甜食
发表于 2025-3-26 21:37:39
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看法等
发表于 2025-3-27 01:23:56
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Allege
发表于 2025-3-27 07:50:51
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
发表于 2025-3-27 10:58:31
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
发表于 2025-3-27 14:57:50
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temperate
发表于 2025-3-27 19:59:28
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
发表于 2025-3-28 01:34:25
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
美学
发表于 2025-3-28 02:44:59
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废除
发表于 2025-3-28 06:47:10
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 ℤ/3ℤ, and we get three components. Thus base extension can create additional idempotents. To have a complete theory
唤醒
发表于 2025-3-28 12:06:28
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