Multiplicative structures on minimal free resolutions ofmonomial ideals
Lukas Katthän
Goethe-Universität Frankfurt

August 5, 2016

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

1 / 29

1

General setting
Notation and preliminaries
Multiplications on free resolutions
Existence of multiplications
Existence of associative multiplications

2

The monomial case
Supportive multiplications
Properties of supportive multiplications
Supportive DGA structures and the Taylor resolution

3

Applications
Betti Vectors of monomial ideals
Subadditivity of syzygies

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

2 / 29Section 1
General setting

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Notation and conventions

k: some field.
S = k[x1 , . . . , xn ]: polynomial ring over k, either with the standard
grading or with the fine Zn -grading.
I ⊂ S: homogeneous ideal

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

4 / 29Reminder: Minimal free resolutions
A minimal free resolution of S/I is an exact sequence of free S-modules
F• :

∂

∂

∂

∂

0 → Fp → Fp−1 → · · · → F1 → F0such that F0 /∂(F1 ) ∼= S/I and ∂(Fi ) ⊂ mFi−1 where m = (x1 , . . . , xn ).
Some facts:
A minimal free resolution always exists and it is unique up toisomorphism. So we may speak about “the” minimal free resolutionof S/I.
All maps in F• can be chosen homogeneous with respect to thegrading from S.
L

We consider F• as the free S-module i Fi . Note that we have two
gradings: The homological one, and the one coming from S.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

5 / 29Reminder: Minimal free resolutions
A minimal free resolution of S/I is an exact sequence of free S-modules
F• :

∂

∂

∂

∂

0 → Fp → Fp−1 → · · · → F1 → F0such that F0 /∂(F1 ) ∼= S/I and ∂(Fi ) ⊂ mFi−1 where m = (x1 , . . . , xn ).
Some facts:
A minimal free resolution always exists and it is unique up toisomorphism. So we may speak about “the” minimal free resolutionof S/I.
All maps in F• can be chosen homogeneous with respect to thegrading from S.
L

We consider F• as the free S-module i Fi . Note that we have two
gradings: The homological one, and the one coming from S.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

5 / 29Reminder: Minimal free resolutions
A minimal free resolution of S/I is an exact sequence of free S-modules
F• :

∂

∂

∂

∂

0 → Fp → Fp−1 → · · · → F1 → F0such that F0 /∂(F1 ) ∼= S/I and ∂(Fi ) ⊂ mFi−1 where m = (x1 , . . . , xn ).
Some facts:
A minimal free resolution always exists and it is unique up toisomorphism. So we may speak about “the” minimal free resolutionof S/I.
All maps in F• can be chosen homogeneous with respect to thegrading from S.
L

We consider F• as the free S-module i Fi . Note that we have two
gradings: The homological one, and the one coming from S.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

5 / 29Reminder: Minimal free resolutions
A minimal free resolution of S/I is an exact sequence of free S-modules
F• :

∂

∂

∂

∂

0 → Fp → Fp−1 → · · · → F1 → F0such that F0 /∂(F1 ) ∼= S/I and ∂(Fi ) ⊂ mFi−1 where m = (x1 , . . . , xn ).
Some facts:
A minimal free resolution always exists and it is unique up toisomorphism. So we may speak about “the” minimal free resolutionof S/I.
All maps in F• can be chosen homogeneous with respect to thegrading from S.
L

We consider F• as the free S-module i Fi . Note that we have two
gradings: The homological one, and the one coming from S.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

5 / 29More notation and conventions

F• : minimal free resolution of S/I.
For g ∈ F• :I
I

|g|: homological degree,
deg g: degree with respect to the grading on S.

βi (S/I) := rank Fi : ith Betti number of S/I.
Remark: F0 ∼= S, so β0 (S/I) = 1. Further, β1 (S/I) is the minimal numberof generators of I.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

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6 / 29More notation and conventions

F• : minimal free resolution of S/I.
For g ∈ F• :I
I

|g|: homological degree,
deg g: degree with respect to the grading on S.

βi (S/I) := rank Fi : ith Betti number of S/I.
Remark: F0 ∼= S, so β0 (S/I) = 1. Further, β1 (S/I) is the minimal numberof generators of I.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

6 / 29More notation and conventions

F• : minimal free resolution of S/I.
For g ∈ F• :I
I

|g|: homological degree,
deg g: degree with respect to the grading on S.

βi (S/I) := rank Fi : ith Betti number of S/I.
Remark: F0 ∼= S, so β0 (S/I) = 1. Further, β1 (S/I) is the minimal numberof generators of I.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

6 / 29More notation and conventions

F• : minimal free resolution of S/I.
For g ∈ F• :I
I

|g|: homological degree,
deg g: degree with respect to the grading on S.

βi (S/I) := rank Fi : ith Betti number of S/I.
Remark: F0 ∼= S, so β0 (S/I) = 1. Further, β1 (S/I) is the minimal numberof generators of I.

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DGA Resolutions

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Multiplications on free resolutions
General idea: Turn the whole resolution into an algebra over S.

Definition
A multiplication on F• is a map · : F• ⊗S F• → F• which is
1

S-linear and respects both the homological and the internal grading,2

satisfies a Leibniz rule: ∂(a · b) = (∂a) · b + (−1)|a| a · ∂b.

3and extends the multiplication on F0 = S.

Definition
A DGA structure on F• is a multiplication, which in addition is4

graded-commutative: a · b = (−1)|a|·|b| b · a and5

associative.

DGA: Differential graded algebra.

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Multiplications on free resolutions
General idea: Turn the whole resolution into an algebra over S.

Definition
A multiplication on F• is a map · : F• ⊗S F• → F• which is
1

S-linear and respects both the homological and the internal grading,2

satisfies a Leibniz rule: ∂(a · b) = (∂a) · b + (−1)|a| a · ∂b.

3and extends the multiplication on F0 = S.

Definition
A DGA structure on F• is a multiplication, which in addition is4

graded-commutative: a · b = (−1)|a|·|b| b · a and5

associative.

DGA: Differential graded algebra.

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DGA Resolutions

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Multiplications on free resolutions
General idea: Turn the whole resolution into an algebra over S.

Definition
A multiplication on F• is a map · : F• ⊗S F• → F• which is
1

S-linear and respects both the homological and the internal grading,2

satisfies a Leibniz rule: ∂(a · b) = (∂a) · b + (−1)|a| a · ∂b.

3and extends the multiplication on F0 = S.

Definition
A DGA structure on F• is a multiplication, which in addition is4

graded-commutative: a · b = (−1)|a|·|b| b · a and5

associative.

DGA: Differential graded algebra.

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DGA Resolutions

August 5, 2016

7 / 29

Multiplications on free resolutions
General idea: Turn the whole resolution into an algebra over S.

Definition
A multiplication on F• is a map · : F• ⊗S F• → F• which is
1

S-linear and respects both the homological and the internal grading,2

satisfies a Leibniz rule: ∂(a · b) = (∂a) · b + (−1)|a| a · ∂b.

3and extends the multiplication on F0 = S.

Definition
A DGA structure on F• is a multiplication, which in addition is4

graded-commutative: a · b = (−1)|a|·|b| b · a and5

associative.

DGA: Differential graded algebra.

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DGA Resolutions

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Example 1
Let I = hx 2 , xy , y 2 i ⊂ k[x , y ].
F• :

0→

S2

−→

g12
g23

7→
7
→

S3
−→ S → S/I
g1
7→ x 2g2
7→ xy
g3
7→ y 2xg2 − yg1
xg3 − yg22

13Multiplication on F• ?
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Example 1
Let I = hx 2 , xy , y 2 i ⊂ k[x , y ].
F• :

0→

S2

−→

g12
g23

7→
7
→

S3
−→ S → S/I
g1
7→ x 2g2
7→ xy
g3
7→ y 2xg2 − yg1
xg3 − yg22

13Multiplication on F• ?
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DGA Resolutions

August 5, 2016

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Example 1
Let I = hx 2 , xy , y 2 i ⊂ k[x , y ].
F• :

0→

S2

−→

g12
g23

7→
7
→

S3
−→ S → S/I
g1
7→ x 2g2
7→ xy
g3
7→ y 2xg2 − yg1
xg3 − yg22

13Multiplication on F• ?
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

8 / 29

Example 1
Let I = hx 2 , xy , y 2 i ⊂ k[x , y ].
F• :

0→

S2

−→

g12
g23

7→
7
→

S3
−→ S → S/I
g1
7→ x 2g2
7→ xy
g3
7→ y 2xg2 − yg1
xg3 − yg22

13Multiplication on F• ?
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

8 / 29

Example 1
Let I = hx 2 , xy , y 2 i ⊂ k[x , y ].
F• :

0→

S2

−→

g12
g23

7→
7
→

S3
−→ S → S/I
g1
7→ x 2g2
7→ xy
g3
7→ y 2xg2 − yg1
xg3 − yg22

13Multiplication on F• ?
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Existence of multiplications: Motivation

Why do we care about multiplications?
DGA Structures on F• allow us to infer statements about F• . For example:

Proposition (Eisenbud-Buchsbaum ’77)
If F• admits a DGA structure, thenβi (S/I) ≥gi!

for 0 ≤ i ≤ g, where g = grade I.
Recall: grade I is the maximal length of a regular sequence in I.

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DGA Resolutions

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Existence of multiplications: Motivation

Why do we care about multiplications?
DGA Structures on F• allow us to infer statements about F• . For example:

Proposition (Eisenbud-Buchsbaum ’77)
If F• admits a DGA structure, thenβi (S/I) ≥gi!

for 0 ≤ i ≤ g, where g = grade I.
Recall: grade I is the maximal length of a regular sequence in I.

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Existence of multiplications

Proposition
Every resolution of S/I admits a multiplication, and it is unique up tohomotopy.
Proof: Comparison Theorem (or Homotopy Transfer Theorem).
“up to homotopy”: If µ, µ0 are two multiplications, then there exists a
map σ : F• ⊗ F• → F• of homological degree +1, such thatµ − µ0 = σ ◦ ∂ + ∂ ◦ σ.
(Eisenbud-Buchsbaum ’77) There always exists a multiplication whichis graded-commutative.
From now on: All multiplications are assumed to be graded-commutative.

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Existence of multiplications

Proposition
Every resolution of S/I admits a multiplication, and it is unique up tohomotopy.
Proof: Comparison Theorem (or Homotopy Transfer Theorem).
“up to homotopy”: If µ, µ0 are two multiplications, then there exists a
map σ : F• ⊗ F• → F• of homological degree +1, such thatµ − µ0 = σ ◦ ∂ + ∂ ◦ σ.
(Eisenbud-Buchsbaum ’77) There always exists a multiplication whichis graded-commutative.
From now on: All multiplications are assumed to be graded-commutative.

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DGA Resolutions

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Existence of multiplications

Proposition
Every resolution of S/I admits a multiplication, and it is unique up tohomotopy.
Proof: Comparison Theorem (or Homotopy Transfer Theorem).
“up to homotopy”: If µ, µ0 are two multiplications, then there exists a
map σ : F• ⊗ F• → F• of homological degree +1, such thatµ − µ0 = σ ◦ ∂ + ∂ ◦ σ.
(Eisenbud-Buchsbaum ’77) There always exists a multiplication whichis graded-commutative.
From now on: All multiplications are assumed to be graded-commutative.

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DGA Resolutions

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Existence of multiplications

Proposition
Every resolution of S/I admits a multiplication, and it is unique up tohomotopy.
Proof: Comparison Theorem (or Homotopy Transfer Theorem).
“up to homotopy”: If µ, µ0 are two multiplications, then there exists a
map σ : F• ⊗ F• → F• of homological degree +1, such thatµ − µ0 = σ ◦ ∂ + ∂ ◦ σ.
(Eisenbud-Buchsbaum ’77) There always exists a multiplication whichis graded-commutative.
From now on: All multiplications are assumed to be graded-commutative.

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DGA Resolutions

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Existence of multiplications

What about associative multiplications, i.e. DGA structures?
(Buchsbaum-Eisenbud ’77) Conjecture: One can always find a DGA
structure on F• .
This is not true (Avramov-Khimich ’74)
Remark: For a given ideal I ⊂ S, one can always find a non-minimal freeresolution which does admit a DGA structure.

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DGA Resolutions

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Existence of multiplications

What about associative multiplications, i.e. DGA structures?
(Buchsbaum-Eisenbud ’77) Conjecture: One can always find a DGA
structure on F• .
This is not true (Avramov-Khimich ’74)
Remark: For a given ideal I ⊂ S, one can always find a non-minimal freeresolution which does admit a DGA structure.

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DGA Resolutions

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Existence of multiplications

What about associative multiplications, i.e. DGA structures?
(Buchsbaum-Eisenbud ’77) Conjecture: One can always find a DGA
structure on F• .
This is not true (Avramov-Khimich ’74)
Remark: For a given ideal I ⊂ S, one can always find a non-minimal freeresolution which does admit a DGA structure.

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DGA Resolutions

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An ideal without a DGA resolution
Consider the ideal I ⊂ k[x , y , z, v ] with generators
m1 = x 2 , m2 = xy , m3 = y 2 z 2 , m4 = zv , m5 = v 2This ideal is generic, so its minimal free resolution is supported on its Scarf
complex:
1
2

3

4

5

It does not admit a DGA structure. More precisely, it does not admit a
DGA-structure which respects the multigrading.
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DGA Resolutions

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An ideal without a DGA resolution
Consider the ideal I ⊂ k[x , y , z, v ] with generators
m1 = x 2 , m2 = xy , m3 = y 2 z 2 , m4 = zv , m5 = v 2This ideal is generic, so its minimal free resolution is supported on its Scarf
complex:
1
2

3

4

5

It does not admit a DGA structure. More precisely, it does not admit a
DGA-structure which respects the multigrading.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201612 / 29

An ideal without a DGA resolution
Consider the ideal I ⊂ k[x , y , z, v ] with generators
m1 = x 2 , m2 = xy , m3 = y 2 z 2 , m4 = zv , m5 = v 2This ideal is generic, so its minimal free resolution is supported on its Scarf
complex:
1
2

3

4

5

It does not admit a DGA structure. More precisely, it does not admit a
DGA-structure which respects the multigrading.
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DGA Resolutions

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Existence of multiplications

There are several cases where the minimal free resolution is known to carrya DGA structure:complete intersections (Koszul complex),
pdim S/I ≤ 3 (Buchsbaum-Eisenbud ’77)
pdim S/I ≤ 4 and Gorenstein (Kustin-Miller ’80)
stable monomial ideals (Eliahou-Kervaire resolution, Peeva ’96)

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DGA Resolutions

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Existence of multiplications

There are several cases where the minimal free resolution is known to carrya DGA structure:complete intersections (Koszul complex),
pdim S/I ≤ 3 (Buchsbaum-Eisenbud ’77)
pdim S/I ≤ 4 and Gorenstein (Kustin-Miller ’80)
stable monomial ideals (Eliahou-Kervaire resolution, Peeva ’96)

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DGA Resolutions

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Existence of multiplications

There are several cases where the minimal free resolution is known to carrya DGA structure:complete intersections (Koszul complex),
pdim S/I ≤ 3 (Buchsbaum-Eisenbud ’77)
pdim S/I ≤ 4 and Gorenstein (Kustin-Miller ’80)
stable monomial ideals (Eliahou-Kervaire resolution, Peeva ’96)

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DGA Resolutions

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Existence of multiplications
A little surprisingly, it is enough to modify the last map in F• to ensurethat there is a DGA structure:

Theorem (K. 2016)
Let I ⊂ S be a homogeneous ideal. Then there exists a homogeneous
element s ∈ S, such that the minimal free resolution of sI has a DGA
structure.
This holds for regular local rings as well.
Also, if I is multigraded, then s can be chosen homogeneous with respect
to the multigrading.
Note that grade sI = 1, so unfortunately this result does not provide lower
bounds for Betti numbers.

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DGA Resolutions

August 5, 201614 / 29

Existence of multiplications
A little surprisingly, it is enough to modify the last map in F• to ensurethat there is a DGA structure:

Theorem (K. 2016)
Let I ⊂ S be a homogeneous ideal. Then there exists a homogeneous
element s ∈ S, such that the minimal free resolution of sI has a DGA
structure.
This holds for regular local rings as well.
Also, if I is multigraded, then s can be chosen homogeneous with respect
to the multigrading.
Note that grade sI = 1, so unfortunately this result does not provide lower
bounds for Betti numbers.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201614 / 29

Existence of multiplications
A little surprisingly, it is enough to modify the last map in F• to ensurethat there is a DGA structure:

Theorem (K. 2016)
Let I ⊂ S be a homogeneous ideal. Then there exists a homogeneous
element s ∈ S, such that the minimal free resolution of sI has a DGA
structure.
This holds for regular local rings as well.
Also, if I is multigraded, then s can be chosen homogeneous with respect
to the multigrading.
Note that grade sI = 1, so unfortunately this result does not provide lower
bounds for Betti numbers.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201614 / 29

Existence of multiplications
A little surprisingly, it is enough to modify the last map in F• to ensurethat there is a DGA structure:

Theorem (K. 2016)
Let I ⊂ S be a homogeneous ideal. Then there exists a homogeneous
element s ∈ S, such that the minimal free resolution of sI has a DGA
structure.
This holds for regular local rings as well.
Also, if I is multigraded, then s can be chosen homogeneous with respect
to the multigrading.
Note that grade sI = 1, so unfortunately this result does not provide lower
bounds for Betti numbers.

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DGA Resolutions

August 5, 201614 / 29

Section 2
The monomial case

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Monomial ideals

From now on:
I is a monomial ideal, andF• is multigraded minimal free resolution of S/I.
Moreover, we only consider multiplications on F• which respect themultigrading.

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DGA Resolutions

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Monomial ideals

From now on:
I is a monomial ideal, andF• is multigraded minimal free resolution of S/I.
Moreover, we only consider multiplications on F• which respect themultigrading.

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DGA Resolutions

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Monomial ideals

From now on:
I is a monomial ideal, andF• is multigraded minimal free resolution of S/I.
Moreover, we only consider multiplications on F• which respect themultigrading.

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Supportive Multiplications
The lcm-lattice LI of I is the lattice of all least common multiples ofsubsets of the generators of I.
(Gasharov-Peeva-Welker ’99) The structure of F• is determined by
the isomorphism type of LI .
The possible multiplications on F• are not determined by LI , because thelcm-lattices of I and sI are isomorphic for every monomial s ∈ S.
Therefore we restrict the class of multiplications we consider:

Definition
We call a multiplication on F• supportive if for all a, b ∈ F• , it holds thata · b = mc for a monomial m ∈ S and a c ∈ F• withdeg c ≤ (deg a) ∨ (deg b).
Here, “∨” denotes the componentwise maximum.

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Supportive Multiplications
The lcm-lattice LI of I is the lattice of all least common multiples ofsubsets of the generators of I.
(Gasharov-Peeva-Welker ’99) The structure of F• is determined by
the isomorphism type of LI .
The possible multiplications on F• are not determined by LI , because thelcm-lattices of I and sI are isomorphic for every monomial s ∈ S.
Therefore we restrict the class of multiplications we consider:

Definition
We call a multiplication on F• supportive if for all a, b ∈ F• , it holds thata · b = mc for a monomial m ∈ S and a c ∈ F• withdeg c ≤ (deg a) ∨ (deg b).
Here, “∨” denotes the componentwise maximum.

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DGA Resolutions

August 5, 201617 / 29

Supportive Multiplications
The lcm-lattice LI of I is the lattice of all least common multiples ofsubsets of the generators of I.
(Gasharov-Peeva-Welker ’99) The structure of F• is determined by
the isomorphism type of LI .
The possible multiplications on F• are not determined by LI , because thelcm-lattices of I and sI are isomorphic for every monomial s ∈ S.
Therefore we restrict the class of multiplications we consider:

Definition
We call a multiplication on F• supportive if for all a, b ∈ F• , it holds thata · b = mc for a monomial m ∈ S and a c ∈ F• withdeg c ≤ (deg a) ∨ (deg b).
Here, “∨” denotes the componentwise maximum.

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DGA Resolutions

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Supportive Multiplications: Example
Consider the ideal I generated by m1 = x 2 , m2 = xy and m3 = xz.
Its minimal free resolution looks like this:3
1

2

Possible choices for g2 · g3 :(g2 · g3 =xg23−zg12 + yg13

supportive,
not supportive.

Remark: Note that the multiplication in not unique in this case.

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DGA Resolutions

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Supportive Multiplications: Example
Consider the ideal I generated by m1 = x 2 , m2 = xy and m3 = xz.
Its minimal free resolution looks like this:3
1

2

Possible choices for g2 · g3 :(g2 · g3 =xg23−zg12 + yg13

supportive,
not supportive.

Remark: Note that the multiplication in not unique in this case.

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DGA Resolutions

August 5, 201618 / 29

Supportive Multiplications: Example
Consider the ideal I generated by m1 = x 2 , m2 = xy and m3 = xz.
Its minimal free resolution looks like this:3
1

2

Possible choices for g2 · g3 :(g2 · g3 =xg23−zg12 + yg13

supportive,
not supportive.

Remark: Note that the multiplication in not unique in this case.

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DGA Resolutions

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Supportive Multiplications
Supportive multiplications have desirable properties:

Proposition
1

If two ideals have isomorphic lcm-lattices, then every supportive
multiplication for one ideal induces a supportive multiplication for theother one. Thus, the minimal free resolution of on the ideals admits a
DGA structure if and only if the other one does.

2

If I is squarefree (i.e. generated by squarefree monomials), then every
multiplication on F• is supportive.

Corollary
1

2

Every monomial ideal admits a supportive multiplication on its
minimal free resolution.
If the polarization I pol of an ideal has a DGA minimal free resolution,then the same holds for I and it is automatically supportive.

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August 5, 201619 / 29

Supportive Multiplications
Supportive multiplications have desirable properties:

Proposition
1

If two ideals have isomorphic lcm-lattices, then every supportive
multiplication for one ideal induces a supportive multiplication for theother one. Thus, the minimal free resolution of on the ideals admits a
DGA structure if and only if the other one does.

2

If I is squarefree (i.e. generated by squarefree monomials), then every
multiplication on F• is supportive.

Corollary
1

2

Every monomial ideal admits a supportive multiplication on its
minimal free resolution.
If the polarization I pol of an ideal has a DGA minimal free resolution,then the same holds for I and it is automatically supportive.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201619 / 29

Supportive Multiplications
Supportive multiplications have desirable properties:

Proposition
1

If two ideals have isomorphic lcm-lattices, then every supportive
multiplication for one ideal induces a supportive multiplication for theother one. Thus, the minimal free resolution of on the ideals admits a
DGA structure if and only if the other one does.

2

If I is squarefree (i.e. generated by squarefree monomials), then every
multiplication on F• is supportive.

Corollary
1

2

Every monomial ideal admits a supportive multiplication on its
minimal free resolution.
If the polarization I pol of an ideal has a DGA minimal free resolution,then the same holds for I and it is automatically supportive.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201619 / 29

Supportive Multiplications
Supportive multiplications have desirable properties:

Proposition
1

If two ideals have isomorphic lcm-lattices, then every supportive
multiplication for one ideal induces a supportive multiplication for theother one. Thus, the minimal free resolution of on the ideals admits a
DGA structure if and only if the other one does.

2

If I is squarefree (i.e. generated by squarefree monomials), then every
multiplication on F• is supportive.

Corollary
1

2

Every monomial ideal admits a supportive multiplication on its
minimal free resolution.
If the polarization I pol of an ideal has a DGA minimal free resolution,then the same holds for I and it is automatically supportive.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201619 / 29

Supportive Multiplications
Supportive multiplications have desirable properties:

Proposition
1

If two ideals have isomorphic lcm-lattices, then every supportive
multiplication for one ideal induces a supportive multiplication for theother one. Thus, the minimal free resolution of on the ideals admits a
DGA structure if and only if the other one does.

2

If I is squarefree (i.e. generated by squarefree monomials), then every
multiplication on F• is supportive.

Corollary
1

2

Every monomial ideal admits a supportive multiplication on its
minimal free resolution.
If the polarization I pol of an ideal has a DGA minimal free resolution,then the same holds for I and it is automatically supportive.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201619 / 29

Supportive Multiplications
Proposition (K. 2016)
If · is a supportive multiplication on F• , then for every homogeneous
element a ∈ F• , there exists a monomial m ∈ S such thatma =Xci,1 · (ci,2 · (· · · (ci,r −1 · ci,r ) · · · ))ifor elements ci,j ∈ F1 . In other words, F• is generated in degree 1, up to
multiplication with monomials.
As a consequence, the differential ∂ on F• is determined by its part indegree 1 and by the multiplication (Leibniz rule).
The proposition does not hold for non-suportive multiplications.
If I is a non-monomial ideal, then F• might not admit a multiplication
which is generated in degree 1.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

20 / 29

Supportive Multiplications
Proposition (K. 2016)
If · is a supportive multiplication on F• , then for every homogeneous
element a ∈ F• , there exists a monomial m ∈ S such thatma =Xci,1 · (ci,2 · (· · · (ci,r −1 · ci,r ) · · · ))ifor elements ci,j ∈ F1 . In other words, F• is generated in degree 1, up to
multiplication with monomials.
As a consequence, the differential ∂ on F• is determined by its part indegree 1 and by the multiplication (Leibniz rule).
The proposition does not hold for non-suportive multiplications.
If I is a non-monomial ideal, then F• might not admit a multiplication
which is generated in degree 1.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

20 / 29

Supportive Multiplications
Proposition (K. 2016)
If · is a supportive multiplication on F• , then for every homogeneous
element a ∈ F• , there exists a monomial m ∈ S such thatma =Xci,1 · (ci,2 · (· · · (ci,r −1 · ci,r ) · · · ))ifor elements ci,j ∈ F1 . In other words, F• is generated in degree 1, up to
multiplication with monomials.
As a consequence, the differential ∂ on F• is determined by its part indegree 1 and by the multiplication (Leibniz rule).
The proposition does not hold for non-suportive multiplications.
If I is a non-monomial ideal, then F• might not admit a multiplication
which is generated in degree 1.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

20 / 29

Supportive Multiplications
Proposition (K. 2016)
If · is a supportive multiplication on F• , then for every homogeneous
element a ∈ F• , there exists a monomial m ∈ S such thatma =Xci,1 · (ci,2 · (· · · (ci,r −1 · ci,r ) · · · ))ifor elements ci,j ∈ F1 . In other words, F• is generated in degree 1, up to
multiplication with monomials.
As a consequence, the differential ∂ on F• is determined by its part indegree 1 and by the multiplication (Leibniz rule).
The proposition does not hold for non-suportive multiplications.
If I is a non-monomial ideal, then F• might not admit a multiplication
which is generated in degree 1.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

20 / 29

Supportive DGA resolutions and the Taylor resolution
The Taylor resolution is a particular non-minimal DGA resolution. Let us
recall the details. For I = hm1 , . . . , mk i ⊂ S and N ⊆ [k] let
mN := lcm(mi i ∈ N).
The Taylor resolution of S/I is the complex
0 → Tk → Tk−1 → · · · → T1 → S → S/I → 0

T• :
where
Ti :=

M

SeN

and

∂eN :=

X
i∈N

N⊆[k]
#N=i

±

mN
e
mN\i N\iand deg eN := deg mN .
This is a free DGA resolution of S/I (typically far from being minimal)
with the following multiplication:eN · eM = ±L. Katthän (Goethe-Universität Frankfurt)mN mM
eN∪M , if N ∩ M = ∅mN∪M
DGA Resolutions

August 5, 2016

21 / 29

Supportive DGA resolutions and the Taylor resolution
The Taylor resolution is a particular non-minimal DGA resolution. Let us
recall the details. For I = hm1 , . . . , mk i ⊂ S and N ⊆ [k] let
mN := lcm(mi i ∈ N).
The Taylor resolution of S/I is the complex
0 → Tk → Tk−1 → · · · → T1 → S → S/I → 0

T• :
where
Ti :=

M

SeN

and

∂eN :=

X
i∈N

N⊆[k]
#N=i

±

mN
e
mN\i N\iand deg eN := deg mN .
This is a free DGA resolution of S/I (typically far from being minimal)
with the following multiplication:eN · eM = ±L. Katthän (Goethe-Universität Frankfurt)mN mM
eN∪M , if N ∩ M = ∅mN∪M
DGA Resolutions

August 5, 2016

21 / 29

Supportive DGA resolutions and the Taylor resolution
The Taylor resolution is a particular non-minimal DGA resolution. Let us
recall the details. For I = hm1 , . . . , mk i ⊂ S and N ⊆ [k] let
mN := lcm(mi i ∈ N).
The Taylor resolution of S/I is the complex
0 → Tk → Tk−1 → · · · → T1 → S → S/I → 0

T• :
where
Ti :=

M

SeN

and

∂eN :=

X
i∈N

N⊆[k]
#N=i

±

mN
e
mN\i N\iand deg eN := deg mN .
This is a free DGA resolution of S/I (typically far from being minimal)
with the following multiplication:eN · eM = ±L. Katthän (Goethe-Universität Frankfurt)mN mM
eN∪M , if N ∩ M = ∅mN∪M
DGA Resolutions

August 5, 2016

21 / 29

Supportive DGA resolutions and the Taylor resolution
The Taylor resolution is a particular non-minimal DGA resolution. Let us
recall the details. For I = hm1 , . . . , mk i ⊂ S and N ⊆ [k] let
mN := lcm(mi i ∈ N).
The Taylor resolution of S/I is the complex
0 → Tk → Tk−1 → · · · → T1 → S → S/I → 0

T• :
where
Ti :=

M

SeN

and

∂eN :=

X
i∈N

N⊆[k]
#N=i

±

mN
e
mN\i N\iand deg eN := deg mN .
This is a free DGA resolution of S/I (typically far from being minimal)
with the following multiplication:eN · eM = ±L. Katthän (Goethe-Universität Frankfurt)mN mM
eN∪M , if N ∩ M = ∅mN∪M
DGA Resolutions

August 5, 2016

21 / 29

Supportive DGA resolutions and the Taylor resolution

The Taylor resolution is a non-minimal supportive DGA resolution of S/I.
In fact it is “free” with this properties in the following sense:

Theorem (K. 2016)
Let I ⊂ S be a monomial ideal with minimal free resolution F• and Taylor
resolution T• . If F• admits a supportive DGA structure, then F• ∼= T• /J
as DGAs for some DG-ideal J ⊂ T• .
This can be used to determine whether F• admits a supportive DGA
structure by considering DG-ideals in T• .

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201622 / 29

Supportive DGA resolutions and the Taylor resolution

The Taylor resolution is a non-minimal supportive DGA resolution of S/I.
In fact it is “free” with this properties in the following sense:

Theorem (K. 2016)
Let I ⊂ S be a monomial ideal with minimal free resolution F• and Taylor
resolution T• . If F• admits a supportive DGA structure, then F• ∼= T• /J
as DGAs for some DG-ideal J ⊂ T• .
This can be used to determine whether F• admits a supportive DGA
structure by considering DG-ideals in T• .

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201622 / 29

Supportive DGA resolutions and the Taylor resolution

The Taylor resolution is a non-minimal supportive DGA resolution of S/I.
In fact it is “free” with this properties in the following sense:

Theorem (K. 2016)
Let I ⊂ S be a monomial ideal with minimal free resolution F• and Taylor
resolution T• . If F• admits a supportive DGA structure, then F• ∼= T• /J
as DGAs for some DG-ideal J ⊂ T• .
This can be used to determine whether F• admits a supportive DGA
structure by considering DG-ideals in T• .

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201622 / 29

Section 3
Applications

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201623 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) :=i
i bi t . Then (1 + t) divides b(t) andP(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) :=i
i bi t . Then (1 + t) divides b(t) andP(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) :=i
i bi t . Then (1 + t) divides b(t) andP(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) :=i
i bi t . Then (1 + t) divides b(t) andP(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) := i bi t i . Then (1 + t) divides b(t) and the coefficients of
1
1+t b(t) satisfy the Kruskal-Katona inequalities.
P(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
The fact that supportive DGA resolutions are generated in degree 1 allows
us to characterize the possible Betti numbers of monomial ideals with such
resolutions.

Theorem (K., Welker 2016)
For a vector b = (b0 , b1 . . . , bn ) ∈ Nn , the following are equivalent:1

There exists a monomial ideal I ⊂ S, whose minimal free resolution isa supportive DGA and bi = βi (S/I) for all i.

2b is the f -vector of some simplicial complex ∆ which is a cone.

3

Let b(t) := i bi t i . Then (1 + t) divides b(t) and the coefficients of
1
1+t b(t) satisfy the Kruskal-Katona inequalities.
P(Kalai ’85) These vectors are also exactly the f -vectors of acyclic simplicial
complexes.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201624 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors
Idea of the proof:“1 ⇒ 2”: Macaulay-type theorem for DGAs generated in degree 1:(Aramova, Herzog and Hibi ’97) Hilbert series ofgraded-commutative algebras generated in degree 1 arecharacterized by the Kruskal-Katona inequalities. We modify
their proof to show that the existence of the differential
restricts the possible Hilbert series to f -vectors of cones.
“2 ⇒ 1”: For a given cone ∆, their exists a monomial ideal whose
Scarf complex equals ∆, so in particular its Betti vector
equals the f -vector of ∆.
We use algebraic discrete Morse theory to find a suitable
DG-ideal in the Taylor resolution.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201625 / 29

Betti vectors

Does the Betti vector of any monomial ideal satisfy these conditions?

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

26 / 29

Betti vectors

Does the Betti vector of any monomial ideal satisfy these conditions?
Answer: No.
The Betti vector of I = (ab, bc, cd, de, ef , fa) is (1, 6, 9, 6, 2). This vector
is not even the f -vector of any simplicial complex.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

26 / 29

Betti vectors

Does the Betti vector of any monomial ideal satisfy these conditions?
Answer: No.
The Betti vector of I = (ab, bc, cd, de, ef , fa) is (1, 6, 9, 6, 2). This vector
is not even the f -vector of any simplicial complex.
Nevertheless, there exist a monomial ideal with the Betti vectors whose
minimal free resolution is a DGA, but not supportive.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 2016

26 / 29

Subadditivity of syzygies
For a monomial ideal I ⊂ S letti := max { j βi,j (S/I) 6= 0 }the maximal shift in the i-th step of the resolution of S/I.

Definition
We say that subadditivity of syzygies holds for I if
ta+b ≤ ta + tb
for all 1 ≤ a, b ≤ pdim S/I such that a + b ≤ pdim S/I.
This does not hold in general for non-monomial ideals(Avramov-Conca-Iyengar ’15), but it is open whether every monomial ideal
has this property.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201627 / 29

Subadditivity of syzygies
For a monomial ideal I ⊂ S letti := max { j βi,j (S/I) 6= 0 }the maximal shift in the i-th step of the resolution of S/I.

Definition
We say that subadditivity of syzygies holds for I if
ta+b ≤ ta + tb
for all 1 ≤ a, b ≤ pdim S/I such that a + b ≤ pdim S/I.
This does not hold in general for non-monomial ideals(Avramov-Conca-Iyengar ’15), but it is open whether every monomial ideal
has this property.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201627 / 29

Subadditivity of syzygies
For a monomial ideal I ⊂ S letti := max { j βi,j (S/I) 6= 0 }the maximal shift in the i-th step of the resolution of S/I.

Definition
We say that subadditivity of syzygies holds for I if
ta+b ≤ ta + tb
for all 1 ≤ a, b ≤ pdim S/I such that a + b ≤ pdim S/I.
This does not hold in general for non-monomial ideals(Avramov-Conca-Iyengar ’15), but it is open whether every monomial ideal
has this property.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201627 / 29

Subadditivity of syzygies
For a monomial ideal I ⊂ S letti := max { j βi,j (S/I) 6= 0 }the maximal shift in the i-th step of the resolution of S/I.

Definition
We say that subadditivity of syzygies holds for I if
ta+b ≤ ta + tb
for all 1 ≤ a, b ≤ pdim S/I such that a + b ≤ pdim S/I.
This does not hold in general for non-monomial ideals(Avramov-Conca-Iyengar ’15), but it is open whether every monomial ideal
has this property.
L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201627 / 29

Subadditivity of syzygies

The fact that supportive multiplications are generated in degree one
implies (with some work):

Proposition (K. 2016)
The subadditivity of syzygies holds for all monomial ideals whose minimal
free resolution admits a supportive DGA structure.
If F• does not have a supportive DGA resolution, then one can still use
these techniques to conclude that ti+1 ≤ ti + t1 for all i. This was shown
before by Herzog and Srinivasan by other techniques.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201628 / 29

Subadditivity of syzygies

The fact that supportive multiplications are generated in degree one
implies (with some work):

Proposition (K. 2016)
The subadditivity of syzygies holds for all monomial ideals whose minimal
free resolution admits a supportive DGA structure.
If F• does not have a supportive DGA resolution, then one can still use
these techniques to conclude that ti+1 ≤ ti + t1 for all i. This was shown
before by Herzog and Srinivasan by other techniques.

L. Katthän (Goethe-Universität Frankfurt)

DGA Resolutions

August 5, 201628 / 29

The end.

Thank you