the cohomology of coherent sheaves - unirioja...mohamed barakat & markus lange-hegermann university...
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OverviewAffine Schemes
Projective SchemesThe Standard Module
The Cohomology of Coherent Sheaves
Mohamed Barakat & Markus Lange-Hegermann
University of Kaiserslautern & RWTH-Aachen University
Mathematics Algorithms and Proofs 2010, Logroño
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://homalg.math.rwth-aachen.de/index.php/extensions/gradedmoduleshttp://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://www.mathematik.uni-kl.dehttp://www.mathematik.rwth-aachen.dehttp://www.unirioja.es/dptos/dmc/MAP2010/http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
1 Affine SchemesFrom rings to schemesFrom modules to quasi-coherent sheaves
2 Projective SchemesFrom graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
3 The Standard ModuleThe module of global sections
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
Overview
1 Affine SchemesFrom rings to schemes
From modules to quasi-coherent sheaves
2 Projective SchemesFrom graded rings to projective schemes
From graded modules to quasi-coherent sheaves
3 The Standard ModuleThe module of global sections
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The spectrum of a ring
DefinitionLet R be a commutative ring with one. The set
Spec(R) := {p ⊳ R | p prime}
is called the (prime) spectrum of R. Define the vanishinglocus of an ideal I E R
V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).
The set {V (√I) | I E R} forms the set of closed subsets of a
topology on X = Spec(R), the so-called ZARISKI topology :PROOF. V (R) = ∅, V (〈0〉) = Spec(R), V (I) ∪ V (J) = V (IJ),and
⋂V (Ii) = V (
∑Ii).
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The spectrum of a ring
DefinitionLet R be a commutative ring with one. The set
Spec(R) := {p ⊳ R | p prime}
is called the (prime) spectrum of R. Define the vanishinglocus of an ideal I E R
V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).
The set {V (√I) | I E R} forms the set of closed subsets of a
topology on X = Spec(R), the so-called ZARISKI topology :PROOF. V (R) = ∅, V (〈0〉) = Spec(R), V (I) ∪ V (J) = V (IJ),and
⋂V (Ii) = V (
∑Ii).
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The spectrum of a ring
DefinitionLet R be a commutative ring with one. The set
Spec(R) := {p ⊳ R | p prime}
is called the (prime) spectrum of R. Define the vanishinglocus of an ideal I E R
V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).
The set {V (√I) | I E R} forms the set of closed subsets of a
topology on X = Spec(R), the so-called ZARISKI topology :PROOF. V (R) = ∅, V (〈0〉) = Spec(R), V (I) ∪ V (J) = V (IJ),and
⋂V (Ii) = V (
∑Ii).
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The spectrum of a ring
DefinitionLet R be a commutative ring with one. The set
Spec(R) := {p ⊳ R | p prime}
is called the (prime) spectrum of R. Define the vanishinglocus of an ideal I E R
V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).
The set {V (√I) | I E R} forms the set of closed subsets of a
topology on X = Spec(R), the so-called ZARISKI topology :PROOF. V (R) = ∅, V (〈0〉) = Spec(R), V (I) ∪ V (J) = V (IJ),and
⋂V (Ii) = V (
∑Ii).
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Definition
For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.
In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:
OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Example
The affine space Ank := Spec(k[x1, . . . , xn]).
For I E k[x1, . . . , xn] the affine subschemeV (I) := Spec(k[x1, . . . , xn]/I).
The two ingredients “prime spectrum” and “sheaf” preexistedthe definition of an affine scheme (U,OU ) but it wasGROTHENDIECK who put them together.
Corollary
{rings}op ≃ {affine schemes}.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Example
The affine space Ank := Spec(k[x1, . . . , xn]).
For I E k[x1, . . . , xn] the affine subschemeV (I) := Spec(k[x1, . . . , xn]/I).
The two ingredients “prime spectrum” and “sheaf” preexistedthe definition of an affine scheme (U,OU ) but it wasGROTHENDIECK who put them together.
Corollary
{rings}op ≃ {affine schemes}.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Example
The affine space Ank := Spec(k[x1, . . . , xn]).
For I E k[x1, . . . , xn] the affine subschemeV (I) := Spec(k[x1, . . . , xn]/I).
The two ingredients “prime spectrum” and “sheaf” preexistedthe definition of an affine scheme (U,OU ) but it wasGROTHENDIECK who put them together.
Corollary
{rings}op ≃ {affine schemes}.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The affine scheme associated to a ring
Example
The affine space Ank := Spec(k[x1, . . . , xn]).
For I E k[x1, . . . , xn] the affine subschemeV (I) := Spec(k[x1, . . . , xn]/I).
The two ingredients “prime spectrum” and “sheaf” preexistedthe definition of an affine scheme (U,OU ) but it wasGROTHENDIECK who put them together.
Corollary
{rings}op ≃ {affine schemes}.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
Schemes
Definition
A scheme is a locally ringed space (X,OX ) in which each pointx ∈ X has an open neighborhood U such that the locally ringedspace (U,OX |U ) is an affine scheme.
We don’t know how to “glue” rings
{rings} ⊂ ? (algebra)
but we know how to glue topological spaces and how to gluetheir structure sheaves
{rings}op ≃ {affine schemes} ⊂ {schemes} (geometry)
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
Schemes
Definition
A scheme is a locally ringed space (X,OX ) in which each pointx ∈ X has an open neighborhood U such that the locally ringedspace (U,OX |U ) is an affine scheme.
We don’t know how to “glue” rings
{rings} ⊂ ? (algebra)
but we know how to glue topological spaces and how to gluetheir structure sheaves
{rings}op ≃ {affine schemes} ⊂ {schemes} (geometry)
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
Schemes
Definition
A scheme is a locally ringed space (X,OX ) in which each pointx ∈ X has an open neighborhood U such that the locally ringedspace (U,OX |U ) is an affine scheme.
We don’t know how to “glue” rings
{rings} ⊂ ? (algebra)
but we know how to glue topological spaces and how to gluetheir structure sheaves
{rings}op ≃ {affine schemes} ⊂ {schemes} (geometry)
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The quasi-coherent sheaf associated to a module
For an R-module M define the sheafification M̃ to be thesheaf on Spec(R) satisfying
M̃p := Mp := (R \ p)−1M = Rp ⊗R M
and
M̃(D(f)) := Mf := Rf ⊗R M
for any p ∈ Spec(R) and any f ∈ R. M̃ is the prototype of aquasi-coherent sheaf.
Corollary
{R-modules} ≃˜
//{quasi-coherent sheaves on Spec(R)}
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The quasi-coherent sheaf associated to a module
For an R-module M define the sheafification M̃ to be thesheaf on Spec(R) satisfying
M̃p := Mp := (R \ p)−1M = Rp ⊗R M
and
M̃(D(f)) := Mf := Rf ⊗R M
for any p ∈ Spec(R) and any f ∈ R. M̃ is the prototype of aquasi-coherent sheaf.
Corollary
{R-modules} ≃˜
//{quasi-coherent sheaves on Spec(R)}
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
The quasi-coherent sheaf associated to a module
For an R-module M define the sheafification M̃ to be thesheaf on Spec(R) satisfying
M̃p := Mp := (R \ p)−1M = Rp ⊗R M
and
M̃(D(f)) := Mf := Rf ⊗R M
for any p ∈ Spec(R) and any f ∈ R. M̃ is the prototype of aquasi-coherent sheaf.
Corollary
{R-modules} ≃˜
//{quasi-coherent sheaves on Spec(R)}
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From rings to schemesFrom modules to quasi-coherent sheaves
Quasi-coherent sheaves
Definition
Let (X,OX ) be a scheme. A sheaf of OX-modules F is calledquasi-coherent if X can be covered by open affine subsetsUi := Spec(Ri) with F |Ui∼= M̃i (where Mi is some Ri-module).
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
Overview
1 Affine SchemesFrom rings to schemes
From modules to quasi-coherent sheaves
2 Projective SchemesFrom graded rings to projective schemes
From graded modules to quasi-coherent sheaves
3 The Standard ModuleThe module of global sections
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space Pnk := Proj(k[x0, . . . , xn])
We would like to construct Pnk as the scheme
Pnk “:=” (Ank \ {0})/k∗.
So consider the graded polynomial ring S := k[x0, . . . , xn]=
⊕i≥0 Si. Define the n+ 1 polynomial rings
Ri := k[x0xi, . . . , xn
xi] < k(Pnk) := k(
x1x0, . . . , xn
x0).
Define the projective space
Pnk := Proj(S)
by gluing together the affine spaces Ui := Spec(Ri) along theiraffine intersections Uij := Ui ∩ Uj := Spec(〈Ri, Rj〉), where〈Ri, Rj〉 is the subring of k(Pnk) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space Pnk := Proj(k[x0, . . . , xn])
We would like to construct Pnk as the scheme
Pnk “:=” (Ank \ {0})/k∗.
So consider the graded polynomial ring S := k[x0, . . . , xn]=
⊕i≥0 Si. Define the n+ 1 polynomial rings
Ri := k[x0xi, . . . , xn
xi] < k(Pnk) := k(
x1x0, . . . , xn
x0).
Define the projective space
Pnk := Proj(S)
by gluing together the affine spaces Ui := Spec(Ri) along theiraffine intersections Uij := Ui ∩ Uj := Spec(〈Ri, Rj〉), where〈Ri, Rj〉 is the subring of k(Pnk) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space Pnk := Proj(k[x0, . . . , xn])
We would like to construct Pnk as the scheme
Pnk “:=” (Ank \ {0})/k∗.
So consider the graded polynomial ring S := k[x0, . . . , xn]=
⊕i≥0 Si. Define the n+ 1 polynomial rings
Ri := k[x0xi, . . . , xn
xi] < k(Pnk) := k(
x1x0, . . . , xn
x0).
Define the projective space
Pnk := Proj(S)
by gluing together the affine spaces Ui := Spec(Ri) along theiraffine intersections Uij := Ui ∩ Uj := Spec(〈Ri, Rj〉), where〈Ri, Rj〉 is the subring of k(Pnk) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space Pnk := Proj(k[x0, . . . , xn])
We would like to construct Pnk as the scheme
Pnk “:=” (Ank \ {0})/k∗.
So consider the graded polynomial ring S := k[x0, . . . , xn]=
⊕i≥0 Si. Define the n+ 1 polynomial rings
Ri := k[x0xi, . . . , xn
xi] < k(Pnk) := k(
x1x0, . . . , xn
x0).
Define the projective space
Pnk := Proj(S)
by gluing together the affine spaces Ui := Spec(Ri) along theiraffine intersections Uij := Ui ∩ Uj := Spec(〈Ri, Rj〉), where〈Ri, Rj〉 is the subring of k(Pnk) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space P1k := Proj(k[x0, x1])
Example
Define the two polynomial rings R0 := k[x1x0 ] = k[t] andR1 := k[
x0x1] = k[t−1]. The projective line
P1k := Proj(k[x0, x1])
as the gluing of the affine spaces
U0 := Spec(R0) = Spec(k[t]) and
U1 := Spec(R1) = Spec(k[t−1])
along their affine intersection U01 := U0 ∩ U1 := Spec(〈R0, R1〉),where 〈R0, R1〉 = k[t, t−1] is LAURENT polynomial ring, which isthe subring of k(P1k) = k(
x1x0) = k(t) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The projective space P1k := Proj(k[x0, x1])
Example
Define the two polynomial rings R0 := k[x1x0 ] = k[t] andR1 := k[
x0x1] = k[t−1]. The projective line
P1k := Proj(k[x0, x1])
as the gluing of the affine spaces
U0 := Spec(R0) = Spec(k[t]) and
U1 := Spec(R1) = Spec(k[t−1])
along their affine intersection U01 := U0 ∩ U1 := Spec(〈R0, R1〉),where 〈R0, R1〉 = k[t, t−1] is LAURENT polynomial ring, which isthe subring of k(P1k) = k(
x1x0) = k(t) generated by Ri and Rj.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Proj construction
One can analogously define X := Proj(S) for an arbitrarygraded rings S =
⊕i≥0 Si. Let m := S>0 :=
⊕i>0 Si denote the
maximal graded ideal in S.
Define the set
Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set
V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).
Finally use the above to construct the structure sheaf OX .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
Projective schemes
Definition
A projective scheme X is a scheme of the form
X := Proj(S)
for some graded ring S =⊕
i≥0 Si.
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The quasi-coh. sheaf associated to a graded module
For a graded S-module M =⊕
i∈ZMi define the sheafificationM̃ to be the quasi-coherent sheaf on Proj(S) satisfying
M̃p := M(p) :=((S \ p)−1homM
)0
and
M̃(D(f)) := M(f) := (Mf )0 := (Sf ⊗S M)0for any p ∈ Proj(S) and any homogeneous f ∈ m.
Theorem
Any quasi-coherent sheaf on a projective scheme X := Proj(S)is the sheafification M̃ of some graded S-module M .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The quasi-coh. sheaf associated to a graded module
For a graded S-module M =⊕
i∈ZMi define the sheafificationM̃ to be the quasi-coherent sheaf on Proj(S) satisfying
M̃p := M(p) :=((S \ p)−1homM
)0
and
M̃(D(f)) := M(f) := (Mf )0 := (Sf ⊗S M)0for any p ∈ Proj(S) and any homogeneous f ∈ m.
Theorem
Any quasi-coherent sheaf on a projective scheme X := Proj(S)is the sheafification M̃ of some graded S-module M .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
Up to finite length
Unfortunately the sheafification does not yield an equivalenceof categories
{graded S-modules} 6≃˜
//{quasi-coh. sheaves onProj(S)}
Theorem
Two graded S-modules M and N define the samequasi-coherent sheaf iff M≥d ∼= N≥d for some d ∈ Z.
Q: Is there a way to find a canonical representative in anequivalence class of graded modules “isomorphic in highdegrees”?
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
Up to finite length
Unfortunately the sheafification does not yield an equivalenceof categories
{graded S-modules} 6≃˜
//{quasi-coh. sheaves onProj(S)}
Theorem
Two graded S-modules M and N define the samequasi-coherent sheaf iff M≥d ∼= N≥d for some d ∈ Z.
Q: Is there a way to find a canonical representative in anequivalence class of graded modules “isomorphic in highdegrees”?
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
Up to finite length
Unfortunately the sheafification does not yield an equivalenceof categories
{graded S-modules} 6≃˜
//{quasi-coh. sheaves onProj(S)}
Theorem
Two graded S-modules M and N define the samequasi-coherent sheaf iff M≥d ∼= N≥d for some d ∈ Z.
Q: Is there a way to find a canonical representative in anequivalence class of graded modules “isomorphic in highdegrees”?
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
Overview
1 Affine SchemesFrom rings to schemes
From modules to quasi-coherent sheaves
2 Projective SchemesFrom graded rings to projective schemes
From graded modules to quasi-coherent sheaves
3 The Standard ModuleThe module of global sections
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The global sections of the twisted sheaves
Let S be a graded NOETHERIAN ring and M a finitely generatedgraded S-module. Denote by M [i] the shifted S-module with
M [i]j := Mi+j
and by
H0(M̃ [i]) := M̃ [i](Proj(S))
the global sections of the coherent twisted sheaf M̃ [i].
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The global sections of the twisted sheaves
Let S be a graded NOETHERIAN ring and M a finitely generatedgraded S-module. Denote by M [i] the shifted S-module with
M [i]j := Mi+j
and by
H0(M̃ [i]) := M̃ [i](Proj(S))
the global sections of the coherent twisted sheaf M̃ [i].
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The global sections of the twisted sheaves
Let S be a graded NOETHERIAN ring and M a finitely generatedgraded S-module. Denote by M [i] the shifted S-module with
M [i]j := Mi+j
and by
H0(M̃ [i]) := M̃ [i](Proj(S))
the global sections of the coherent twisted sheaf M̃ [i].
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The standard module
DefinitionThe truncated direct sum
H0≥0(M̃ ) :=⊕
i≥0H0(M̃ [i])
has a natural S-module structure. We call it the standardmodule of the coherent sheaf M̃ (truncated at 0).
Theorem
The sheafification of the standard module of a sheaf M̃ is againisomorphic to M̃ :
˜H0≥0(M̃ )
∼= M̃ .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The standard module
DefinitionThe truncated direct sum
H0≥0(M̃ ) :=⊕
i≥0H0(M̃ [i])
has a natural S-module structure. We call it the standardmodule of the coherent sheaf M̃ (truncated at 0).
Theorem
The sheafification of the standard module of a sheaf M̃ is againisomorphic to M̃ :
˜H0≥0(M̃ )
∼= M̃ .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
The standard module
DefinitionThe truncated direct sum
H0≥0(M̃ ) :=⊕
i≥0H0(M̃ [i])
has a natural S-module structure. We call it the standardmodule of the coherent sheaf M̃ (truncated at 0).
Theorem
The sheafification of the standard module of a sheaf M̃ is againisomorphic to M̃ :
˜H0≥0(M̃ )
∼= M̃ .
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
software demo
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
-
OverviewAffine Schemes
Projective SchemesThe Standard Module
The module of global sections
Mohamed Barakat and Markus Lange-Hegermann, AnAxiomatic Setup for Algorithmic Homological Algebra andan Alternative Approach to Localization, to appear inJournal of Algebra and its Applications(arXiv:1003.1943).
Robin Hartshorne, Algebraic geometry, Springer-Verlag,New York, 1977, Graduate Texts in Mathematics, No. 52.MR MR0463157 (57 #3116)
Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves
http://arxiv.org/abs/1003.1943http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves
OverviewAffine SchemesFrom rings to schemesFrom modules to quasi-coherent sheaves
Projective SchemesFrom graded rings to projective schemesFrom graded modules to quasi-coherent sheaves
The Standard ModuleThe module of global sections
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