The main purpose of the book is to provide material for an introductory postgraduate course of one or two semesters. The four chapters that make up the second part of the book relate to this idea.
The Algebra Geometry Lexicon
Maximal Ideals
Let's say A is an algebra over a field K. a) If A is an integral domain and algebraic over K, then A is a field. Since K[x] is a principal ideal domain, it follows that I= (f) where f ∈K[x] is irreducible, so is a maximal ideal. b) As a contradiction, assume that A has an element a1 that is not algebraic.
Jacobson Rings
There we also find the following: If α is a homomorphism that makes A an R-algebra, then for every m ∈ Specmax(A) the original image α−1(m) is also maximal. A typical example of a non-Jacobson ring is the formal power ring K[[x]] over the field K (see Exercise 1.2).
Coordinate Rings
The coordinate ring is sometimes also called the ring of regular functions on X. a) Every element of the coordinate ring K[X] of an affine variety is a class f +I with f ∈ K[x1,. Indeed, in the proof we have given, X depends on the choice of the generators of A.
The Noether and Artin Property for Rings and Modules
Since there are infinitely many strictly decreasing chains of subsets of X, there are also infinitely many strictly increasing chains of ideals in R. 4) The ringsZandK[x] discussed above are not Artinian. We need the following definition to advance the theory. a) The product of I and M is defined as the abelian group generated by all products of elements of I and elements of M. B).
Noetherian Rings and Modules
Every polynomial in A defines a function Kn → K. Loosely speaking, this means that S has the same point-separating capabilities as A.). a) Show that if S⊆A generates A as an algebra, then S is separable. Give reasons for your answers. a) Every finitely generated module over an Artin ring is Artin.
Affine Varieties
The Zariski topology is therefore the coarsest topology for which single elements (i.e. sets with one element) are closed. This illustrates how coarser the Zariski topology is compared to the regular topology on Ror C.
Spectra
We have a pair of inverse bijections between the set of radical ideals ofR and the set of closed subsets ofSpec(R), given byVSpec(R)andIR. It may also be interesting to note that a ringRi is a Jacobson ring if and only if for every closed subset Y ⊆ Spec(R) we say that Specmax(R)∩Y is dense in Y. Recall that a subset in a topological space is said to be compact if the closure is the entire space.) In fact, this is nothing but a translation of the Jacobson property.
Noetherian and Irreducible Spaces
However, the following two theorems show that the situation is much better if we consider spaces with the Zariski topology. Assume that R is noetherian and Spec(R) = Specmax(R), so we need to show that R is artinian.
True Geometry: Affine Varieties
The second part is more general and links algebraic objects (such as rings) to objects that are geometric in a more abstract sense (such as spectra of rings). with the ring of regular functions onX. By assigning an affineK manifold X its coordinate ring, a map is created. affineK manifolds −→ reduced affineK algebras. Conversely, every reduced affine K-algebra is isomorphic to the coordinate ring of an affine K-manifold, which is unique up to isomorphism.
Abstract Geometry: Spectra
We should mention that some parts of the lexicon remain intact if we reject the hypothesis that K is algebraically closed. This is expressed by saying that Spec(R) together with fi is acoproduct in the category of ring spectra. b) Spec(R) is the disjoint concatenation of thefi images.
Dimension
Nakayama’s Lemma and the Principal Ideal Theorem
This is a generalization of implication (c) ⇒ (b) of Theorem 5.13 (which talks about polynomial rings) to arbitrary Noetherian rings. As in the proof of Theorem 7.4, we can assume that R is a local ring with a maximal ideal P. So P is a prime ideal that is minimal over Q+ (a1), and since P is a unique maximal ideal of R, it is the only prime ideal over Q+ (a1).
In the case that R is an affine domain, Theorem 7.5 translates into a statement about the dimension, which is given in Theorem 8.25.
The Dimension of Fibers
Readers can already take a look at inequality (10.8) in Corollary10.6 on page 152, which provides a translation of Theorem7.12 into geometric terms. This exercise provides a more abstract description of the fiber ring. a) Show that the fiber ring S[P] is the extension of ϕ and ψ. Let R 6={0} be a Noetherian ring and R[[x]] the formal power series over R. The proof can be divided into the following steps.
Define the “locus of freedom”, ie. setXfree⊆Spec(R) of all P ∈Spec(R) such that (R\P)−1S is free as an RP-module.
Integral Closure
The elements are integral over R. b) The subalgebra R[s] ⊆ S generated by s is ultimately generated as an R-module. Then the following statements are equivalent. a) All he are integral over R. c) S is finitely generated as anR-module. If T is integral over S and S is integral over R, then T is integral over R. By Lemma 8.3, S0[t] is finitely generated as an S0-module, and by Theorem 8.4, S0 is finitely generated as an R-module.
It follows that S0[t] is finitely generated as an R-module, so applying Lemma8.3 again shows that it is integral over R. Corollary8.5 requests the following definition.
Lying Over, Going Up and Going Down
But what we need to be able to do this is the subtraction property which was discussed in Section 7.2 (see page 95). Let R⊆S be an integral extension of rings such that the descent down holds for the inclusion R ,→S. Unfortunately, reduction does not always hold for integral ring extensions, as Exercise 8.8 shows.
Let N be a field with characteristic p ≥ 0 and let K ⊆ N be a subfield such that N is finite and normal over K (see Lang [33, Ch. VII, Theorem 3.3] for the definition of a normal field extension) .
Noether Normalization
In fact, Theorem 8.19 tells us that for an affine variety X of dimension over a field K there is a morphism. In the case of the largest ideal m∈Specmax(K[X]) belonging to a point x∈X of the affine variety, it says that the height m is equal to the largest dimension of the irreducible component X containing x. In particular, if A is an affine domain or, more generally, equidimensional, then all maximal ideals have ht(m) = dim(A).
Let A be an affine domain or, more generally, an equidimensional affine algebra, and let I= (a1, . . . , an)⊆Abe an ideal generated by nelements.
Computational Methods
Buchberger’s Algorithm
In general, it depends on the choice of monomial ordering whether a given generating subset G⊆I is a Gr¨obner basis or not. Now that we have proven Buchberger's criterion, we are ready to present an algorithm for calculating Gröbner bases. Reduced Gröbner bases are uniquely determined by the ideal I they generate and of course by the choice of monomial ordering (see Exercise 9.4).
Obviously, this requires extending the definitions of the monomial order and the Gr¨obner basis.
First Application: Elimination Ideals
The following theorem tells us how elimination ideals can be calculated using Gr¨obner bases. According to Theorem 9.16, all I{xk,..,xn} can be computed from a single lexicographic Gr¨obner basis of I. It answers the question of the meaning of the part of a Gröbner basis that is 'thrown away' when calculating an elimination ideal.
This provides the desired test for radical membership, since the condition 1∈J can be tested by computing a Gr¨obner basis of J. Solution on page 271) 9.7 (Testing Affine Algebras for Dimension Zero).
The Generic Freeness Lemma
Before we explain the algorithm, we derive the 'existence version' of the generic lemma of freedom. Problem 10.3 contains a surprising application of the generic freedom lemma: A special case of this application says that a subalgebra of an affine K-domain has a localization that is again an affine K-domain. This contradicts the fact that all g ∈ Gy have normal form w.r.t.Gx, which demonstrates the termination of the algorithm.
By induction on the recursion depth, we can assume that for each f ∈M the recursive call of the algorithm computes the image of ϕ∗f, where ϕf:K[x1,.
Fiber Dimension and Constructible Sets
Then the image im(ϕ∗) of the induced mapping ϕ∗: Spec(S)→Spec(R) is a constructive subset of Spec(R). By specializing Corollary 10.8 to the example of coordinate rings of affine varieties, we obtain that the image of the morphism f:X →Y of affine varieties over an algebraically closed field is constructible. The natural embedding example ϕ:Z→Q shows that the finite generation hypothesis cannot be dropped from Corollary 10.8: the image ϕ∗ consists only of the null ideal, and it is easy to check that it is not a constructive subset of Spec(Z ).
Corollary 10.8 and Exercise 10.7 imply that the image of a morphism has a subset that is open and dense in the image closure.
Application: Invariant Theory
According to a famous theorem by Rosenlicht (see Popov and Vinberg [44, Theorem 2.3] or Springer [49, Satz 2.2]), this behavior is universal: one can always restrict fiber of F is exactly one G-path (provided one chooses enough invariants to form F). Note that K(X)G is not always the field of fractions of K[X]G; but if so, the above equation implies equality in (10.11). From which of the following hypotheses does it follow that the image im(f) has a subset U that is open and closed in the image termination im(f). a) X and Y are affine varieties (over a field that need not be algebraically closed), andf is a morphism.
In the second section we show that the degree of the Hilbert polynomial is equal to the Krull dimension of A.
The Hilbert-Serre Theorem
The Hilbert series is then the power series whose coefficients are the dimensions of the pieces. It is tempting to define the Hilbert function and the Hilbert series of an affine algebra. Because the Hilbert function and the Hilbert series of the zero ideal depend on the number of n indefinite values, we will write them ashn(d) and Hn(t) respectively in this proof.
A consequence of the correctness of Algorithm 11.8 is that the Hilbert series HI(t) can be written as a rational function with (1−t)m+1 as denominator.
Hilbert Polynomials and Dimension
Exercise 12.5 gives a presentation of the associated graded ring of the coordinate ring of an affine manifold, localized at a point. Since m/Rais is the maximum ideal of R/Ra, hR/Ra(d) is the length of the module. We need the following statement, which is not part of the standard curriculum of an abstract algebra course.
The kernel of the Jacobi matrix moduloP can be interpreted as the tangent space at the point P.
Local Rings
The Length of a Module
In particular, a finite chain M0 $ M1 $· · · $Mn submodules of M is maximal if no further submodules can be added to the chain by insertion or addition at both ends. Let M be a module over the ring R. So, in particular, all maximal chains have the same length. In particular, R has finite length as a module over itself if and only if it is artinian.
If we take the largest chains of submodules N and M/N, raise the latter to M, and put these chains together, we get the largest chain of submodules M.
The Associated Graded Ring
In fact, we will interpret the function d7→length R/md+1. as the Hilbert function of the associated graded ring, which will be defined in the next section. as an Abelian group), so that Si·Sj⊆Si+j for all, j∈N0. First, we need to prove the following theorem, which is a consequence of the Artin-Rees lemma and Nakayama's lemma. Since any point in V(I) can be moved by changing the coordinates, we obtain the representation of the associated graded ring of the coordinate ring of the affine variety localized at the point.
If R is a coordinate ring of an affine variety localized at a point, gr(R) can be interpreted as a coordinate ring of a tangent cone at that point (see Exercise 12.5).
Basic Properties of Regular Local Rings
As previously mentioned, local rings serve for the study of the local behavior of a global object, such as an affine manifold. Before dealing with more examples, it is useful to establish the following regularity criterion in terms of the associated graded ring. Eacha ∈A can be written as a polynomial over K in the bi, and it follows thatπ(a) is a K-linear combination of theπ(bi).
By changing coordinates, one can also determine the location's associated graduated ring at other points.
The Jacobian Criterion
The main goal of this section is to prove the Jacobian criterion for the regularity of the local ring AP/I. The matrix that appears in the Jacobian criterion is constructed from the (formal) partial derivatives of polynomials that generate I. The singular points are those where the size of the tangent space exceeds the lower bound.
The following lemma gives an interpretation of the rank of the Jacobian determinant off1,. fmmodulo prime ideal in terms of the ideal generated by fi. xn] prime ideal containing I. b) If L is a separable extension of the field K, then the equality in (a) holds.
Regular Rings and Normal Rings
Multiplicative Ideal Theory
Dedekind Domains
