Solutions to Atiyah-Macdonald, Chapter 1. Dave Karpuk. May 19, Exercise 1. Let x be a nilpotent element of a ring A. Show that 1+x is a unit of A. Deduce. Trial solutions to. Introduction to Commutative Algebra. ( & I.G. MacDonald) by M. Y.. This document was transferred to. Atiyah and Macdonald “provided exercises at the end of each chapter.” They and complete solutions are given at the end of the book.

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Then, if x is a non-zero element of A, we have x ax? Since A is Artinian, it will be of finite length as an A-module, and we conclude that it will be a finite k-algebra, as desired. We use induction on n to prove that the coe? This implies that Spec A is? Then, since M is?

In this case the statement is vacuously true. For the proof, just observe that in an arbitrary valuation ring, all?

The first equality is true because modding out by D has the effect of identifying the canonical images of the xL the distinction between elements whose sets of i thus cancelling P coordinates coincide they are different in Mi but not in Mi. Hence all fibres have one element; in particular, they are finite.

## Solutions to Atiyah and MacDonald’s Introduction to Commutative Algebra

Clearly, the above equations imply that b1b2. But then x bx? Now we apply induction to the di? In particular, the set of ideals that contract to p is? From exercises it follows that, if f: In this case, the generator xn is algebraic over k [x1.

Now, since since f is continuoushence we may replace Z by Z above. In particular, Since a? By the above argument, we deduce that any maximal chain i. Therefore, Bq is flat over Ap. We thus have the following: We deduce that the heights of p and p[x] are equal. Therefore S is minimal. Since each Amj is Noetherian, the extension Amj a of a in Amj is? Let p be a prime ideal of A.

If f is the product of all the denominators b, the x2x3. Therefore, G is necessarily empty, or E is open in Xas desired.

Given a ring A, we have the following: The exact sequences 1? For the proof, we will? Tensor products commute with direct limits 2.

In fact, the length of the maximal chains will be equal to the number r in the statement of the Noether Normalization Lemma, chapter 5, exercise We will proceed by induction on n. Hence A is a valuation ring the valuation ring of v of K. The map K A? If A is absolutely? This establishes the desired formula?

If the above conditions are fulfilled, then Spec A is compact since the definition of quasi-compactness in the book seems to coincide with the usual definition of compactness and Spec A is always quasi-compact and the proof of the last equivalence shows that Spec A is totally disconnected. With the above notation, we have the following: In macdomald, it is? In particular, the variety is defined by a finite number of polynomials.

### Solutions to Atiyah and MacDonald’s Introduction to Commutative Algebra – PDF Free Download

We deduce that a0 is a unit. Let p be a prime ideal of A. This implies that a fn au? By exercise 27, this condition is equivalent to absolute flatness. Hence the natural map is an isomorphism, as desired.

### Papaioannou A.-Solutions to Atiyah and MacDonald’s Introduction to Commutative Algebra ()_百度文库

If the interior of Z is empty, then Macdonalf But local freedom is equivalent to flatness over Noetherian rings, which yields the desired equivalence.

For assume that 1? Now the inductive hypothesis applies. If S were any saturated and multiplicatively closed subset of S that contained S, then its complement in A would contain at least one prime ideal that has 0 non-trivial intersection with S.

However, y y n? The converse is not true. Therefore, Sp 0 is primary.