Computations in Local Rings using Macaulay2 Mahrud Sayrafi University of California, Berkeley Northern California Undergraduate Mathematics Conference, Sonoma State University, March 2017 Mahrud Sayrafi LocalRings.m2 Table of Contents 1 Definitions 2 My Work 3 Applications 4 Conclusion Mahrud Sayrafi LocalRings.m2 Table of Contents 1 Definitions 2 My Work 3 Applications 4 Conclusion Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Rings Let’s review some definitions that I hope everyone is familiar with. Definition: Ring(R,+, ) · SetR equipped with operations + and satisfying 3 axioms. · Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Rings Let’s review some definitions that I hope everyone is familiar with. Definition: Ring(R,+, ) · SetR equipped with operations + and satisfying 3 axioms. · Examples Ring of integersZ. FieldsR,Q,C,F, etc. Matrix ringsM n(k) Polynomial ringsk[x 1,...,x n]. Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Rings Let’s review some definitions that I hope everyone is familiar with. Definition: Ring(R,+, ) · SetR equipped with operations + and satisfying 3 axioms. · Examples Ring of integersZ. FieldsR,Q,C,F, etc. Matrix ringsM n(k) Polynomial ringsk[x 1,...,x n]. My work is on commutative rings with 1R . For this talk let’s focus on polynomial rings. Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Ideals Definition: Ideals SubsetA R such that for anya A andr R we havea r A. ⊂ ∈ ∈ · ∈ Examples 0 = 0 Z � � { }⊂ 6 = 0, 6, 12,... Z � � { ± ± }⊂ (x + 1) = (x + 1)p(x,y) p(x,y) k[x,y] k[x,y] | ∈ ⊂ � � Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Prime Ideals Definition: Prime ideals An idealp�R is prime if: for anya,b R such that ab p, eithera p orb p. ∈ ∈ ∈ ∈ Examples (2) Z ⊂ (x) k[x,y] ⊂ (x 1) k[x,y] − ⊂ (x+y) k[x,y] ⊂ (x2 +y 2 1) k[x,y] − ⊂ Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Maximal Ideals Definition: Maximal ideals An idealm�R is maximal if: for any idealI R such thatm I , eitherm=I orI=R. ⊂ ⊂ Examples (2) Z ⊂ (x,y) k[x,y] ⊂ (x a,y b,z c) k[x,y,z] − − − ⊂ Mahrud Sayrafi LocalRings.m2 Abstract Algebra: Quotients Quotient operation Given idealI R, consider the mapR R/I sendingI to 0. ⊂ → Examples Z Z/(5) then: 7 2 → �→ k[x,y,z] k[x,y,z]/(z) = k[x,y] then: 2x+z 2x → ∼ �→ Mahrud Sayrafi LocalRings.m2 Localizations: Definition Now, instead of mapping elements to zero, we send them to 1 . Mahrud Sayrafi LocalRings.m2 Localizations: Definition Now, instead of mapping elements to zero, we send them to 1 make them invertible. Mahrud Sayrafi LocalRings.m2 Localizations: Definition Now, instead of mapping elements to zero, we send them to 1 make them invertible. Localization Given a multiplicatively closed setS, denote: 1 r S− R= r R,s S / . { s | ∈ ∈ } ∼ r r Where � if we havet S such thatt(rs � r �s) = 0. s ∼ s� ∈ − Mahrud Sayrafi LocalRings.m2 Localizations: Definition Now, instead of mapping elements to zero, we send them to 1 make them invertible. Localization Given a multiplicatively closed setS, denote: 1 r S− R= r R,s S / . { s | ∈ ∈ } ∼ r r Where � if we havet S such thatt(rs � r �s) = 0. s ∼ s� ∈ − Examples Fixf R and letS= 1,f,f 2,... Then elements of ∈ 1 { } Rf :=S − R have powers off in their denominator. Mahrud Sayrafi LocalRings.m2 Localization with respect to Prime Ideal Remark For a prime idealp, we have a multiplicative systemS p :=R p. 1 \ We callR p :=S p− R the localization ofR with respect to prime idealp. Mahrud Sayrafi LocalRings.m2 Localization with respect to Prime Ideal Remark For a prime idealp, we have a multiplicative systemS p :=R p. 1 \ We callR p :=S p− R the localization ofR with respect to prime idealp. Examples (0) Z is prime, thenS =Z 0 and we have a localizaion: ⊂ (0) \{ } Z := a a Z,b Z 0 / (0) { b | ∈ ∈ \{ }} ∼ Mahrud Sayrafi LocalRings.m2 Localization with respect to Prime Ideal Remark For a prime idealp, we have a multiplicative systemS p :=R p. 1 \ We callR p :=S p− R the localization ofR with respect to prime idealp. Examples (0) Z is prime, thenS =Z 0 and we have a localizaion: ⊂ (0) \{ } Z := a a Z,b Z 0 / = Q (0) { b | ∈ ∈ \{ }} ∼ ∼ Mahrud Sayrafi LocalRings.m2 Localization: A Geometric Example Examples Consider the ringR=k[x,y] and ideals: I= x2 + (y + 1)2 1 (y x 2) andp= x+1,y+1 . � − − � � � � � Mahrud Sayrafi LocalRings.m2 Localization: A Geometric Example Examples Consider the ringR=k[x,y] and ideals: I= x2 + (y + 1)2 1 (y x 2) andp= x+1,y+1 . � − − � � � 1 Quotient� R byI to get:� R/I=k[x,y]/ x2 + (y + 1)2 1 (y x 2) . � − − � � � Mahrud Sayrafi LocalRings.m2 Localization: A Geometric Example Examples Consider the ringR=k[x,y] and ideals: I= x2 + (y + 1)2 1 (y x 2) andp= x+1,y+1 . � − − � � � 1 Quotient� R byI to get:� R/I=k[x,y]/ x2 + (y + 1)2 1 (y x 2) . � − − � � � 2 LocalizeR/I with respect top to get: (R/I) = R /I =k[x,y] / x2 + (y + 1)2 1 p ∼ p p (x+1,y+1) � − � Mahrud Sayrafi LocalRings.m2 Localization: A Geometric Example Examples Consider the ringR=k[x,y] and ideals: I= x2 + (y + 1)2 1 (y x 2) andp= x+1,y+1 . � − − � � � 1 Quotient� R byI to get:� R/I=k[x,y]/ x2 + (y + 1)2 1 (y x 2) . � − − � � � 2 LocalizeR/I with respect top to get: (R/I) = R /I =k[x,y] / x2 + (y + 1)2 1 p ∼ p p (x+1,y+1) � − � Remark This is equivalent to quotienting byJ= x 2 + (y + 1)2 1 and � − � then localizing with respect top. Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Affine Variety Affine Variety For a “nice” idealI=(f ,...,f ) C[x,y], its affine variety is: 1 n ⊂ V(I)= (x,y) C 2 f (x,y)= =f (x,y)=0 . { ∈ | 1 ··· n } Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Affine Variety Affine Variety For a “nice” idealI=(f ,...,f ) C[x,y], its affine variety is: 1 n ⊂ V(I)= (x,y) C 2 f (x,y)= =f (x,y)=0 . { ∈ | 1 ··· n } Ideal of a Diagonal Line I= x y � − � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Affine Variety Affine Variety For a “nice” idealI=(f ,...,f ) C[x,y], its affine variety is: 1 n ⊂ V(I)= (x,y) C 2 f (x,y)= =f (x,y)=0 . { ∈ | 1 ··· n } Ideal of a Diagonal Line Variety I= x y � − � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Affine Variety Affine Variety For a “nice” idealI=(f ,...,f ) C[x,y], its affine variety is: 1 n ⊂ V(I)= (x,y) C 2 f (x,y)= =f (x,y)=0 . { ∈ | 1 ··· n } Variety Maximal Ideal Point -1 ⇐⇒ p= x+1,y+1 . � � -1 Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Localization Prime Ideal of Circle I= x 2 + (y + 1)2 1 . � − � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Localization Prime Ideal of Circle Variety I= x 2 + (y + 1)2 1 . � − � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Localization Variety Prime Ideal of Parabola J= y x 2 . � − � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Localization Prime Ideal of Circle Variety I= x 2 + (y + 1)2 1 . � − � Prime Ideal of Parabola J= y x 2 . � − � Product Ideal IJ= x2 +(y +1)2 1 (y x 2) . � − − � � � Mahrud Sayrafi LocalRings.m2 Algebraic Geometry: Localization Prime Ideal of Circle Variety I= x 2 + (y + 1)2 1 . � − � Prime Ideal of Parabola J= y x 2 . � − � Product Ideal IJ= x2 +(y +1)2 1 (y x 2) . � − − � � � Localized Ideal (IJ) = x 2 + (y + 1)2 1 . p � − � Mahrud Sayrafi LocalRings.m2 Table of Contents 1 Definitions 2 My Work 3 Applications 4 Conclusion Mahrud Sayrafi LocalRings.m2 Writing a package for Macaulay2 computer algebra system to handle computations in local rings. Mahrud Sayrafi LocalRings.m2 Writing a package for Macaulay2 computer algebra system to handle computations in local rings. Main methods: Gr¨obnerbases Syzygies Free resolutions Mahrud Sayrafi LocalRings.m2 Writing a package for Macaulay2 computer algebra system to handle computations in local rings. Main methods: Gr¨obnerbases Syzygies Free resolutions Developing a “monomial order” for efficient algorithms Mahrud Sayrafi LocalRings.m2 Writing a package for Macaulay2 computer algebra system to handle computations in local rings. Main methods: Gr¨obnerbases Syzygies Free resolutions Developing a “monomial order” for efficient algorithms for localization at maximal ideals: Mora’s tangent cone algorithm is efficient. for localization at prime ideals: no standard basis is known. Mahrud Sayrafi LocalRings.m2 Table of Contents 1 Definitions 2 My Work 3 Applications 4 Conclusion Mahrud Sayrafi LocalRings.m2 Applications Generally, many problems in commutative algebra and algebraic geometry are easier to solve if you look at the problem locally. Mahrud Sayrafi LocalRings.m2 Applications Generally, many problems in commutative algebra and algebraic geometry are easier to solve if you look at the problem locally. Testing local properties Mahrud Sayrafi LocalRings.m2 Applications Generally, many problems in commutative algebra and algebraic geometry are easier to solve if you look at the problem locally. Testing local properties: is my curve Cohen-Macaulay? Mahrud Sayrafi LocalRings.m2 Applications Generally, many problems in commutative algebra and algebraic geometry are easier to solve if you look at the problem locally. Testing local properties: is my curve Cohen-Macaulay? is it a complete intersection? Mahrud Sayrafi LocalRings.m2 Applications Generally, many problems in commutative algebra and algebraic geometry are easier to solve if you look at the problem locally.