By Robert G. Underwood

ISBN-10: 0387727655

ISBN-13: 9780387727653

The research of Hopf algebras spans many fields in arithmetic together with topology, algebraic geometry, algebraic quantity idea, Galois module idea, cohomology of teams, and formal teams and has wide-ranging connections to fields from theoretical physics to desktop technological know-how. this article is exclusive in making this attractive topic available to complex graduate and starting graduate scholars and specializes in purposes of Hopf algebras to algebraic quantity idea and Galois module concept, supplying a soft transition from glossy algebra to Hopf algebras.

After supplying an advent to the spectrum of a hoop and the Zariski topology, the textual content treats presheaves, sheaves, and representable workforce functors. during this approach the coed transitions easily from uncomplicated algebraic geometry to Hopf algebras. the significance of Hopf orders is underscored with purposes to algebraic quantity idea, Galois module idea and the speculation of formal teams. by means of the top of the publication, readers might be accustomed to confirmed ends up in the sphere and able to pose examine questions in their own.

An workout set is incorporated in each one of twelve chapters with questions ranging in hassle. Open difficulties and learn questions are offered within the final bankruptcy. must haves comprise an figuring out of the fabric on teams, jewelry, and fields quite often coated in a uncomplicated path in glossy algebra.

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**Additional info for An Introduction to Hopf Algebras**

**Example text**

Now, % W M ! M ˝S T is an injection by the faithful flatness of S ! T . Consequently, % is an injection, and so Q ! Q ˝S T is faithfully flat. 4. Let S ! T be faithfully flat, and let x 2 Spec S . Then the induced map Sx ! T ˝S Sx is faithfully flat. Proof. Let M be an Sx -module, and let ' W M ! T ˝S Sx / be the map defined as m 7! 1 ˝ 1/. M ˝S T / ˝S Sx : Since S ! T is faithfully flat, there is an injection M ! M ˝S T given as m 7! m ˝ 1. Consequently, ' is also an injection. u t Faithful flatness is a critical condition in view of the following.

1; 1/ D fx W jxj < 1g Â R, then . p/g. Observe that a subset S Â X is closed if and only if S D S c . We can determine precisely when singleton subsets fxg of Spec A are closed. 7. The subset fxg Â Spec A is closed if and only if x is a maximal ideal of A. Proof. Suppose fxg Â Spec A is closed. E/ for some subset E Â A. Let I be an ideal of A for which x Â I A. 1, there exists a prime ideal y for which x Â I Â y A. Now, since fxgc D fxg, x D y D I . Thus x is maximal. We leave the converse as an exercise.

Suppose that ˛ W M ! N has non-zero kernel L. 1, L˝S T 6D 0. By the flatness of %, L˝S T ! M ˝S T is an injection. Since L˝S T is in the kernel of ˛ 0 , ˛ 0 is not an injection, which proves the lemma. 2. A flat map S ! T is faithfully flat if the map ' W M ! m/ D m ˝ 1 is an injection for all S -modules M . The localization map S ! Sf may not be faithfully flat, though it can be used to build a faithfully flat map. Let ff1 ; f2 ; : : : ; fn g be a finite set of non-nilpotent elements of S , and suppose that the ideal generated by ff1 ; f2 ; : : : ; fn g is S .

### An Introduction to Hopf Algebras by Robert G. Underwood

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