Families of schemes
Introduction to Flatness and the Quot scheme
Sheffield, Reading seminar on DT theory, 308/04/2020
Anna Barbieri
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Today
0. intro1. flatness of sheaves/schemes2. the Quot scheme
Bibliography- Huybrechts, Lehn, The geometry of moduli spaces of sheaves, Ch. 2
- Eisenbud, Harris, The geometry of schemes
- Nitsure, Construction of Hilbert and Quot scheme, in “Fundamentalalgebraic geometry: Grothendieck’s FGA explained”
- Cristina!
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Introduction
X a scheme, X LOC=(
Spec R,OX
)(AffSch)op ' (Rings)
(Sch) (Schk ) (SchS)
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Introduction
X ∈ (SchS)f : T → S, morphism of schemes
Base-change
XT := X ×S Tfibres over T
XT πX//
πT��
X
��T // S
If S = Spec R, X = Spec R1, T = Spec R2, then XT = Spec(R1 ⊗R R2).
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1. Flatness
Flatness for modulesDef.: M ∈ R-mod flat iff
∀ M1 ↪→ M2 injectivethen M1 ⊗R M ↪→ M2 ⊗R M injective.
Examples- free modules are flat;- flat modules are torsion-free, silly ex:
0→ Z ·2−→Z→ Z/2Z→ 0 ⊗ZZ/2Z
Z/2Z ·2−→Z/2Z
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Flatness for coh sheaves
Def.: F ∈ Coh(X ) is flat if
it is a sheaf of OX -flat modules.
Equiv.:every stalk Fx is a flat OX ,x -module.
Example: Locally free sheaves are flat
OX (D) is a flat OX -sheaf
Non-example: torsion sheaves
D a divisor on X , OD is not flat over X .
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Flatness for schemes / sheaves on S-schemes
Def.: A map f : X → S is flat if
OX is flat as an OS-module.
Recall: a map of sch.s f : X → S induces a map of sheaves f # : OS → OX .
By the (non)example above, pt → CP1 is not flat.
Def.: A sheaf F on X ∈ (SchS) is flat over S if
Fx is flat as an Of (x),S-module.
Remark: Given f : X → S, a sheaf F ∈ Coh(X ) which is flatover S is also called a
flat family of coherent sheaves on the fibres of f .
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Some general facts
• Open embeddings are flat (maps)
• X a Noetherian scheme, then
F flat over X ⇐⇒ locally free.
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Some general factsf : X → S, F ∈ Coh(X ) S-flat
• Base change
• transitivity: g : S → S′ flat⇒ F is S′-flat too• behaviour in short exact sequences
0→ A→ B → FF → 0
A is S-flat⇔ B is S-flat.
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Flat families of schemes
X (flat) S-scheme, akaπ : X → S (flat) family of schemes
Flat families provide a notion of “continuously varying” family.
This will be better understood if we look at one-parameter families⇒ geometric characterization of flatness.
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Geometric characterization of flatness for maps ofschemes, 1
Dimension 1
Propositionf : X → S a morphism of schemes,where S = Spec R, smooth, 1-dimensional affine scheme.
map f flat ⇐⇒ no irreducible nor embedded componentsare contracted.
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Proof
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Proof
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Flatness is a local property!
Dimension 1
PropositionS = Spec R 1-dimensional affine smooth scheme,X o ⊂ An
B a flat family of closed schemes over So = S \ {b∗}.
There is a unique way of completing X o to a flat family over B.
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Flatness is a local property!
Obtained by taking the closure X o inside AnB.
Added fibre: limit of X o at b∗.
Depends very much on the embedding.
Does not admit generalization to any dimensional bases.
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Geometric characterization of flatness for maps ofschemes, 2
Any dimension
PropositionS be a reduced scheme
f : X → S is flat ⇐⇒ ∀C ↪→ S a curve,fC is flat
XC //
fC��
X
f��
C // S
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Geometric meaning of flatness
Consequence: all fibres must have the same dimension. So
1. SpecC[x , y ]/(xy , x2)→ SpecC[x , y ]/(x) is not flat
2. BlpP2 → P2 is not flat
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RemarkWe are not able to speak about limit fibre of a punctured sheafis S is not of dimension 1.
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Open condition
PropositionX ,S Noetherian schemes.
Flatness of f : X → S (resp. of a sheaf F ∈ Coh(X )) is an opencondition.
I.e.
∀V ⊂ S, there is a maximal open U ⊂ V ⊂ Xwhere f (resp. F ) is flat.
As a corollary, being locally free is an open condition as well.
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Hilbert polynomial of flat families
f : X → S a projective family
Def.: L = O(1) is an f -ample line bundle on X if L|Xs is amplefor any s ∈ S.
Hilbert polynomial of a projective family f : X → S{φL|Xs (Xs) : s ∈ S
}
• it is locally constant as a function of s ∈ S;• it is constant⇔ if f is flat and B is reduced;• finitely many possible Hilbert polynomials of the fibres;
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Hilbert polynomial of flat families
Hilbert polynomial of a projective family f : X → S{φL|Xs (Xs) : s ∈ S
}
• induces a stratification of S by
φs = φL|Xs (Xs).
This decomposition is called a flattening stratification of S forf and it is unique.
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2. The Quot functor
Recall the Grassmannian of a vector space. . .
V v.s. of dimension n > 0, /k ,0 ≤ r ≤ n
Gr (r ,n) = {v.s. W ⊆ V , dim W = r} ' Gr (n − r ,n)
via V = W ⊕W⊥ = V ⊕ V/W .
Generalize to trivial v.bundle over a base B, take sub-vectorbundles of a given rank r .
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OB ⊗k V , loc. free, B ∈ (Schk )
defineGrB(V , r )
as the set of all sub-sheaves K ⊂ OB ⊗k V with locally freequotient
F =(OB ⊗k V
)/K
of constant rank r ,i.e.
q : OB ⊗k V → F → 0.
Why quotient sheaves?- in general sequences do not split,- quotient sheaves behaves better than sub-sheaves.
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Sketchy Quot
Vaguely speaking, forf : X → S projective morphismH ∈ Coh(X ),P ∈ Q[z]
we want to parametrise flat coherent quotient sheaves
q : H → F → 0
with Hilbert polynomial P.
Work “relative“, and take Quot as the scheme that represents afunctor Quot.
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The functor of points
X a scheme, the functor of points of the scheme X is
hX : (Sch)op → (Sets)T 7→ Mor(T ,X )(
f : Y → Z)7→(hX (f ) : g 7→ g ◦ f
)g ∈ Mor(Z , X ), g ◦ f ∈ Mor(Y , X )
Def.: A covariant functor
H : (Sch)op → (Sets)
is representable if
∃X scheme, s.t. H = hX .
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The Quot functor, Pn
S a (locally) Noetherian scheme, PnS = Pn ×Z S, r ∈ N>0.
A family of quotients of O⊕rPn parametrised by S is⟨
F ,q⟩
(i) F ∈ Coh(PnS) flat over S,and
(ii) q : O⊕rPn
S→ F surjective and OPn
S-linear homomorphism,
up to⟨F ,q
⟩=⟨F ′,q′
⟩if ker(q) = ker(q′).
The equivalence relation is also given byF'→ F ′ fitting the commutative diagram:
F
'��
⊕rOPnS
q 44
q′
))F ′
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The Quot functor, Pn
Controvariant functor
QuotO⊕rPn
: (Sch)→ (Sets)
S 7→{〈F ,q〉 parametrised by S
}
〈F , q〉 family of quotients parametrised by S:F ∈ Coh(Pn
S) flat over S,q : O⊕r
PnS→ F surjective.
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If we wanted
OPnS
q // F
xx
OPn
xxS � // Pn
S//
��
Pn
��S // Z
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The functor QuotE /X/S
S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )
E
wwX
��S
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The functor QuotE /X/S
S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )
∀ T → S an S-scheme
ET := π∗X E
ET
E
xxXT
πT��
πX// X
��T // S
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The functor QuotE /X/S
S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )
∀ T → S an S-schemeET := π∗X E
ET
q // F
ww
E
wwXT πX//
πT��
X
��T // S
choose
〈F ,q〉F T -flat sheaf, q : ET → F surjective
up to equivalence relation given by ker(q) = ker(q′).
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The functor QuotE /X/S
S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )
∀ T → S an S-schemeET := π∗X E
ET
q // F
ww
E
wwXT πX//
πT��
X
��T // S
choose
〈F ,q〉shf T -flat sheaf, q : ET → F surjective
up to equivalence relation given by ker(q) = ker(q′).
QuotE /X/S : (SchS)→ (Sets)
T 7→{〈F ,q〉 parametrised by T
}32 / 38
The Hilbert functorA special Quot
HilbX/S := QuotOX/X/S
HilbX/S : (SchS)→ (Sets)
T 7→{closed Y ⊂ XT flat over T
}In particular
HilbPn := HilbPnZ/SpecZ
associates to S families of subschemes of Pn parametrised byS.
(closed subschemes Y ⊂ PnS flat /S same as OPn → OY on Pn
S flat /S)
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Stratification by Hilbert polynomial
f : X → SL ample line bundle on X relative SRecall the flattening stratification of S for f
Decomposition
QuotE/X/S =⊔φ,L
Quotφ,LE/X/S,
where, for any φ ∈ Q[λ],
Quotφ,LE/X/S : (T → S) 7→{〈F ,q〉 | ∀ t ∈ T φL∗T (Ft ) = φ
}.
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The Grassmannian as a Quot scheme
V v.s. dim V = n, 0 ≤ d ≤ n.The functor T 7→ GrT (V ,d) is
Grass(n,d) = Quotd ,OZO⊕nZ /Spec(Z)/Spec(Z)
,
abbreviated Quotd ,OZO⊕nZ /Z/Z. It maps
T 7→⟨F ∈ Coh(T ) l.f. rk d , q : O⊕n
T → F⟩
When d = 1, we know this is PnZ = ProjZ[x0, . . . , xn]. In
particular it is representable.
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The Grassmannian as a Quot scheme
E locally free OS-module of rank r
Grass(E ,d) = Quotd ,OSE/S/S.
is also representable. Grass(E ,d) parametrises sub-vectorbundles of E .
When d = 1,
Grass(E ,d) = P(E) = Proj SymOS(E).
The construction of the scheme Quot make sense even ifE ∈ Coh(S) is not locally free.
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Representability of Quot
The functor Quot is proven to be representable.
π : X → S, Noetherian.In the special case when
X = P(V ),E = π∗(W )
V ,W vector bundles /S
the representability is obtained by
Quotφ,LE /X/S → Grass(W ⊗OS Symr V , φ(r )
)for some r ∈ N>0
The general case then follows from base change and usingsome properties of flat families.
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Thanks
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