Free resolutions talk at the ACAGM Summer School Leuven p. 1

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Free resolutions

talk at the

ACAGM Summer School Leuven 2011

-Hans Sch ¨onemann

hannes@mathematik.uni-kl.de

Department of Mathematics University of Kaiserslautern

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Syzygies

Definition

Let I = {g1, . . . , gq} ⊆ K[x]r.

The module of syzygies syz(I) is

ker(K[x]q → K[x]r),Xwiei 7→

X

wigi

Lemma The module of syzygies of I is

(g1(x) − er+1, . . . , gq(y) − er+q) ∩ {0}r × K[x]q

in (K[x1, . . . , xm]/J)q .

ring R=0,(x,y,z),(c,dp); ideal I=maxideal(1);

// the syzygies of the (x,y,z) syz(I);

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Generalization of Gröbner bases to Modules

monomials me_i ∈ Rr, i = 1..r, m monomial in R monomial orderings

L(f) leading monomial, L(M) leading submodule NF(f|M) normal form

GB(M) Gröbner basis Buchbergers algorithm

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Generalization of Monomial Orderings to Modules

In what follows, let F be the free R-module F = Rs with its canonical basis e1, . . . , e_s.

Definition. A monomial in F is a monomial in R times a basis vector

of F, that is, an element of the form xαei. A term in F is a monomial

in F times a scalar, that is, an element of type axαei, where a ∈ K. A

submodule of F which is generated by monomials is called a

monomial submodule. A monomial order on F may be defined in the

same way as a monomial ordering on R. That is, it is a total order > on the set of monomials in F satisfying

xαei > xβej =⇒ xγxαei > xγxβej for each γ ∈ Nn.

We require in addition that

xαei > xβei ⇐⇒ xαej > xβej ,∀i, j = 1, . . . , s.

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Generalization of Monomial Ordering to Modules

Important orderings: term over position

ring R=....,(dp,c); ring R=....,(dp,C);

position over term

ring R=....,(c,dp); ring R=....,(C,dp);

Capital C sorts generators in ascending order, i.e., gen(1) < gen(2) < ....

A small c sorts in descending order, i.e., gen(1) > gen(2) > .... Ordering, ..., C ) is the default.

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Generalization of Gröbner Bases to Modules

Finally, given a monomial order on F, we define the leading term, the leading coefficient, the leading monomial, and the tail of an element of

F as we did for a polynomial in R.

With this basic notation, the whole concept of Gröbner bases including its fundamental algorithms extend.

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Free Resolutions

Let I = (g1, ..., gs) ⊆ Rr and M = Rr/I. A free resolution of M is a

long exact sequence

... −→ F2 −→ F1 −→ F0 −→ M −→ 0,

where the columns of the matrix A1generate I,the columns of A2

generate the kernel of A1 etc. Note that resolutions need not to be

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Hilbert Syzygy Theorem

Hilbert Syzygy Theorem

For an ideal I ∈ K[x1...xn] there exists a free resolution of length

smaller or equal than n.

For a module M = K[x1...xn]r/N there exists a free resolution of

length smaller or equal than n.

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Free Resolutions

By iterating the process of computing syzygies, starting from a finitely generated R-module N, we get what is called a free resolution of M = Rr/N:

ring r=0,(x,y,z),dp; ideal I=x,y,z;

list Ir=res(I,0);

// print the results: Ir;

list Im=mres(I,0);

// print the results: Im;

list Is=sres(std(I),0); // print the results: Is;

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Intersection of Modules

Syzygies can also be used to compute ideal intersections and ideal quotients.

Given ideals I = hf1, . . . , f_ri and J = hg1, . . . , g_si of R, compute the

syzygies on the columns of the matrix

1 f1 . . . fr 0 . . . 0

1 0 . . . 0 g1 . . . gs

! .

The entries of the first row of the resulting syzygy matrix generate I ∩ J.

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The Geometry-Algebra Dictionary

Ideal quotient I

In general, given two algebraic sets, it is not their difference but the smallest algebraic set containing the difference which can be

described by polynomial equations.

Definition. If I, J are two ideals of a ring R, the set

I : J = {f ∈ R | f g ∈ I for all g ∈ J}

is an ideal of R containing I, the ideal quotient of I by J. Similarly, the

set I : J∞ := {f ∈ R | f Jm ⊂ I for some m ≥ 1} = ∞ [ m=1 (I : Jm)

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The Geometry-Algebra Dictionary

Ideal quotient II

Theorem. If K = K, and I, J are ideals of R, then

V (I) \ V (J) = V (I : J∞

). If I is a radical ideal, then

V (I) \ V (J) = V (I : J).

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The Geometry-Algebra Dictionary

Saturation

Since the polynomial ring R is Noetherian by Hilbert’s basis theorem, and since I : Jm = (I : Jm−1

) : J for any two ideals I, J ⊂ R, the computation of the saturation

I : J∞ = ∞ [ m=1 (I : Jm)

just means to iterate the computation of ideal quotients. Indeed, the ascending chain

I : J ⊂ I : J2 ⊂ · · · ⊂ I : Jm ⊂ . . .

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Ideal Quotient: Computation

In the same way as with the intersection, we get generators for I : J from the matrix

      g1 f1 . . . fr 0 . . . 0 g2 0 . . . 0 f1 . . . fr 0 . . . 0 .. . . .. gs 0 . . . 0 f1 . . . fr       .

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Ideal quotient: Example

Example > ring R = 2, (x,y,z), dp; > poly F _{= x5+y5+(x-y)ˆ2*xyz;} > ideal J = jacob(F); > J;

//-> J[1]=x4+x2yz+y3z J[2]=y4+x3z+xy2z J[3]=x3y+xy3 > maxideal(2);

//-> _[1]=z2 _[2]=yz _[3]=y2 _[4]=xz _[5]=xy _[6]=x2 > ideal K = quotient(J,maxideal(2));

> K;

//-> K[1]=y4+x3z+xy2z K[2]=x3y+xy3 K[3]=x4+x2yz+y3z //-> K[4]=x3z2+x2yz2+xy2z2+y3z2 K[5]=x2y2z+x2yz2+y3z2 //-> K[6]=x2y3

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Ideal quotient: Example

> K = quotient(K,maxideal(2)); > K; K[1]=x3+x2y+xy2+y3 K[2]=y4+x2yz+y3z K[3]=x2y2+y4 > K = quotient(K,maxideal(2)); > K; K[1]=x3+x2y+xy2+y3 K[2]=y4+x2yz+y3z K[3]=x2y2+y4

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Saturation: Example

> LIB "elim.lib";

> int p = printlevel;

> printlevel = 2; // print more information while computing > sat(J,maxideal(2));

// compute quotient 1 // compute quotient 2 // compute quotient 3

// saturation becomes stable after 2 iteration(s) [1]: _[1]=x3+x2y+xy2+y3 _[2]=y4+x2yz+y3z _[3]=x2y2+y4 [2]: 2

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Summary of Operations with Ideals

sum of ideals (intersection of algebraic sets)

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Summary of Operations with Ideals

sum of ideals (intersection of algebraic sets) intersection of ideals (union of algebraic sets)

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Summary of Operations with Ideals

elimination of variables (projection of algebraic sets)

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Summary of Operations with Ideals

elimination of variables (projection of algebraic sets) ideal quotient/saturation (“difference“ of algebraic sets)

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Summary of Operations with Ideals

elimination of variables (projection of algebraic sets) ideal quotient/saturation (“difference“ of algebraic sets) Hilbert function

dimension of the ideals (dimension of the algebraic set)