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Brandeis MATH 101A - Math101a_notesC4a

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12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA4. Tensor productHere is an outline of what I did:(1) categorical definition(2) construction(3) list of basic properties(4) distributive property(5) right exactness(6) localization is flat(7) extension of scalars(8) applications4.1. definition. First I gave the categorical definition and then I gavean explicit construction.4.1.1. universal condition. Tensor product is usually defined by thefollowing universal condition.Definition 4.1. If E, F are two modules over a commutative ring R,their tensor product E ⊗ F is defined to be the R-module having thefollowing universal property. First, there exists an R-bilinear mappingf : E × F → E ⊗ F.Second, this mapping is universal in the sense that, for any other R-module M and bilinear mapping g : E × F → M, there exists a uniqueR-module homomorphism h : E ⊗ F → M making the following dia-gram commute.E × Ff!!g""!!!!!!!!!E ⊗ F∃!h##"""""MAs with all universal conditions, this definition only gives the unique-ness of E ⊗ F up to isomorphism. For the existence we need a con-struction.4.1.2. construction of E ⊗ F . The mapping f : E × F → E ⊗ F isnot onto. However, the image must generate E ⊗ F otherwise we geta contradiction. The elements in the image of f are denotedf(x, y) = x ⊗ y.Definition 4.2. The tensor product E ⊗ F is defined to be the Rmodule which is generated by the symbols x ⊗ y for all x ∈ E, y ∈ Fmodulo the following conditionsMATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA 13(1) x ⊗ − is R-bilinear. I.e.(a) x ⊗ ry = r(x ⊗ y) for all r ∈ R(b) x ⊗ (y + z) = (x ⊗ y) + (x ⊗ z)(2) − ⊗ y is R-bilinear. I.e.,(a) rx ⊗ y = r(x ⊗ y) for all r ∈ R(b) (x + y) ⊗ z = (x ⊗ z) + (y ⊗ z)I pointed out that these conditions require R to be commutativesincers(x ⊗ y) = r(sx ⊗ y) = sx ⊗ ry = s(x ⊗ ry) = sr(x ⊗ y).Proposition 4.3. E ⊗ F as given in the second definition satisfiesthe universal condition of the first definitions and therefore, the tensorproduct exists and is unique up to isomorphism.Proof. I said in class that this is obvious. If there is a bilinear mappingg : E × F → M, the induced mapping h : E ⊗ F → M must takethe generators x ⊗ y to g(x, y). Otherwise the diagram will not com-mute. Therefore, h is given on the generators and is thus unique. Theonly thing we need is to show that h is a homomorphism. But this isequivalent to showing that the elements of the formrx ⊗ y − r (x ⊗ y)and elements corresponding to the other three conditions in the seconddefinition go to zero in M. But this element goes tog(rx, y) − rg(x, y) = 0since g is R-bilinear and similarly for the other three elements. So, his an R-module homomorphism and we are done. !4.1.3. functorial properties of tensor product. The first properties Imentioned were the categorical properties which follow directly fromthe definition.Proposition 4.4. For a fixed R-module M, tensor product with M isa functorM ⊗ − : R-Mod → R-Mod.What this means is that, given an homomorphism f : A → B thereis an R-module homomorphism1 ⊗ f : M ⊗ A → M ⊗ Bwhich satisfies two conditions:(1) 1 ⊗ idA= idM⊗A14 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA(2) 1 ⊗ fg = (1 ⊗ f)(1 ⊗ g).The definition is (1⊗f)(x⊗y) = x⊗f(y). This gives a homomorphismsince the mapping M × A → M ⊗ B given by(x, y) &→ x ⊗ f(y)is bilinear and therefore induces the desired mapping 1 ⊗ f .More generally, given two homomorphisms f : M → N, g : A → Bwe get a homomorphismf ⊗ g : M ⊗ A → N ⊗ Bby the formula(f ⊗ g)(x ⊗ y) = f(x) ⊗ g(y).4.2. exact functors and flat modules. Flat modules are those forwhich the functor M ⊗ − is exact. An exact functor is one that takesshort exact sequences to short exact sequences. So, first I explainedthe definitions.Definition 4.5. An exact sequence is a sequence of modules and ho-momorphisms so that the image of each map is equal to the kernelof the next map. A short exact sequence is an exact sequence of thefollowing form:0 → Aα−→ Bβ−→ C → 0.In other words, α : A → B is a monomorphism, β : B → C is anepimorphism and im α = ker β or: C∼=B/αA.Sometimes short exact sequences are written:A " B # C.Definition 4.6. A functor F : R-Mod → R-Mod is called exact if ittakes short exact sequences to short exact sequences. Thus the shortexact sequence above should give the short exact sequence0 → FAF α−→ F BF β−→ F C → 0.Definition 4.7. An R-module M is called flat if M ⊗ − is an exactfunctor. I.e.,0 → M ⊗ A1⊗α−−→ M ⊗ B1⊗β−−→ M ⊗ C → 0is exact for all short exact sequences A " B # C.One of the main results (which we will see is actually trivial) is thatS−1R is flat for any multiplicative set S. I.e., localization is exact.MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA 154.3. list of properties. I explained that the exactness of localizationwas one of the key ideas. However, the explanation required an under-standing of the basic properties of tensor product. So, I went back tothe beginning with this list.(0) (unity) R ⊗ M∼=M.(1) (commutative) M ⊗ N∼=N ⊗ M(2) (distributive) N ⊗ ⊕Mi∼=!(N ⊗ Mi)(3) (associative) (A ⊗ B) ⊗ C∼=A ⊗ (B ⊗ C)(4) (right exactness) M ⊗ − is right exact, i.e., a short exact se-quence A " B # C gives an exact sequenceM ⊗ A1⊗α−−→ M ⊗ B1⊗β−−→ M ⊗ C → 0(5) (localization is exact) I.e., we get an exact sequence:0 → S−1A → S−1B → S−1C → 0.(6) (extension of scalars) Given a ring homomorphism R → S,every R-module M gives an S module S ⊗RM.4.3.1. Grothedieck ring. I did not prove properties (1) and (3). I saidthey were obvious. However, I put the first three conditions into aconceptual framework by pointing out that these are the axioms of aring. The only thing that we don’t have is an additive inverse. Thealgebraic construction is as follows.First, you take the set of all isomorphism classes of finnitely gener-ated R-modules [M]. This set has addition and multiplication givenby[M] + [N] = [M ⊕ N][M][N] = [M ⊗ N]Addition and multiplication are associative and commutative and haveunits: [0] is the additive unit and [R] is the multiplicative unit. It justdoesn’t have additive inverses. So, Grothendieck said to just put informal inverses:[M] − [N]which are defined like fractions:[M] − [N] = [A] − [B]if there exists


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