# An Introduction to Hopf Algebras by Robert G. Underwood

By Robert G. Underwood

The learn of Hopf algebras spans many fields in arithmetic together with topology, algebraic geometry, algebraic quantity thought, Galois module idea, cohomology of teams, and formal teams and has wide-ranging connections to fields from theoretical physics to machine technological know-how. this article is exclusive in making this enticing topic obtainable to complicated graduate and starting graduate scholars and specializes in purposes of Hopf algebras to algebraic quantity concept and Galois module idea, offering a delicate transition from sleek algebra to Hopf algebras.

After offering an creation to the spectrum of a hoop and the Zariski topology, the textual content treats presheaves, sheaves, and representable staff functors. during this approach the scholar transitions easily from uncomplicated algebraic geometry to Hopf algebras. the significance of Hopf orders is underscored with purposes to algebraic quantity thought, Galois module idea and the speculation of formal teams. by means of the tip of the publication, readers should be conversant in validated leads to the sector and able to pose examine questions in their own.

An workout set is integrated in each one of twelve chapters with questions ranging in trouble. Open difficulties and learn questions are awarded within the final bankruptcy. must haves comprise an figuring out of the cloth on teams, earrings, and fields commonly lined in a easy path in sleek algebra.

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Let W F ! G be an epimorphism of R-group schemes. S /; that is, suppose g D B . Then S is a B-module with b s D . b/s. A; S /, f . b// D f . 1/ Db 1 D . b/ D f . b// D . b//; and so S 0 can be taken to be S . A; S /: Thus, the collection of all preimages of W F ! 3. 4 An Example In this section, we present an important example of a short exact sequence of group schemes. Let K be a field, let Gm denote the multiplicative group scheme represented by KŒX; X 1 , and let G0m denote a copy that is represented by KŒY; Y 1 .

A/. a/: t u What condition on guarantees the existence of a multiplicative identity element? P Let m W AP˝R A ! A denote the multiplication map of A defined as m. a ˝ b/ D ab, and let W R ! S denote the structure map of the commutative R-algebra S . 1R / D 1S . 3. Let W A ! a/ D a D m. a/ for a 2 A. Then the R-algebra homomorphism . S /. S /. 3) W A ! S satisfies f Df Df . / is a left and right identity for the binary operation Proof. a/ D X . 3): In a similar manner, one can show that f D f .

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. fj / in the covering. s// for all s 2 A, 1 Ä j Ä m. s/, and so h D k, which is property (i).