an invitation to combinatorial commutative algebra · combinatorial commutative algebra is a very...

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An invitation to Combinatorial Commutative AlgebraPhD Seminar

Alessio D’Alı

Universita degli Studi di Genova

January 20th, 2016

Alessio D’Alı (Univ. di Genova) Combinatorial Commutative Algebra January 20th, 2016 1 / 20

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Introduction

Combinatorial commutative algebra is a very lively branch in modernmathematics. It combines the broad abstract methods of algebra,geometry and topology with the more intuitive ones of combinatorics,”the art of counting”, which is a common ground for mathematicianscoming from many branches.In particular, the interplay between commutative algebra, algebraicgeometry and combinatorics has been proved to be effective also inseemingly distant areas like statistics (giving rise to the realm ofalgebraic statistics) and even in different fields like computer scienceand biology.


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Monomial and toric ideals

Combinatorial commutative algebra often deals with squarefree monomialsinside a polynomial ring S = k[x1, . . . , xn]. Here “squarefree” just meansthat no variable can appear twice inside the same monomial: x1x2 issquarefree, but x2

1 x2 is not.Given a set m1, . . . , mt of squarefree monomials, we can construct twodifferent algebraic-combinatorial objects of interest:

the squarefree monomial ideal of S generated by m1, . . . , mt ;the k-algebra k[m1, . . . , mt ] generated by our set. The kernel of thestandard presentation

k[y1, . . . , yt ] ↠ k[m1, . . . , mt ]yi 7→ mi

is the toric ideal associated with our set.

During the rest of this talk we will focus on squarefree monomial ideals.


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Modules

Let R be a (commutative and unitary) ring. An abelian group (M, +) issaid a (left) R-module if there exists an application

R ×M → M(r , m) 7→ rm

such that (r + r ′)m = rm + r ′m, r(m + m′) = rm + rm′, (rr ′)m = r(r ′m)and 1Rm = m for all r , r ′ ∈ R, m, m′ ∈ M.

When R = Z, we get abelian groups.When R = k, we get k-vector spaces.If I is an ideal of R, then both I and R/I have a natural R-modulestructure.Rn =

⊕ni=1 R is called a free R-module; note that free R-modules are

the only R-modules endowed with a basis.


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Chain complexes and homology

A map of R-modules ϕ : M → N is a map of abelian groups such thatϕ(rm) = rϕ(m) for all r ∈ R, m ∈ M.A chain complex (C•, ∂•) is a collection of R-modules Ci and maps∂i : Ci → Ci−1 (where i ∈ Z) such that ∂i ◦ ∂i+1 = 0 for all i .

C• : . . .→ Ci+1∂i+1−−→ Ci

∂i−→ Ci−1 → . . .

Note that the condition on ∂• means that im(∂i+1) ⊆ ker(∂i) for all i .It then makes sense to consider the R-module

Hi(C•) := ker(∂i)/im(∂i+1).

This is called the i-th homology module of the chain complex C•.When Hi(C•) = 0, i.e. im(∂i+1) = ker(∂i), we say that C• is exact in thei-th homological position.


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From abelian groups to general modules / 1

Let G be a Z-module, i.e. an abelian group. It is well-known that anygroup, in particular an abelian one, can be expressed in terms ofgenerators and relations. For simplicity, assume G is finitely generated.Let us consider, for instance,

G = Z⊕ Z/2Z⊕ Z/9Z.

G has three generators g1 = (1, 0, ¯0), g2 = (0, 1, ¯0) and g3 = (0, 0, ¯1); bydefinition, one has that 2(0, 1, ¯0) = 9(0, 0, ¯1) = 0. This can be translatedby saying that the sequence

0→ Z2

0 02 00 9

−−−−−−→ Z3

(g1 g2 g3

)−−−−−−−−−−→ G → 0

is exact.Alessio D’Alı (Univ. di Genova) Combinatorial Commutative Algebra January 20th, 2016 6 / 20

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Let G be a Z-module, i.e. an abelian group. It is well-known that anygroup, in particular an abelian one, can be expressed in terms ofgenerators and relations. For simplicity, assume G is finitely generated.Let us consider, for instance,

G = Z⊕ Z/2Z⊕ Z/9Z.

G has three generators g1 = (1, 0, ¯0), g2 = (0, 1, ¯0) and g3 = (0, 0, ¯1); bydefinition, one has that 2(0, 1, ¯0) = 9(0, 0, ¯1) = 0. This can be translatedby saying that the sequence

0→ Z2

(relations on thegenerators of G

)−−−−−−−−−−−−−−→ Z3

(generators of G

)−−−−−−−−−−−−−→ G → 0

is exact.


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In the previous slide we have found an alternative way to express theinformation in a finitely generated Z-module. What happens when weconsider something more complicated?Let S = k[x , y , z ] and let I be the ideal generated by x2, xy and xz .Then, starting as before, we can construct generators and relations:

0→ S

z−yx

−−−−−→

S3

(relations on thegenerators of I

)−−−−−−−−−−−−−−→ S3

(generators of I

)−−−−−−−−−−−−−→ I → 0

This time, though, the map on the left is not injective. We need to go onestep further, i.e. consider the relations... on the relations!

This is the idea behind the concept of free resolution.


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0→ S

z−yx

−−−−−→

S3

y z 0−x 0 z0 −x −y

−−−−−−−−−−−−−→ S3

(x2 xy xz

)−−−−−−−−−−→ I → 0




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0→ S

z−yx

−−−−−→ S3

y z 0−x 0 z0 −x −y

−−−−−−−−−−−−−→ S3

(x2 xy xz

)−−−−−−−−−−→ I → 0




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Free resolutions

DefinitionLet R be a ring and M be a finitely generated R-module. We say that thecomplex

F• : . . .→ F3ϕ3−→ F2

ϕ2−→ F1ϕ1−→ F0 → 0

is a free resolution for M as an R-module if:the sequence is exact in all positions except the zeroth andH0(F•) ∼= M;Fi is a free R-module ∀i ∈ N.

If R and M are graded (think of the case when R is a polynomial ring andM is a homogeneous ideal), we say the resolution above is graded if eachFi is a graded module and each map involved is a homomorphismpreserving the degree.

Note that all the maps ϕi can be expressed as matrices.


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Minimal graded free resolution

Some resolutions are better than others!Let S = k[x1, . . . , xn]. A graded free resolution where for each Fi we havethat

ϕi(Fi) ⊆MFi−1,

where M is the homogeneous maximal ideal of S generated by thevariables x1, . . . , xn, is called minimal. This means precisely that all thenonzero elements appearing in the matrices describing the homomorphismsϕi are homogeneous and belong to M.

TheoremMinimal graded free resolutions are unique up to isomorphism ofcomplexes.


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Betti numbers

Since there exists only one minimal graded free resolution of M up toisomorphism of complexes, it makes sense to count the copies of the ringthat are used in each step of this resolution and associate this set ofnumbers with M itself. Let us consider our previous example, where Mwas a homogeneous ideal:

0→ S1

z−yx

−−−−−→ S3

y z 0−x 0 z0 −x −y

−−−−−−−−−−−−−→ S3 → 0

There are three copies of S in the zeroth position, another three copies ofS in the first position, and one copy of S in the second position.

These numbers are the Betti numbers of M over S, denoted by βSi (M).

In this case, β0(M) = 3, β1(M) = 3, β2(M) = 1, βi(M) = 0 for all i ≥ 3.


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Simplicial complexes: a brief introductionLet [n] = {1, 2, . . . , n} and let ∆ be a subset of 2[n]. We call ∆ asimplicial complex if the following property holds:

if F ∈ ∆ and G ⊆ F , then G ∈ ∆.

The elements in ∆ are called faces of the complex.There is a nice geometric way to look at these objects (and this is wherethe name “simplicial complex” comes from): roughly speaking, we can seeevery face of ∆ as a simplex of the “right” dimension.Simplicial complexes turn out to be collections of simplices glued togetherby their faces.

∆ = {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {2, 4},{3, 4}, {1}, {2}, {3}, {4}, ∅}

2

13

4


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Simplicial complexes and squarefree monomial ideals

How can we do (commutative) algebra with simplicial complexes?

Fix a field k. Given a complex ∆ with n vertices, one can consider theideal I∆ of the polynomial ring k[x1, . . . , xn] generated by the nonfaces of∆, i.e. the squarefree monomial ideal defined in the following way:

I∆ := (xi1xi2 . . . xis | {i1, i2, . . . , is} /∈ ∆).

DefinitionI∆ is called the Stanley-Reisner ideal of ∆.

Note that we can restrict ourselves to the minimal (with respect toinclusion) nonfaces of ∆, since nonfaces that are not minimal will giveredundant generators of the ideal.


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An example of a Stanley-Reisner ideal

∆ = {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {2, 4},{3, 4}, {1}, {2}, {3}, {4}, ∅}

2

13

4

Nonfaces of ∆: {1, 4}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}.Stanley-Reisner ideal of ∆:

I∆ = (x1x4, x1x2x4, x1x3x4, x2x3x4, x1x2x3x4).


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∆ = {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {2, 4},{3, 4}, {1}, {2}, {3}, {4}, ∅}

2

13

4

Minimal nonfaces of ∆: {1, 4},��{1, 2, 4},��{1, 3, 4}, {2, 3, 4},��{1, 2, 3, 4}.Stanley-Reisner ideal of ∆:

I∆ = (x1x4,��x1x2x4,��x1x3x4, x2x3x4,(((((x1x2x3x4).


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∆ = {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {2, 4},{3, 4}, {1}, {2}, {3}, {4}, ∅}

2

13

4

Minimal nonfaces of ∆: {1, 4}, {2, 3, 4}.Stanley-Reisner ideal of ∆:

I∆ = (x1x4, x2x3x4).


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The Stanley-Reisner correspondence

We have just seen that, given a simplicial complex ∆, we canassociate a squarefree monomial ideal with it.

On the other hand, given any minimal set of squarefree monomials,we can associate a unique simplicial complex with it.

This is called the Stanley-Reisner correspondence:

squarefree monomial ideals ←→ simplicial complexes

The Stanley-Reisner correspondence is not the only way to associatesquarefree monomial ideals with simplicial complexes and viceversa, but itis in some sense the most natural. More on this later!


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Simplicial homology / 1

Given a simplicial complex ∆ and a field k, one can construct theassociated chain complex C•(∆,k): this is done as follows.

Ci(∆,k) is the k-vector space whose dimension equals the number offaces of dimension i in ∆. The empty set has dimension −1.Given a face F and the corresponding basis element eF , one has that

∂(eF ) =∑v∈F

(−1)[v ,F ]eF\v ,

where (−1)[v ,F ] is a sign depending on the position of v inside theface F (after fixing a total order on the vertices of ∆).

DefinitionThe homology of C•(∆,k) is called reduced homology of ∆ and isdenoted by H•(∆,k).


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Simplicial homology / 2Homology is actually a topological feature of the simplicial complex,i.e. it depends on the “shape” of the object. It is invariant underhomeomorphism and even homotopy equivalence.

Let us consider an example:

y

z

x

∆ = {∅, {x}, {y}, {z}, {x , y}, {x , z}, {y , z}}

C•(∆,k) : 0← k{∅}

(1 1 1

)←−−−−−−−−

k{x}⊕

k{y}⊕

k{z}

1 1 0−1 0 10 −1 −1

←−−−−−−−−−−−−−

k{xy}⊕

k{xz}⊕

k{yz}

← 0

H−1(∆,k) = H0(∆,k) = 0, H1(∆,k) ∼= k.


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Combinatorics is not enough

Attention!The homology of simplicial complexes does depend (in general) on thechoice of the field k!

1

2

3

1

2

3

5 6

4

Standard triangulation ∆ of the real projective plane RP2.H1(∆,k) ∼= k if k has characteristic 2; otherwise, H1(∆,k) = 0.


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Recap

Commutative algebra: free resolutions of modules are importantbecause they encode structure information. Betti numbers arenumerical invariants tied to minimal free resolutions.

Combinatorics: simplicial complexes are collections of simplices gluedtogether by their faces. They are in one-to-one correspondence withsquarefree monomial ideals (Stanley-Reisner correspondence).

Algebraic topology: homology is a set of invariants attached to asimplicial complex (and, more generally, to a topological space).Homotopy equivalent spaces have the same homology.


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Where the worlds meet: Hochster’s formula / 1

In general, computing Betti numbers is possible but costly (since thereare Grobner bases involved). On the other hand, computing homologycan be reduced to linear algebra procedures and is hence easier.

The Stanley-Reisner correspondence is not just formal, but allows usto study algebraic properties of I∆ via an analysis of thecorresponding simplicial complex ∆. The crucial instance of thisphenomenon is Hochster’s formula, which allows a fruitful interplaybetween commutative algebra, combinatorics and algebraic topology.


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Where the worlds meet: Hochster’s formula / 2

Given a simplicial complex ∆, pick a subset R of its vertices. Thesubcomplex of ∆ induced by R (denoted by ∆|R) is the simplicial complexwhose faces are exactly the faces of ∆ contained in R. We will denote by|R| the cardinality of R.

Hochster’s formula (1977)

βi ,R(I∆) = dimk H|R|−i−2(∆|R ;k).

Since homology depends on the choice of k, it turns out that Bettinumbers (even of monomial ideals!) are characteristic-dependent ingeneral.On the other hand, there are lucky situations where one can really useonly the combinatorial structure of the simplicial complex to computeBetti numbers. This has been a very active research topic in the lastdecades.


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Thank you for your attention!


an invitation to combinatorial commutative algebra · combinatorial commutative algebra is a very...

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