March 7, 2017

A Bound on the Real Stability Radius of Continuous-Time by Bobylev N. A., Bulatov V.

By Bobylev N. A., Bulatov V.

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Sample text

Xn ]/I(V ) die affine Algebra von V . Man zeige, dass es Bijektionen V → Max(K[V ]) und Max(K[V ]) → HomAlg (K[V ], K) gibt, wobei HomAlg (K[V ], K) die Menge der K-Algebrahomomorphismen von K[V ] nach K bezeichnet. Aufgabe 16 Sei K ein K¨ orper und A eine affine K-Algebra, die von n Elementen x1 , . . , xn erzeugt werde. Man zeige, dass es eine algebraische Menge V n ∼ in K und einen K-Algebraisomorphismus K[V ] → A gibt, bei dem die i-te Koordinatenfunktion auf V mit xi f¨ ur i = 1, . .

Fm (v)) induziert den KAlgebrahomomorphismus α∗ : K[Y1 , . . , Ym ] → K[V ], g → g ◦ α , wobei g◦α = Def von α g ◦ (f1 , . . , fm ) = g ◦ (γ(y1 ), . . , γ(ym )) = γ(g + I(W )) = γ 0K[W ] = 0K[V ] . siehe folgende Bemerkung falls g ∈ I(W ) Da W = V(I(W )) gilt, folgt α(v) ∈ W f¨ ur alle v ∈ V . Weiter folgt, dass α∗ auch auf K[W ] = K[Y1 , . . , Ym ]/I(W ) wohldefiniert ist und α∗ = γ gilt. rm Y1r1 · . . · Ymrm , woraus folgt: rm und also g + I(W ) = ar1 . . arm y1r1 · . . · ym r1 γ(g + I(W )) = ar1 .

Siehe folgende Bemerkung falls g ∈ I(W ) Da W = V(I(W )) gilt, folgt α(v) ∈ W f¨ ur alle v ∈ V . Weiter folgt, dass α∗ auch auf K[W ] = K[Y1 , . . , Ym ]/I(W ) wohldefiniert ist und α∗ = γ gilt. rm Y1r1 · . . · Ymrm , woraus folgt: rm und also g + I(W ) = ar1 . . arm y1r1 · . . · ym r1 γ(g + I(W )) = ar1 . . arm γ(y1 ) . . γ(ym )rm = g ◦ (γ(y1 ), . . , γ(ym )). Die letzte Behauptung des Satzes folgt leicht: =⇒“ gilt, weil F ein Funktor mit (α ◦ β)∗ = β ∗ ◦ α∗ und (idV )∗ = idK[V ] ” ist.

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