Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, for general smooth connected linear algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book, as it develops tools likely to be used in tackling other problems.

ALMOST RING THEORY 15 2.2.18. Note that the essential image of M → M * is closed under limits. Next recall that the forgetful functor A * -Alg → Set (resp. A * -Mod → Set) has a left adjoint A * [−] : Set → A * -Alg (resp. A (−) : Set → A * -Mod) that assigns to a set S the free A * -algebra A * [S] (resp. the free A * -module A (S) * ) generated by S. If S is any set, it is natural to write A[S] (resp. A (S) ) for the A-algebra (A * [S]) a (resp. for the A-module (A (S) * ) a . This yields a left adjoint, called the free A-algebra functor Set → A-Alg (resp. the free A-module functor Set → A-Mod) to the "forgetful" functor A-Alg → Set (resp. A-Mod → Set) B → B * .2.2.19. Now let R be any V -algebra; we want to construct a left adjoint to the localisation functor R-Mod → R a -Mod. For a given R a -module M, letWe have the natural map (unit of adjunction) R → R a * , so that we can view M ! as an R-module. Proposition 2.2.21. Let R be a V -algebra.(i) The functor R a -Mod → R-Mod defined by (2.2.20) is left adjoint to localisation.(ii) The unit of the adjunction M → M a ! is a natural isomorphism from the identity functor 1 R a -Mod to the composition of the two functors.Proof. (i) follows easily from (2.2.4) and (ii) follows easily from (i).Corollary 2.2.22. Suppose that m is a flat V -module. Then we have :(i) the functor M → M ! is exact;(ii) the localisation functor R-Mod → R a -Mod sends injectives to injectives.Proof. By proposition 2.2.21 it follows that M → M ! is right exact. To show that it is also left exact when m is a flat V -module, it suffices to remark that M → M * is left exact. Now, by (i), the functor M → M a is right adjoint to an exact functor, so (ii) is clear.

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.

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