000			02168 a2200253 4500
003			OSt
005			20210706095630.0
008			210204b xxu||||| |||| 00| 0 eng d
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 con­ 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