then can be considered as the subring of . By the Nullstellensatz, and . And if denotes the inclusion map, then by Chapter 5, Exercise 16, is the induced mapping of . Since is integral over , is finite. By Exercise 8.4, the fibres of are finite.