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Algebroid Curves in Positive Characteristic by A. Campillo PDF

By A. Campillo

ISBN-10: 0387100229

ISBN-13: 9780387100227

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X31 x2 • 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The monoideal (t1 , t2 , . ) ⊆ Tn is generated by {t1 , . . , tN } . In particular, for every ring R , the ideal (t1 , t2 , . ) ⊆ R[x1 , . . , xn ] is finitely generated. 44 1. Foundations αn 1 Proof. The map log : Tn → Nn given by xα → (α1 , . . , αn ) is 1 · · · xn clearly an isomorphism of monoids. The monoideal (log(t1 ), log(t2 ), . ) ⊆ Nn is finitely generated by the previous proposition. Thus there exists a number N > 0 such that this monoideal is generated by {log(t1 ), . . , log(tN )} .

Exercise 3. Show that the map log : Tn −→ Nn is an isomorphism of monoids. Exercise 4. Let v1 = (a11 , a21 , . . , an1 ), . . , vn = (a1n , a2n , . . , ann ) be elements of Zn , and let A = (aij ) ∈ Matn (Z) be the matrix whose columns are the coordinates of v1 , . . , vn . Show that the set {v1 , . . , vn } is a Z -basis of Zn if and only if det(A) ∈ {1, −1} . Exercise 5. Let S be the set of functions from Z to Z . a) Show that S with the usual sum and product of functions is a Z algebra.

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Algebroid Curves in Positive Characteristic by A. Campillo


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