Description of admissibility Definition J is I -admissible if J v ( - PowerPoint PPT Presentation

Description of admissibility Definition J is I -admissible if J ≤ v ( I ). Lemma This is equivalent to IO Y ′ = E ℓ I ′ , with J = ¯ J ℓ and I ′ an ideal. Indeed, on Y ′ the center J becomes E ℓ , in particular principal. Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

Description of admissibility Definition J is I -admissible if J ≤ v ( I ). Lemma This is equivalent to IO Y ′ = E ℓ I ′ , with J = ¯ J ℓ and I ′ an ideal. Indeed, on Y ′ the center J becomes E ℓ , in particular principal. E ℓ ⊇ IO Y ′ . ♠ So on Y ′ , we have J ≤ v ( I ) ⇔ This is more subtle in Quek’s theorem! Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

Description of admissibility Definition J is I -admissible if J ≤ v ( I ). Lemma This is equivalent to IO Y ′ = E ℓ I ′ , with J = ¯ J ℓ and I ′ an ideal. Indeed, on Y ′ the center J becomes E ℓ , in particular principal. E ℓ ⊇ IO Y ′ . ♠ So on Y ′ , we have J ≤ v ( I ) ⇔ This is more subtle in Quek’s theorem! Write J = ( x a 1 1 , . . . , x a k k ) and I = ( f 1 , . . . , f m ). Expand f i = � c α x α 1 1 · · · x α 1 n . J < v ( I ) ⇔ v J ( f i ) ≥ 1 for all i Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

Description of admissibility Definition J is I -admissible if J ≤ v ( I ). Lemma This is equivalent to IO Y ′ = E ℓ I ′ , with J = ¯ J ℓ and I ′ an ideal. Indeed, on Y ′ the center J becomes E ℓ , in particular principal. E ℓ ⊇ IO Y ′ . ♠ So on Y ′ , we have J ≤ v ( I ) ⇔ This is more subtle in Quek’s theorem! Write J = ( x a 1 1 , . . . , x a k k ) and I = ( f 1 , . . . , f m ). Expand f i = � c α x α 1 1 · · · x α 1 n . J < v ( I ) ⇔ v J ( f i ) ≥ 1 for all i � α j a j ≥ 1 for all i and α such that c α � = 0. ⇔ Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 1 / 2

Consequences J is I 1 , I 2 -admissible ⇒ J is I 1 + I 2 -admissible. J is I -admissible ⇒ J a is I a -admissible. J is I -admissible ⇒ J 1 − 1 a 1 is D ( I )-admissible. Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2

Consequences J is I 1 , I 2 -admissible ⇒ J is I 1 + I 2 -admissible. J is I -admissible ⇒ J a is I a -admissible. J is I -admissible ⇒ J 1 − 1 a 1 is D ( I )-admissible. Proof. = � α i � � ∂ x α a i − 1 a j ≥ 1 − 1 a 1 . v J ∂ x j Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2

Consequences J is I 1 , I 2 -admissible ⇒ J is I 1 + I 2 -admissible. J is I -admissible ⇒ J a is I a -admissible. J is I -admissible ⇒ J 1 − 1 a 1 is D ( I )-admissible. Proof. = � α i � � ∂ x α a i − 1 a j ≥ 1 − 1 a 1 . ♠ v J ∂ x j Combining: Proposition A center J is I -admissible if and only if J ( a 1 − 1)! is C ( I , a 1 ) -admissible. Dan Abramovich Admissibility and coefficient ideals New York, July 27, 2020 2 / 2