Computations in Local Rings Using Macaulay2

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

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

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: Gröbnerbases 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? Computing local cohomology. Resolution of Singularities. Applications in applied math: Solving “inverse systems” of diﬀerential equations. Local invariants are related various open problems in algebraic vision.

Mahrud Sayrafi LocalRings.m2 Singular Varieties

Ideal of a Nodal Cubic Ideal of a Cuspidal Cubic p=(x 3 +x 2 y 2) q=(x 3 y 2) − −

Mahrud Sayrafi LocalRings.m2 Singular Varieties

Ideal of a Nodal Cubic Ideal of a Cuspidal Cubic p=(x 3 +x 2 y 2) q=(x 3 y 2) − − Variety Variety

Mahrud Sayrafi LocalRings.m2 Blowing up Singularities!!

Ideal of a Nodal Cubic p=(x 3 +x 2 y 2) − Singular Variety

Mahrud Sayrafi LocalRings.m2 Blowing up Singularities!!

Ideal of a Nodal Cubic Resolving the Singularity p=(x 3 +x 2 y 2) − Singular Variety

Mahrud Sayrafi LocalRings.m2 Table of Contents

1 Definitions

2 My Work

3 Applications

4 Conclusion

Mahrud Sayrafi LocalRings.m2 Interested? Read here:

Mahrud Sayrafi LocalRings.m2 Interested? Read here: Final thoughts ... Thank you! My code is available on GitHub: https://goo.gl/ZoIY84 https://github.com/mahrud

Mahrud Sayrafi LocalRings.m2