Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity - PowerPoint PPT Presentation

Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity William D. Taylor University of Arkansas KUMUNUjr, April 8, 2017 William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 1 / 12

Setting and Notation ( R , m ) is a local ring of characteristic p > 0 and dimension d I ⊆ R is an m -primary ideal of R λ ( M ) denotes the length of the R -module M Definition The Hilbert-Samuel multiplicity of I is defined to be d ! · λ ( R / I n ) e ( I ) = lim . n d n →∞ The Hilbert-Kunz multiplicity of I is defined to be R / I [ p e ] � � λ e HK ( I ) = lim . p ed e →∞ William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 2 / 12

Properties of e ( I ) and e HK ( I ) 1 If I = J , then e ( I ) = e ( J ) 2 If I ∗ = J ∗ , then e HK ( I ) = e HK ( J ) 3 If R is regular then e ( m ) = e HK ( m ) = 1 4 There is an Associativity Formula relating each multiplicity to the multiplicity after quotienting by the set of primes of maximal dimension. Theorem (Rees ‘61, Hochster-Huneke ‘90) If R is a complete domain and I ⊆ J, then the converses of 1 and 2 hold. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 3 / 12

Definition of s -multiplicity Remark Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈ sp e ⌉ ⊇ I [ p e ] , William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 4 / 12

Definition of s -multiplicity Remark Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈ sp e ⌉ ⊇ I [ p e ] , hence I ⌈ sp e ⌉ + I [ p e ] = I ⌈ sp e ⌉ , William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 4 / 12

Definition of s -multiplicity Remark Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈ sp e ⌉ ⊇ I [ p e ] , hence I ⌈ sp e ⌉ + I [ p e ] = I ⌈ sp e ⌉ , R / ( I ⌈ sp e ⌉ + I [ p e ] ) R / I ⌈ sp e ⌉ � � � � = s d λ = λ hence lim d ! e ( I ). p ed p ed e →∞ William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 4 / 12

Definition of s -multiplicity Remark Let s be a positive real number. If s is small, then for e ≫ 0, I ⌈ sp e ⌉ ⊇ I [ p e ] , hence I ⌈ sp e ⌉ + I [ p e ] = I ⌈ sp e ⌉ , R / ( I ⌈ sp e ⌉ + I [ p e ] ) R / I ⌈ sp e ⌉ � � � � = s d λ = λ hence lim d ! e ( I ). p ed p ed e →∞ Similarly, If s is large, then for e ≫ 0, I ⌈ sp e ⌉ ⊆ I [ p e ] , hence I ⌈ sp e ⌉ + I [ p e ] = I [ p e ] , R / ( I ⌈ sp e ⌉ + I [ p e ] ) R / I [ p e ] � � � � λ λ hence lim = lim = e HK ( I ). p ed p ed e →∞ e →∞ William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 4 / 12

Definition of s -multiplicity Definition (-) For a positive real number s , the s-multiplicity of I is R / ( I ⌈ sp e ⌉ + I [ p e ] ) � � λ e s ( I ) = lim p ed H s ( d ) e →∞ where ⌊ s ⌋ ( − 1) i � d � � ( s − i ) d . H s ( d ) = d ! i i =0 William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 5 / 12

Properties of s -multiplicity Proposition (-) The function e s ( I ) has the following properties: 1 e s ( I ) is a continuous function of s. 2 If s is sufficiently small, then e s ( I ) = e ( I ) . 3 If s is sufficiently large, then e s ( I ) = e HK ( I ) . 4 If R is regular, then e s ( m ) = 1 for all s. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 6 / 12

Properties of s -multiplicity Proposition (-) The function e s ( I ) has the following properties: 1 e s ( I ) is a continuous function of s. 2 If s ≤ 1 , then e s ( I ) = e ( I ) . 3 If s ≥ d, then e s ( I ) = e HK ( I ) . 4 If R is regular, then e s ( m ) = 1 for all s. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 7 / 12

The Associativity Formula Theorem (-) If s is any positive real number, then e R / p � e s ( I ) = ( I ( R / p )) λ R p ( R p ) s p ∈ Assh R where Assh R = { p ∈ Spec R | dim R / p = dim R } . This theorem generalizes the Associativity Formulae for the Hilbert-Samuel and Hilbert-Kunz multiplicities. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 8 / 12

Closures Recall: If I = J , then e ( I ) = e ( J ) If I ∗ = J ∗ , then e HK ( I ) = e HK ( J ) If I ⊆ J and R is “nice”, then the converse holds. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 9 / 12

Closures Recall: If I = J , then e ( I ) = e ( J ) If I ∗ = J ∗ , then e HK ( I ) = e HK ( J ) If I ⊆ J and R is “nice”, then the converse holds. Question: Are there closures that similarly relate to s -multiplicity? William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 9 / 12

Closures Recall: If I = J , then e ( I ) = e ( J ) If I ∗ = J ∗ , then e HK ( I ) = e HK ( J ) If I ⊆ J and R is “nice”, then the converse holds. Question: Are there closures that similarly relate to s -multiplicity? Answer: Yes, we call them s -closures. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 9 / 12

s -closures Definition (-) Let s ≥ 1. We say x ∈ I cl s , the s -closure of I , if there exists c not in any minimal prime of R such that cx p e ∈ I ⌈ sp e ⌉ + I [ p e ] . for all e ≫ 0 , Remark When s = 1, s -closure is integral closure. When s ≥ d , s -closure is tight closure. As s increases, the s -closures get smaller. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 10 / 12

s -closures Question: Do any new closures actually appear? William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 11 / 12

s -closures Question: Do any new closures actually appear? Short Answer: Yes! Example Let R = k [[ x , y ]], and let I = ( x 3 , y 3 ). Then ( x , y ) 3 = I  if s = 1   I cl s = ( x 3 , x 2 y 2 , y 3 ) if 1 < s ≤ 4 3  ( x 3 , y 3 ) = I ∗ if s > 4 3 .  William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 11 / 12

s -closures Question: Do any new closures actually appear? Short Answer: Yes! Example Let R = k [[ x , y ]], and let I = ( x 3 , y 3 ). Then ( x , y ) 3 = I  if s = 1   I cl s = ( x 3 , x 2 y 2 , y 3 ) if 1 < s ≤ 4 3  ( x 3 , y 3 ) = I ∗ if s > 4 3 .  Long Answer: In many cases, there are uncountably many distinct s -closures on a fixed ring R . William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 11 / 12

s -closures Theorem (-) Let I and J be m -primary ideals of R. If I cl s = J cl s , then e s ( I ) = e s ( J ) . If R is an F-finite complete domain and I ⊆ J, then the converse holds. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 12 / 12

s -closures Theorem (-) Let I and J be m -primary ideals of R. If I cl s = J cl s , then e s ( I ) = e s ( J ) . If R is an F-finite complete domain and I ⊆ J, then the converse holds. Remark The proof of this theorem uses a nice result of Polstra and Tucker on limits related to positive characteristic rings. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 12 / 12

s -closures Theorem (-) Let I and J be m -primary ideals of R. If I cl s = J cl s , then e s ( I ) = e s ( J ) . If R is an F-finite complete domain and I ⊆ J, then the converse holds. Remark The proof of this theorem uses a nice result of Polstra and Tucker on limits related to positive characteristic rings. Remark Extending the converse result to the non-domain case seems difficult. William D. Taylor (U. Arkansas) Interpolating Between e ( I ) and e HK ( I ) KUMUNUjr 2017 12 / 12