Titlebook: Introduction to Affine Group Schemes; William C. Waterhouse Textbook 1979 Springer-Verlag New York Inc. 1979 Abelian group.Algebra.Algebra

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Allege

Faithful Flatnessthen . ?. . → . ?. . is also an injection. For example, any localization . →. . is flat. Indeed, an element .?. in .?. .= . . is zero iff . = 0 for some . in .; if . injects into . and . is zero in ., it is zero in .. What we really want, however, is a condition stronger than flatness and not satisfied by localizations.

不成比例

Affine Group Schemesa familiar process for constructing a group from a ring. Another such process is GL., where GL.(.) is the group of all 2 × 2 matrices with invertible determinant. Similarly we can form SL. and GL.. In particular there is GL., denoted by the special symbol G.; this is the ., with G.(.) the set of inv

Acupressure

temperate

Representationsl come up later for general ., but the only case of interest now is . = .? ., where . is a fixed .-module. If the action of . here is also .-linear, we say we have a . of . on .. The functor . = Aut.(. ?.) is a group functor; a linear representation of . on . clearly assigns an automorphism to each

APO

Algebraic Matrix Groupser only a fixed field .. We call a subset S of . if it is the set of common zeros of some polynomials {.} in .[.,…,.]. Clearly an intersection of closcd sets is closed. Also, if S is the zeros of {.} and . the zeros of {.}} then . ∪ . is the zeros of {. .}, so finite unions of closed sets are closed

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Connected Components and Separable Algebrasted by . = .[.]/(. ? 1). Over the reals there are two points in Spec ., reflecting the decomposition . ? 1 = (. – 1)(. + . + 1). But over the complex numbers the group is isomorphic to ?/3?, and we get three components. Thus base extension can create additional idempotents. To have a complete theory

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Groups of Multiplicative Types. One calls an . × . matrix . . if the subalgebra .[.] of End(.) is separable. We have of course .[.] ? .[.]/.(.) where .(.) is the minimal polynomial of . Separability then holds iff .[.]?. = .[.] ? .[.]/.(.) is separable over .. This means that . has no repeated roots over ., which is the familia