By Robert G. Underwood
Only booklet on Hopf algebras geared toward complicated undergraduates
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Extra info for An Introduction to Hopf Algebras
Example text
R-alg /. The law of composition is ordinary function composition. Let =sets denote the category of sets. =R-alg /, and define a map F W =R-alg ! =R-alg /. 1. F defined as above is a functor from the category of commutative R-algebras to the category of sets. Proof. =R-alg /. S /, s 2 A. 1 hold. We prove (i) here and leave (ii) as an exercise. =R-alg /. F . s/ D . F . s/ D F . F . s/; t u and so (i) holds. f / Spec A? f // ! g/ W Af ! Af / ! Af /, s 2 A. g/ . fj /gm j D1 be a finite open covering of Spec A.
1. Let % W S ! T be flat, and suppose that P T D 6 T for every maximal ideal P of S . If M is a non-zero S -module, then M ˝S T is non-zero. Proof. Let m 6D 0 be an element of M , and let I be the annihilator ideal of m. Then S m Š S=I . S=I / ˝S T ! M ˝S T . I T /. P T / 6D 0. It follows that M ˝S T 6D 0. 2. Let % W S ! T be flat, and suppose that P T 6D T for every maximal ideal P of S . Let ˛ W M ! N be a map of S -modules, and let ˛ 0 W M ˝S T ! N ˝S T be the induced map defined by m ˝ t 7!
We next show that a natural transformation W G ! F can be written in the form D a for some R-algebra homomorphism W A ! B. B/ ! B/. Let IB W B ! B denote the identity map. B/. Then W A ! B is a map of R-algebras. Now, for an R-algebra map z W B ! S / ! z/. z/. z/: Consequently, D a . Thus the map 7! a is surjective. Suppose that a D a !. IB / D IB ! D !. , and so the map 7! a is a bijection. 1 1. Let S be a subset of a topological space X . Suppose that for each a 2 S there exists an open set U Â S for which a 2 U .