From Algebraic Geometry to Homological Algebra Sepehr Jafari Università Degli Studi di Genova Dipartimento Di Matematica November 8, 2016 November 8, 2016 1 / 24
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From Algebraic Geometry to Homological Algebra
Sepehr Jafari
Università Degli Studi di GenovaDipartimento Di Matematica
November 8, 2016
November 8, 2016 1 / 24
Outline
1 Historical Events
Algebraic Geometry
Homological Algebra
2 Modern Concepts
3 Some open Problems
November 8, 2016 2 / 24
Outline
1 Historical Events
Algebraic Geometry
Homological Algebra
2 Modern Concepts
3 Some open Problems
November 8, 2016 2 / 24
Outline
1 Historical Events
Algebraic Geometry
Homological Algebra
2 Modern Concepts
3 Some open Problems
November 8, 2016 2 / 24
Outline
1 Historical Events
Algebraic Geometry
Homological Algebra
2 Modern Concepts
3 Some open Problems
November 8, 2016 2 / 24
The One Who Started All..
”... algebraic geometry andnumber theory have more openproblems than solved ones...”
David Hilbert (1862-1943)
November 8, 2016 3 / 24
Historical Events in Geometry (400 B.C-1630 A.D)
Greeks invented geometry as a deductive science, they never made anattempt to divorce it from algebra.
The study of straight lines, cycles and circles.
Geometric construction of the roots for x2 = ab.
Greeks, in particular , used coordinates without, however, reachingthe general point of view of Descartes and Fermat.
November 8, 2016 4 / 24
Historical Events in Geometry (400 B.C-1630 A.D)
Greeks invented geometry as a deductive science, they never made anattempt to divorce it from algebra.
The study of straight lines, cycles and circles.
Geometric construction of the roots for x2 = ab.
Greeks, in particular , used coordinates without, however, reachingthe general point of view of Descartes and Fermat.
November 8, 2016 4 / 24
Historical Events in Geometry (400 B.C-1630 A.D)
Greeks invented geometry as a deductive science, they never made anattempt to divorce it from algebra.
The study of straight lines, cycles and circles.
Geometric construction of the roots for x2 = ab.
Greeks, in particular , used coordinates without, however, reachingthe general point of view of Descartes and Fermat.
November 8, 2016 4 / 24
Historical Events in Geometry (400 B.C-1630 A.D)
Greeks invented geometry as a deductive science, they never made anattempt to divorce it from algebra.
The study of straight lines, cycles and circles.
Geometric construction of the roots for x2 = ab.
Greeks, in particular , used coordinates without, however, reachingthe general point of view of Descartes and Fermat.
November 8, 2016 4 / 24
Historical Events in Geometry (1630-1795)
Contributions of Descartes, Fermat, Newton, Euler and Leibniz.
Invention of analytic Geometry and birth of algebraic geometry.
The possibility of studying arbitrary equations. (Where the Greekscould not go beyond third or fourth degree.)
All curves of degree two are conics. (Euler)
Classification of all cubics with respect to change of coordinates andprojections. (Newton)
November 8, 2016 5 / 24
Historical Events in Geometry (1630-1795)
Contributions of Descartes, Fermat, Newton, Euler and Leibniz.Invention of analytic Geometry and birth of algebraic geometry.
The possibility of studying arbitrary equations. (Where the Greekscould not go beyond third or fourth degree.)
All curves of degree two are conics. (Euler)
Classification of all cubics with respect to change of coordinates andprojections. (Newton)
November 8, 2016 5 / 24
Historical Events in Geometry (1630-1795)
Contributions of Descartes, Fermat, Newton, Euler and Leibniz.Invention of analytic Geometry and birth of algebraic geometry.
The possibility of studying arbitrary equations. (Where the Greekscould not go beyond third or fourth degree.)
All curves of degree two are conics. (Euler)
Classification of all cubics with respect to change of coordinates andprojections. (Newton)
November 8, 2016 5 / 24
Historical Events in Geometry (1630-1795)
Contributions of Descartes, Fermat, Newton, Euler and Leibniz.Invention of analytic Geometry and birth of algebraic geometry.
The possibility of studying arbitrary equations. (Where the Greekscould not go beyond third or fourth degree.)
All curves of degree two are conics. (Euler)
Classification of all cubics with respect to change of coordinates andprojections. (Newton)
November 8, 2016 5 / 24
Historical Events in Geometry (1630-1795)
Contributions of Descartes, Fermat, Newton, Euler and Leibniz.Invention of analytic Geometry and birth of algebraic geometry.
The possibility of studying arbitrary equations. (Where the Greekscould not go beyond third or fourth degree.)
All curves of degree two are conics. (Euler)
Classification of all cubics with respect to change of coordinates andprojections. (Newton)
November 8, 2016 5 / 24
Historical Events in Geometry (1630-1795)
The concept of dimension is known to Fermat.
The concept of parametric representation of a curve is Newton’sapproach to calculus.
The problem of intersection of two plane curves is tackled by Newtonand Leibniz using elimination.
November 8, 2016 6 / 24
Historical Events in Geometry (1630-1795)
The concept of dimension is known to Fermat.
The concept of parametric representation of a curve is Newton’sapproach to calculus.
The problem of intersection of two plane curves is tackled by Newtonand Leibniz using elimination.
November 8, 2016 6 / 24
Historical Events in Geometry (1795-1850)
The Golden Age of Projective Geometry
Monge and Ponclete introduced points at infinity and imaginarypoints.
For almost a century, geometry in complex projective plane is themain topic of study for complex geometers.With Mobius, Plucker and Cayley, projective geometry received analgebraic basis by suing homogeneous coordinates.
November 8, 2016 7 / 24
Historical Events in Geometry (1795-1850)
The Golden Age of Projective Geometry
Monge and Ponclete introduced points at infinity and imaginarypoints.
For almost a century, geometry in complex projective plane is themain topic of study for complex geometers.
With Mobius, Plucker and Cayley, projective geometry received analgebraic basis by suing homogeneous coordinates.
November 8, 2016 7 / 24
Historical Events in Geometry (1795-1850)
The Golden Age of Projective Geometry
Monge and Ponclete introduced points at infinity and imaginarypoints.
For almost a century, geometry in complex projective plane is themain topic of study for complex geometers.With Mobius, Plucker and Cayley, projective geometry received analgebraic basis by suing homogeneous coordinates.
November 8, 2016 7 / 24
Historical Events in Geometry
”... a mathematician, then, willbe defined in what follows as
someone who has published theproof of at least one nun-trivial
theorem...”
Jean Dieudonne (1906-1992)
November 8, 2016 8 / 24
Historical Events in Homological Algebra (1857-presenet)
Riemann’s work on connectedness” number was the starting point ofthe study of genus.
Enrico Betti was interested in the study of simply connectedmanifolds.
DefinitionA simply connected manifold is a path-connected topological space whichis closed under continues transformations of paths between to given point.
November 8, 2016 9 / 24
Historical Events in Homological Algebra (1857-presenet)
Riemann’s work on connectedness” number was the starting point ofthe study of genus.
Enrico Betti was interested in the study of simply connectedmanifolds.
DefinitionA simply connected manifold is a path-connected topological space whichis closed under continues transformations of paths between to given point.
November 8, 2016 9 / 24
Historical Events in Homological Algebra (1857-presenet)
Poincaré was inspired by Betti’s work and contributed to homologicalalgebra in
The notion of homology for linear manifolds
DefinitionA family of n−dimensional sub manifolds of V , denoted by Vi , are linearlyindependent if there is no homology connecting them.
He defined βn to be the size of a maximal independent family of submanifolds of V .
November 8, 2016 10 / 24
Historical Events in Homological Algebra (1857-presenet)
Poincaré was inspired by Betti’s work and contributed to homologicalalgebra in
The notion of homology for linear manifolds
DefinitionA family of n−dimensional sub manifolds of V , denoted by Vi , are linearlyindependent if there is no homology connecting them.
He defined βn to be the size of a maximal independent family of submanifolds of V .
November 8, 2016 10 / 24
Historical Events in Homological Algebra (1857-presenet)
Poincaré was inspired by Betti’s work and contributed to homologicalalgebra in
The notion of homology for linear manifolds
DefinitionA family of n−dimensional sub manifolds of V , denoted by Vi , are linearlyindependent if there is no homology connecting them.
He defined βn to be the size of a maximal independent family of submanifolds of V .
November 8, 2016 10 / 24
Historical Events in Homological Algebra (1857-presenet)
Poincaré was inspired by Betti’s work and contributed to homologicalalgebra in
The notion of homology for linear manifolds
DefinitionA family of n−dimensional sub manifolds of V , denoted by Vi , are linearlyindependent if there is no homology connecting them.
He defined βn to be the size of a maximal independent family of submanifolds of V .
November 8, 2016 10 / 24
Historical Events in Homological Algebra (1857-presenet)
November 8, 2016 11 / 24
Historical Events in Homological Algebra (1857-presenet)
First evidences of existence of an algebraic notion.
Emmy Noether pointed out that homologies are abelian groups.
Inspired by Noether, L.Mayer introduced the algebraic notion of chaincomplex,cycles and homology groups.
Mathematicians started a movement in generalizing Poincaré’s ideas,which led to more variations of homology.
November 8, 2016 12 / 24
Historical Events in Homological Algebra (1857-presenet)
First evidences of existence of an algebraic notion.
Emmy Noether pointed out that homologies are abelian groups.
Inspired by Noether, L.Mayer introduced the algebraic notion of chaincomplex,cycles and homology groups.
Mathematicians started a movement in generalizing Poincaré’s ideas,which led to more variations of homology.
November 8, 2016 12 / 24
Historical Events in Homological Algebra (1857-presenet)
First evidences of existence of an algebraic notion.
Emmy Noether pointed out that homologies are abelian groups.
Inspired by Noether, L.Mayer introduced the algebraic notion of chaincomplex,cycles and homology groups.
Mathematicians started a movement in generalizing Poincaré’s ideas,which led to more variations of homology.
November 8, 2016 12 / 24
Historical Events in Homological Algebra (1857-presenet)
The rise of algebraic methodes
Hassler Whiteny discovered the tensor product of abelian groups.(1938)
The concept of an exact sequence first appeared in Hurewicz work.(1941)
Eilenberg an Mac Lane defined Hom and Ext for the first time.(1942)
November 8, 2016 13 / 24
Historical Events in Homological Algebra (1857-presenet)
The rise of algebraic methodes
Hassler Whiteny discovered the tensor product of abelian groups.(1938)
The concept of an exact sequence first appeared in Hurewicz work.(1941)
Eilenberg an Mac Lane defined Hom and Ext for the first time.(1942)
November 8, 2016 13 / 24
Historical Events in Homological Algebra (1857-presenet)
The rise of algebraic methodes
Hassler Whiteny discovered the tensor product of abelian groups.(1938)
The concept of an exact sequence first appeared in Hurewicz work.(1941)
Eilenberg an Mac Lane defined Hom and Ext for the first time.(1942)
November 8, 2016 13 / 24
Historical Events in Homological Algebra (1857-presenet)
Sheaves and Spectral Sequences
Jean Leray was a prisoner of WW2.
He organized a university in his prison camp and taught a course intopological algebra.
He invented sheaves, sheaf cohomology and spectral sequences.
Koszul discovered the algebraic sides of spectral sequences.
November 8, 2016 14 / 24
Historical Events in Homological Algebra (1857-presenet)
Sheaves and Spectral Sequences
Jean Leray was a prisoner of WW2.
He organized a university in his prison camp and taught a course intopological algebra.
He invented sheaves, sheaf cohomology and spectral sequences.
Koszul discovered the algebraic sides of spectral sequences.
November 8, 2016 14 / 24
Historical Events in Homological Algebra (1857-presenet)
Sheaves and Spectral Sequences
Jean Leray was a prisoner of WW2.
He organized a university in his prison camp and taught a course intopological algebra.
He invented sheaves, sheaf cohomology and spectral sequences.
Koszul discovered the algebraic sides of spectral sequences.
November 8, 2016 14 / 24
Historical Events in Homological Algebra (1857-presenet)
The Cartan-Eilenberg revelution
They published the first book on ”Homological algebra” and united thetheory in 1956.
Concepts of Torn and Extn for category of modules.
Projective modules.
Drived functors.
November 8, 2016 15 / 24
Historical Events in Homological Algebra (1857-presenet)
The Cartan-Eilenberg revelution
They published the first book on ”Homological algebra” and united thetheory in 1956.
Concepts of Torn and Extn for category of modules.
Projective modules.
Drived functors.
November 8, 2016 15 / 24
Historical Events in Homological Algebra (1857-presenet)
The Cartan-Eilenberg revelution
They published the first book on ”Homological algebra” and united thetheory in 1956.
Concepts of Torn and Extn for category of modules.
Projective modules.
Drived functors.
November 8, 2016 15 / 24
Historical Events in Homological Algebra (1857-presenet)
The Cartan-Eilenberg revelution
They published the first book on ”Homological algebra” and united thetheory in 1956.
Concepts of Torn and Extn for category of modules.
Projective modules.
Drived functors.
November 8, 2016 15 / 24
Historical Events in Geometry
”... the reason for the division isthat on the one hand it is
necessary to have general culture,on the other hand it is necessary
to have deep knowledge of aparticular field...”
Guido Castelnuovo (1865-1952 )
November 8, 2016 16 / 24
Historical Events in Geometry
”... algebraic geometry seems tohave acquired the reputation of
being esoteric, exclusive and veryabstract, with adherents who aresecretly plotting to take over allthe rest of the mathematics. In
one respect, this last point isaccurate...”
David Mumford (1937- )
November 8, 2016 17 / 24
Modern Concepts
Definition (Chain Complex)Let R be a commutative ring. A chain complex (C•,d•) is a family ofR−modules {Ci}i∈Z and R−homomorphisms {di}i∈Z where di ◦ di+1 = 0.
· · · → Ci+1di+1−−−→ Ci
di−→ Ci−1di−1−−→ · · ·
In other words,Img di+1 ⊆ Ker di
For the case of equality, (C•,d•) is defined to be exact.The i−th homology module of C• is defined as
Hi(C•) = Ker di/ Img di+1
November 8, 2016 18 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n Sn
Si Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+j
If S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n Mn
Si Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Setting
Let S = K [x1, . . . , xn] be a polynomial ring.There exists a family {Sn}n∈Z of subgroups of S such that
S =⊕
n SnSi Sj ⊆ Si+jIf S1 be the subgroup generated by variables, S has a standard grading.
Let M be a finitely generated S−module. M is called graded moduleif there exists a family of subgroups of M such that
M =⊕
n MnSi Mj ⊆ Mi+j
November 8, 2016 19 / 24
Graded Minimal Free Resolution
Let M be a finitely generated graded S− module.
F• : · · · →⊕
jSβ(2,j)(−j) d2−→
⊕j
Sβ(1,j)(−j) d1−→⊕
jSβ(0,j)(−j) π−→ M → 0
is a graded minimal free resolution.
DefinitionThe i−th syzygy of M is defined as Syzi = Ker di .
Theorem (Hilbert’s Syzygy Theorem)For i > n, Syzi = 0.
November 8, 2016 20 / 24
Graded Minimal Free Resolution
Let M be a finitely generated graded S− module.
F• : · · · →⊕
jSβ(2,j)(−j) d2−→
⊕j
Sβ(1,j)(−j) d1−→⊕
jSβ(0,j)(−j) π−→ M → 0
is a graded minimal free resolution.
DefinitionThe i−th syzygy of M is defined as Syzi = Ker di .
Theorem (Hilbert’s Syzygy Theorem)For i > n, Syzi = 0.
November 8, 2016 20 / 24
Graded Minimal Free Resolution
Let M be a finitely generated graded S− module.
F• : · · · →⊕
jSβ(2,j)(−j) d2−→
⊕j
Sβ(1,j)(−j) d1−→⊕
jSβ(0,j)(−j) π−→ M → 0
is a graded minimal free resolution.
DefinitionThe i−th syzygy of M is defined as Syzi = Ker di .
Theorem (Hilbert’s Syzygy Theorem)For i > n, Syzi = 0.
November 8, 2016 20 / 24
The Functor Tori
TheoremFor a finitely generated graded module M, minimal free resolutions areunique up to isomorphism.
DefinitionLet M and N be modules and F• and Q• be a minimal free resolutions ofM and N resp. The i−th torsion module of M and N is defined as
Tori(M,N) = Hi(F• ⊗ N) = Hi(M ⊗Q•)
Theorem
dimK Tori(K ,M)j = βi ,j(M)
November 8, 2016 21 / 24
The Functor Tori
TheoremFor a finitely generated graded module M, minimal free resolutions areunique up to isomorphism.
DefinitionLet M and N be modules and F• and Q• be a minimal free resolutions ofM and N resp. The i−th torsion module of M and N is defined as
Tori(M,N) = Hi(F• ⊗ N) = Hi(M ⊗Q•)
Theorem
dimK Tori(K ,M)j = βi ,j(M)
November 8, 2016 21 / 24
The Functor Tori
TheoremFor a finitely generated graded module M, minimal free resolutions areunique up to isomorphism.
DefinitionLet M and N be modules and F• and Q• be a minimal free resolutions ofM and N resp. The i−th torsion module of M and N is defined as
Tori(M,N) = Hi(F• ⊗ N) = Hi(M ⊗Q•)
Theorem
dimK Tori(K ,M)j = βi ,j(M)
November 8, 2016 21 / 24
The Functor Tori
TheoremFor a finitely generated graded module M, minimal free resolutions areunique up to isomorphism.
DefinitionLet M and N be modules and F• and Q• be a minimal free resolutions ofM and N resp. The i−th torsion module of M and N is defined as
Tori(M,N) = Hi(F• ⊗ N) = Hi(M ⊗Q•)
Theorem
dimK Tori(K ,M)j = βi ,j(M)
November 8, 2016 21 / 24
Castelnuovo-Mumford Regularity
DefinitionThe Castelnuovo-Mumford regularity of M is
reg(M) = sup{j − i : βi ,j(M) 6= 0}
DefinitionLinear free resolution:
F• : · · · → Sβ(2,j+2)(−j − 2) d2−→ Sβ(1,j+1)(−j − 1) d1−→ Sβ(0,j)(−j) π−→ M → 0
November 8, 2016 22 / 24
Castelnuovo-Mumford Regularity
DefinitionThe Castelnuovo-Mumford regularity of M is
reg(M) = sup{j − i : βi ,j(M) 6= 0}
DefinitionLinear free resolution:
F• : · · · → Sβ(2,j+2)(−j − 2) d2−→ Sβ(1,j+1)(−j − 1) d1−→ Sβ(0,j)(−j) π−→ M → 0
November 8, 2016 22 / 24
Some Interesting Problems
The study of the properties of the regularity of Koszul rings.
Properties of regularity of multi graded rings?
Finding an upper bound for the regularity of product of ideals.
November 8, 2016 23 / 24
Some Interesting Problems
The study of the properties of the regularity of Koszul rings.
Properties of regularity of multi graded rings?
Finding an upper bound for the regularity of product of ideals.
November 8, 2016 23 / 24
Some Interesting Problems
The study of the properties of the regularity of Koszul rings.
Properties of regularity of multi graded rings?
Finding an upper bound for the regularity of product of ideals.
November 8, 2016 23 / 24
Bibliography
Weibel, Charles A.”History of homological algebra.”na, 1999.Dieudonne, Suzanne C.” History Algebraic Geometry.”CRC Press, 1985.Peeva, Irena, and Mike Stillman.”Open problems on syzygies and Hilbert functions.”Journal of Commutative Algebra 1.1 (2009): 159-195.
November 8, 2016 24 / 24
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