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020 _a9781461262176
040 _cIIT Kanpur
041 _aeng
082 _a510
_bW292i
100 _aWaterhouse, William C.
245 _aIntroduction to affine group schemes [Perpetual]
_cWilliam C. Waterhouse
260 _bSpringer-Verlag
_c1979
_aNew York
300 _axi,167p
440 _aGraduate texts in mathematics; no.66
490 _a/ edited by F. W. Gehring
520 _aAh Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then conĀ­ struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.
650 _aGroup schemes (Mathematics)
650 _aMathematics
856 _uhttps://link.springer.com/book/10.1007/978-1-4612-6217-6
942 _cEBK
999 _c563542
_d563542