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Basic Vocabulary 2 Generators and Relations for an R-Module

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Basic Vocabulary 2 Generators and Relations for an R-Module Modules as given by matrices November 23 Material covered in Artin Ed 1: Chapter 12 section 1-6; and Artin Ed 2: 14.1-14.8; and also more generally covered in Dummit and Foote Abstract Algebra. 1 Recall modules; set up conventions; basic vocabulary We will work with commutative rings with unit R. The category of R-modules. If U; V are R- modules we have the direct sum U ⊕ V as an R-module, and HomR(U; V ). Some basic vocabulary, some of which we have introduced, and some of which we will be introducing shortly. • submodule, quotient module • finitely generated • cyclic R-modules and Ideals in R • free • \short" exact sequence of R-modules 0 ! U ! V ! W ! 0: • split \short" exact sequence of R-modules • vector spaces; dimensions, ranks. 2 Generators and relations for an R-module; presentations Recall that our rings are commutative with unit, and we're talking about modules over such rings. Recall what it means for a set of elements fx1; x2; : : : ; xng ⊂ M to be a \a system of generators" for M over R: it means that any element of M is a linear combo of these elements with coefficients in R. Definition 1 If M is an R-module and fx1; x2; : : : ; xng ⊂ M a subset of elements that has the property that any linear relation of the form n Σi=1rixi = 0 in the ring R is trivial in the sense that all the ri's are 0, we'll say that the elements fx1; x2; : : : ; xng are linearly independent over R. 1 Recall also that we say that F is a free R-module of rank n if there is an isomorphism of R-modules F ≡ Rn := R ⊕ R ⊕ · · · ⊕ R (n times). Equivalently, if there is a system of n generators of the R-module F that are linearly independent over R. (Such a system that generates F and is linear independent over R we'll call a basis of F over the ring R.) Exercise 1 Let F be free R-module of rank n with fx1; x2; : : : ; xng a basis. Let M be any R- module and fm1; m2; : : : ; mng ⊂ M any ordered subset of n elements. Show that there is a unique R-homomorphism from F to M sending xi to mi for all i. Now show \the reverse." Consider any R-module E with an ordered subset of n elements fe1; e2; : : : ; eng ⊂ E that has this marvelous property: For M any R-module and fm1; m2; : : : ; mng ⊂ M any ordered subset of n elements there is a unique R-homomorphism from E to M sending ei to mi for all i. Prove that E is a free R-module of rank n and fe1; e2; : : : ; eng ⊂ E is a basis of E over the ring R. Exercise 2 Let 0 ! A ! B ! C ! 0 be a short exact sequence of R-modules. Show that if C is free (say of rank n) over R then the short exact sequence of R-modules is \split" (meaning that it is \splittable" in the sense defined in today's lecture). Exercise 3 Let 0 ! A ! B ! C ! 0 be a short exact sequence of R-modules. Show that if A is free (say of rank n1) C is free (say of rank n2) over R then B is free (of rank n1 + n2). Exercise 4 Suppose I have a ring R that has the property that any cyclic R-module is free over R. What does that say about the ring R? Note|by the way| that, if in addition to the above, I know that every finitey generated R-module is a direct sum of cyclic R-modules, I get that every finitely generated R-module is free. Exercise 5 Find a ring R and 0 ! A ! B ! C ! 0 a short exact sequence of R-modules that is not splittable. 2 3 Presentations as given by matrices Discuss. Emphasize importance of this idea! A finite presentation of an R-module M is a \system" (i.e., set) of generators of the R-module M, fx1; x2; : : : ; xng and a system of relations fy1; y2; : : : ; ymg, so that we can write: n yi = Σj=1aijxj for some \scalars" aij 2 R, and so that we can use the m × n matrix A := (aij) 2 Matm×n(R) to give us an exact sequence Rm−!A Rn −! M −! 0 where the R-homomorphism A is what you think it is. As discussed, the A is not determined by the R-isomorphism class of M (not by a wide wide margin) but the isomorphism class of M is indeed determined by any finite presentation matrix A for it. 3.1 Presentations given by diagonal matrices Examples: • Cyclic modules presented by a 1 × 1 diagonal matrix::: • If a presentation is given by a diagonal matrix, then the R-module is a sum of cyclic R- modules. • The Chinese Remainder Theorem (again!) expresses finite cyclic Z-modules as direct sums of cyclic Z-modules of prime-power order. 3.2 Certain standard changes of presentation 1. Add a scalar multiple of one row to another row; or add a scalar multiple of one column to another column. 2. Interchange two rows; or interchange two columns. 3. Multiply a row or column by a unit (i.e., an invertible element) in R. If x1; x2; : : : ; xn are the generators and y1; y2; : : : ; ym are the relations, the above operations corre- spond to: 3 1. Replacing a generator xi by xi + r · xj for r 2 R and i 6= j or replacing a relation yi by yi + r · yj. 2. Interchanging two generators or interchanging two relations. 3. Multiplying a generator or a relation by a unit (i.e., an invertible element) in R. 3.3 \Diagonalizing" matrix presentations for R = Z: Start with these matrix presentations and say what you end up with as (a) \diagonal" matrix, and (b) presented module: 5 3 ; 2 1 3 3 ; 2 1 1 2 3 : 4 5 6 4 Diagonalizing presentation matrices As mentioned above, a \diagonalizable presentation" is the presentation of a direct sum of cyclic R- modules. I.e., when you diagonalize a presentation matrix by standard row-and-column operations as enumerated in the last lecture, you've guaranteed that your module is a sum of cyclic R-modules. 4.1 Diagonalizing matrix presentations for R = a field Scale and clear. Exercise 6 Let R = a field. Show that any finitely generated (say) R-module is free. Show that two finitely generated R-modules are isomorphic if and only if they have the same dimension. (I.e., the single number dimension determines the ≡-class of a (finitely generated) vector space over R.) 4 4.2 Fundamental Theorem Exercise 7 \Diagonalizing" matrix presentations for R = Z: Start with these matrix presenta- tions and say what you end up with as (a) \diagonal" matrix, and (b) presented module: 5 3 ; 2 1 3 3 ; 2 1 1 2 3 : 4 5 6 Exercise 8 * Find a module over the ring R = Z[X] that is not the direct sum of cyclic modules over R. Exercise 9 * Find a ring R and a module over R that is finitely generated but not finitely presented. (Smallish hint: find a nonfinitely generatable ideal in some humungous ring.) Theorem 1 Any m × n matrix with entries in Z can be diagonalized by a sequence of standard row-and-column operations. Corollary 2 ∗ Any finitely generated abelian group is a (finite) direct sum of cyclic groups. Abbreviated Proof: Do it by induction. In fact the \move" we describe will reduce m and n (simultaneously) or the smallest nonzero absolute value of any entry: bring the entry with the smallest nonzero absolute value to the upper right hand corner (making it positive) and use the archimedean principle to either clear the first row and column thereby reducing our problem to an (m − 1) × (n − 1) matrix, or else will lead to a reduction of the smallest nonzero absolute value of any entry. QED Postmortum: 5 • The famous corollaries to be drawn from this!! • Proof analysis: The algorithm above uses the archimedean principle. Note that this holds, say, in polynomial rings R = k[t] in one variable over fields. So we get the same famous corollaries for these. • What if m or n is infinite (but not both)? • A similar, but easier, abbreviated proof does this when R = a field. Discuss vector spaces and dimension. Exercise 10 * Generalize the proof we gave for R = Z to show that if k is a field and R = k[X], then any matrix over R can be \diagonalized" (and in the corresponding \specific way" that was be described in the formulation of the proof). Given the corollary corresponding to the above corollary. 5 Applications to Linear Operators • Description of \shift operator." • Discuss the dictionary: operator T on a vector space $ k[T ] − module: • Examples. 6 Homework set due Dec 2 • Do all the unstarred numbered exercises in this handout. • For Artin Ed 1: Do exercises 1, 5 (from the set of exercises 4: Diagonalization of Integer Matrices) on page 485; for Artin Ed 2: page 438 4.1(a), (b), and 4.3 • For Artin Ed 1: 6: Structure of abelian groups Ex. 1,2, 3(a),(b), (c) (e); for Artin Ed 2: page 439 7.1. 7.2, 7.3 • How many isomorphism classes of abelian groups of order 400 are there? (Explain) 6.
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