However, any book with the goals of the present must include a selection of topics that reach further into deeper waters. The discussion of the basics of Galois theory is strongly influenced by the presentation in the own monograph.
Separable and regular extensions 363
Algebraic Spaces
Spec of a ring 405
Noetherian Rings and Modules
Primary decomposition 421
Nakayama's lemma 424
Indecomposable modules 439
Real Fields
Real zeros and homomorphisms
Definitions, dependence, and independence 465
Completions 468
Completions and valuations 486
Zeros of polynomials in complete fields 491
Duality 522
Representation of One Endomorphism 553
Representations 553
Decomposition over one endomorphism 556
Preliminaries, orthogonal sums 571
Symmetric forms, orthogonal bases 575
The spectral theorem (hermitian case) 581 7. The spectral theorem (symmetric case) 584
Basic properties 607
Tensor product of algebras 629
Symmetric products 635
Conditions defining semisimplicity 645
The density theorem 646
Semisimple rings 651
Simple rings 654
The Jacobson radical, base change, and tensor products 657
Representations and semisimplicity 663
Positive decomposition of the regular character 699
Supersolvable groups 702 10. Brauer's theorem 704
The Alternating Product
Fitting ideals 738
General Homology Theory
Complexes 761
Homotopies of morphisms of complexes 787
Appendix 2
- GIH -. Imf
Let S be a subset of G, and let N = Ns be the set of all elementsx EG such that xSx-t = S. Second, let G be the set of all maps Ta,b: R R such that Ta,b(X) = ax+b,witha=t=0 and barbit trees. Then G is a group subset.
1 GIN
GORENSTEIN, The Classification of Finite Simple Groups, Plenum Press, 1983
- The kernel is trivial. Thenfis an isomorphism of Z onto the cyclic subgroup
Conversely, any automorphism of G is mapped onto some generator of G. Then there exists a unique subgroup of G of order d. v) Let G1, G2 be cyclic of orders m, n respectively. Conversely, it is clear that mnZ is contained in the nucleus. so the kernel is mnZ. The image of Z A x B is surjective by the Chinese remainder theorem.
1 will provide an important specific application
- and (pml
If S is the set of letters Xl'..'xn, we say that Fab(S) is the free abelian group with free generatorsXl'.·.,xn. Given an abelian group A and a subgroup B, it is sometimes desirable to find the subgroup Csuch thatA = BEt>C.
Let A be a finite abelian group, B a subgroup, A A the dual
Let p be a prime instead of , and let p. be the family of normal subgroups of finite index equal to a power of p. The full Galois group of F'/F is the inverse limit with respect to subgroupsoffinite index, as in the above general situation.
Two sets Mor(A, B) and Mor(A', B') are disjoint unless A = A'
For each object A of CI there is a morphism id A E Mor(A, A) which acts as left and right identity for the elements of Mor(A, B) and
The CI category consists of a collection of Ob(Ci) objects; and for two objects A, BEOb(CI) the set Mor(A, B), which is called the set of morphisms of A into B; and for. A morphism fi is called an isomorphism if there exists a morphism g:B-+ A such that g0fandfog are identities in Mor(A, A) and Mor(B, B) or Determining the group of automorphisms in each category is one of the basic problems of the field of mathematics dealing with this category.
Such topological theories are not independent of the algebraic theories, because by functoriality an action of G on the manifold causes a. Next consider the case where C1 is the category of bebelic groups, which we can denote by Ab. Late Abeanabelse group and Gagroup.
A Universal objects
In general, given a family of objects {AJiel in C1, the product for that family consists of (P, {};}iel)', where P is an object in C1 and {};}iel is a family of morphisms. As an element of note, we will usually write A x B for the product of two objects in a category and n Ai for the product of any family in a. In the product and coproduct, we will say that the morphism h is the morphism caused by the family {gi}.
Then the coproduct of (S, x) and (S', x') exists in this category and can be constructed as follows. Then the triple «V, x), f}, f2) is the isoproduct for (S, x) and (S', x') in the category of point sets. Given two such rings A, B, one can form the tensor product, and there are homomorphisms of the natural ring A.
FREE GROUPS
If (S) (oras wealsosay,f) generates F, then it is clear that there exists at most one homomorphism t/JofF intoG that makes the following diagram. There exists a family and a family of groups {G;hEI such that, if g: S G is a map of S into a group G, and 9 generates G, then G is isomorphic to some G;. Let F be the subgroup iFo generated by the image i/o, and weletI simply beequalto10'viewed as the amap of Sinto F.
For each set S we select one free group, determined by S, and denote it by (F(s),ls) or simply F(S). SL2(F) therefore denotes the group of 2 x 2 matrices with components in a field F and a determinant equal to 1.
Let r be the set of group structures on S and for each YEr let
1 and Proposition 12.3 to a more explicit representation of the free group or coproduct, as the case may be.
EXERCISES
GIN G'IN',
The only partitions of S which are stable are the two partitions mentioned above
Let a finite group G act transitively and precisely on a set S with at least 2 elements and let H be an isotropic group of some elements s from S. All other isotropic groups are conjugates of H.) Prove the following:. Fiber products and co-products Pull-backs and push-out. a) Show that fiber products exist in the category of abelian groups. The maps Pt, P2 are projections onto the first and second factors. b) Prove that the backward withdrawal homomorphism is surjective.
Suppose G operates on asset S. Foreach i E I, suppose assubsetSiofS, and let sbeapointof S - l) Si. Suppose that for each 9E G; - {e}, we have. By fixed point of MonCwe means a complex number zsuchtwatM(z)= z. Assume that M has two distinct fixed points =1= 00. a) Show that there cannot be more than two fixed points and that these are fixed.
Rings
The multiplication is associative, and has a unit element
Therefore U satisfies all the axioms of a multiplicative group, and is called the group of units of A. If this is the case, then B itself is a ring, the laws of action in B are the same as. The additive unit is the constant map whose value is the additive unit of A, namely O.
Let M be an additive abelian group, and let A be the set End(M) of group homomorphisms of M within itself. We will now give examples of rings whose product is given by what is called convolution.
- below.)
Let R be the ring of algebraic integers in the number field K. For definitions see Chapter VII.). Let S be a subset of B that commutes with A; in other words, we have as= sa for all aEA and sES. For simplicity, it is common. write modp instead of modpZ and similarly write modn instead of . modnZ for any integer n.) Similarly, given an integer n> 1, units v.
This is proved by the ZjnZ End(A) isomorphism. then there exists k' such that kk' = 1 modn, so A has an inverse h and h is. For example, let Jln be the group of the nth root function in C. Then all automorphisms of fin are given by.
Let A be a principal entire ring. Then A is factorial
- MATSUMURA, Commutative Algebra, second edition, Benjamin-Cummings, New York, 1980
- MATSUMURA, Commutative Rings, Cambridge University Press, Cambridge, UK, 1986
Similarly, we shall later show that the ring of polynomials in one variable over a field is factorial, and one chooses representatives of the prime elements to be the irreducible polynomials with leading. Roughly speaking, regular local rings arise in the following context of algebraic or complex geometry. Consider the ring of regular functions near some point on a complex or algebraic manifold.
The ring R can be viewed as a ring of regular functions on the curve y2 =x3, which has a singularity at the origin, as you can see if you draw its real graph. This ring is a subring of the ring of all functions or all differentiable functions.
EXERC1SES
- Suppose that 1 :F 0 in A. Let S be a multiplicative subset of A not containing o
- Let f: A A' be a surjective homomorphism of rings, and assume that A is local,
- Le! A be a factorial ring and p a prime element. Show that the local ring A(p) is principal
- Let R be the ring of trigonometric polynomials as defined in the text. Show that
- Every ideal has a factorization as a product of prime ideals, uniquely determined
- M-.O is exact, we say that u is an epimorphism
- THE GROUP OF HOMOMORPHISMS
Let p be a prime number, and let A there is Z/prz (r=integer > 1). the group of units in A, that is, the group of integers prime to p, modulo pro Show that G is cyclic, except in the case when. In the general case, show that G is the product of a cyclic group generated by 1 +p, and a cyclic group of order p- 1. In the exceptional case, show that G is the product of the group {+1} is. with the cyclic group generated by the residue class of 5mod2r.].
Let D be an integer >1 and let R be the set of all elements a+b with a,bEZ. Kn denotes the vector space of (vertical) n-tuples of elements of K, and R is their of x nmatrices. with components in K, then V is a module over R. these lines, see the example at the end of 2. Let S be a non-empty set and M an A-module.
We see that the Horn function in either variable need not be exact if the other variable is held fixed. A similar situation exists in several variables, when let V be the vector space of variables of functions Coo inn in the open set of R n. By representing G in an A-M module, we mean a homomorphism p: G EndA(M) of G into the multiplicative monoid ofEndA(M).
Suppose that K is the Galois extension of k with the Galois group G (see Chapter VI). Then we can see K itself as amodule over the group k[G]. ThenAut(X/Xo) operates on the homology eX, so this homology is a module over the group ring.
The law of composition of morphisms is bilinear, and there exists
- FREE MODULES
It is immediately verified that kernels and kernels are universal misfit categories, and thus are uniquely determined to a unique isomorphism as them.
So in terms of abelian groups, given cp E A(S) there exist elements ai E A and Xi E S such that. Let us first assume that there exists a basis of V with a finite number of elements, e.g. {VI'..'Vrn}, m > 1. We leave the general case of an infinite basis as an exercise to the reader.
If the Vadmitson vector space is based on a certain number of elements, then we will say that Vis is finite dimensional and that m is its dimension. Given Theorem 5.2, we see that m is the number of elements in each basis of V.
Then x' gives rise to a functionality on V, according to the rulex1---+(x, x'), and this functional obviously depends only on the coset of x' modulo W'; in. For example, EV will be considered in Chapter we can call E" the Pontryagin. double.
Let Q be the set consisting of valelementesa ER such that there exists an elementX EM that can be written. IfxEMr+1 then the coefficient of x with respect to Xr+1 is divisible by ar+l' and therefore there exists CER such that x - cw in Mr.
Let E be a finitely generated module and E' a submodule
- Let R be an entire ring containing a field k as a subring, Suppose that R is a finite
Choose AI such that AI(M) is maximal in the set of ideals {J,\}, that is, there is no greater ideal in the set {J,\}. We say that this system satisfies the Mittag-LefflerML condition if for each, . the decreasing sequence um,n{Am) (m > n) stabilizes, i.e. Hint: Induction on the maximum number of linearly independent elements of A over R. Let Vb..., Vm be the maximum radius set of such elements, and let Ao be the subset of A contained in the space R generated by.
Show that b= can and thatfi is given by the mappingme:X cx(multiplication by c). map mb: a 0 is an element of the dual aV. a) Let M be a finite aprojective module over the Dedekind ring o. Prove that the inverse limit of a system of simple groups in which there are homomorphisms. are surjective is either the trivial set or the simple set. a) Let n be a sequence over the positive integers and let p be a prime number, Show that. the abelian groups An = Z/pnz form a subcanonical projective system. the homomorphism ifn > m, LeZp be its inverse limit.
Polynomials
IN ONE VARIABLE
- Let k be a field. Then the polynomial ring in one variable
- The ring k [X] is factorial
- Let k be a field, and let S1' ..., Sn be infinite subsets of k
- Let k be an infinite field and f a polynomial in n variables
- Let k be a finite field with q elements. Let f be a polynomial in n variables over k such that the degree of f in each variable
- Let k be a field and let U be a finite multiplicative sub-
- If k is a finite field, then k* is cyclic
- Gauss Lemma). Let A be a factorial ring, and let K be
- Let f(X) E A [X] have a factorization f(X) = g(X)h(X) in K[X]. If C g
- Let A be a factorial ring. Then the polynomial ring A[X]
- Let A be a factorial ring. Then the ring of polynomials in
If k is a field, then every non-zero element is a unit in k and we see. immediately that the units of k[X] are simply the units of k. If we perform the above operation a finite number of times, for all monomials appearing inf and all variables Xl' .., Xn we obtain some polynomial f* which results in the same function as f, but with degree v . each variable is An element ,in a field k such that there exists an integer n > 1 such that ,n= 1 is called a root of unity or more precisely an nth root of unity. Proof IfK has characteristic 0, then the derivative of amonomial ayXY such that v > 1 and ay =F 0 is not zero since it is vayXy-l.
