phd defense, may 24, 2007 intersection multiplicities and...
TRANSCRIPT
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PhD defense, May 24, 2007
Intersection multiplicities andGrothendieck spacesEsben Bistrup Halvorsen
Department of Mathematical Sciences
University of Copenhagen
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm.
– p.2/15
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define
D(R) = the derived category ofR.
– p.2/15
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define
D(R) = the derived category ofR.
Df2(R) = the full subcategory of finite complexes.
– p.2/15
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define
D(R) = the derived category ofR.
Df2(R) = the full subcategory of finite complexes.
Pf(R) = the full subcategory of finite complexes
isomorphic to a bounded complex of projectives.
– p.2/15
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define
D(R) = the derived category ofR.
Df2(R) = the full subcategory of finite complexes.
Pf(R) = the full subcategory of finite complexes
isomorphic to a bounded complex of projectives.
Let X, Y ∈ Df2(R) with Supp X ∩ Supp Y = {m}.
– p.2/15
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INTERSECTION MULTIPLICITIES
Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define
D(R) = the derived category ofR.
Df2(R) = the full subcategory of finite complexes.
Pf(R) = the full subcategory of finite complexes
isomorphic to a bounded complex of projectives.
Let X, Y ∈ Df2(R) with Supp X ∩ Supp Y = {m}. The
intersection multiplicityof X andY is defined as
χ(X, Y) = ∑i
(−1)i length Hi(X ⊗LR Y)
wheneverX ∈ Pf(R) or Y ∈ P
f(R). – p.2/15
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INTERSECTION MULTIPLICITIES
The ringR satisfies vanishingif
χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.
– p.3/15
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INTERSECTION MULTIPLICITIES
The ringR satisfies vanishing if
χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.
This generalizes the vanishing conjecture by Serre.
– p.3/15
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INTERSECTION MULTIPLICITIES
The ringR satisfies vanishing if
χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.
This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishingif this holds when bothX ∈ P
f(R) and Y ∈ Pf(R).
– p.3/15
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INTERSECTION MULTIPLICITIES
The ringR satisfies vanishing if
χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.
This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishing if this holds when bothX ∈ P
f(R) and Y ∈ Pf(R).
Theorem (Dutta–Hochster–McLaughlin). Thevanishing conjecture does not hold, so not all ringssatisfy vanishing!
– p.3/15
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INTERSECTION MULTIPLICITIES
The ringR satisfies vanishing if
χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.
This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishing if this holds when bothX ∈ P
f(R) and Y ∈ Pf(R).
Theorem (Dutta–Hochster–McLaughlin). Thevanishing conjecture does not hold, so not all ringssatisfy vanishing!
So far, all rings satisfy weak vanishing.
– p.3/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field.
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R.
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. TheDuttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y).
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y),
whereLF is the left-derived Frobenius functor.
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y),
whereLF is the left-derived Frobenius functor, given by
X = · · · −→ Xi
∂Xi−→ Xi−1 −→ · · ·
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y),
whereLF is the left-derived Frobenius functor, given by
X ≃ · · · −→ Rm(aij)−→ Rn −→ · · ·
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y),
whereLF is the left-derived Frobenius functor, given by
LF(X) ≃ · · · −→ Rm(a
pij)
−→ Rn −→ · · · .
– p.4/15
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INTERSECTION MULTIPLICITIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX, Y ∈ D
f2(R) with
Supp X ∩ Supp Y = {m}, X ∈ Pf(R) and
dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ(LFe(X), Y),
whereLF is the left-derived Frobenius functor, given by
LF(X) ≃ · · · −→ Rm(a
pij)
−→ Rn −→ · · · .
The Dutta multiplicity satisfies vanishing. – p.4/15
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GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes.
– p.5/15
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GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. TheGrothendieck groupof C is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
– p.5/15
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GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. The Grothendieck group ofC is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup.
– p.5/15
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GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. The Grothendieck group ofC is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup. We can compute
χ(X, Y)– p.5/15
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GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. The Grothendieck group ofC is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup. We can compute
χ(X, Y) = χ([X], Y)– p.5/15
![Page 26: PhD defense, May 24, 2007 Intersection multiplicities and ...esben.bistruphalvorsen.dk/papers/phddefense.pdf · PhD defense, May 24, 2007 Intersection multiplicities and Grothendieck](https://reader034.vdocuments.net/reader034/viewer/2022050505/5f9705fac1ebe926543dcfdd/html5/thumbnails/26.jpg)
GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. The Grothendieck group ofC is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup. We can compute
χ(X, Y) = χ([X], Y) = χ([X′], Y)– p.5/15
![Page 27: PhD defense, May 24, 2007 Intersection multiplicities and ...esben.bistruphalvorsen.dk/papers/phddefense.pdf · PhD defense, May 24, 2007 Intersection multiplicities and Grothendieck](https://reader034.vdocuments.net/reader034/viewer/2022050505/5f9705fac1ebe926543dcfdd/html5/thumbnails/27.jpg)
GROTHENDIECK GROUPS
Let C be a full subcategory of the category of finitecomplexes. The Grothendieck group ofC is the AbeliangroupK0(C) presented by generators[X], one for eachisomorphism class of a complexX in C, and relations
[X] = 0 wheneverX is exact
[X] = [X′] + [X′′] whenever0 → X′ → X → X′′ → 0is a short exact sequence inC.
Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup. We can compute
χ(X, Y) = χ([X], Y) = χ([X′], Y) = χ(X′, Y).– p.5/15
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GROTHENDIECK GROUPS
We can also compute
χ∞(X, Y)
– p.6/15
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GROTHENDIECK GROUPS
We can also compute
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ([LFe(X)], Y)
where[LFe(X)] ∈ K0(C).
– p.6/15
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GROTHENDIECK GROUPS
We can also compute
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ([LFe(X)], Y)
= lime→∞
χ(1
pe codim(Supp X)[LFe(X)], Y)
wherep−e codim(Supp X)[LFe(X)] ∈ K0(C)⊗Z Q.
– p.6/15
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GROTHENDIECK GROUPS
We can also compute
χ∞(X, Y) = lime→∞
1
pe codim(Supp X)χ([LFe(X)], Y)
= lime→∞
χ(1
pe codim(Supp X)[LFe(X)], Y)
= χ( lime→∞
1
pe codim(Supp X)[LFe(X)], Y)
where???
– p.6/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R.
– p.7/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R: that is,
p ∈ X and p ⊆ q implies q ∈ X.
– p.7/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R: that is,
p ∈ X and p ⊆ q implies q ∈ X.
Define
Df2(X) = the full subcategory ofDf
2(R) of complexes
with support contained inX.
– p.7/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R: that is,
p ∈ X and p ⊆ q implies q ∈ X.
Define
Df2(X) = the full subcategory ofDf
2(R) of complexes
with support contained inX.
Pf(X) = the full subcategory ofPf(R) of complexes
with support contained inX.
– p.7/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R: that is,
p ∈ X and p ⊆ q implies q ∈ X.
Define
Df2(X) = the full subcategory ofDf
2(R) of complexes
with support contained inX.
Pf(X) = the full subcategory ofPf(R) of complexes
with support contained inX.
Let Xc denote the maximal subset ofSpec R such that
X∩Xc = {m} and dim X + dim Xc ≤ dim R.
– p.7/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R.
– p.8/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R. TheGrothendieck spaceof P
f(X) is theQ-vector spaceGPf(X)
presented by generators[X], one for each isomorphismclass of a complexX in P
f(X), and relations
[X] = [X′] when χ(X,−) = χ(X′,−)
as metafunctionsDf2(Xc) → Q.
– p.8/15
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GROTHENDIECK SPACES
Let X be a specialization-closed subset ofSpec R. TheGrothendieck space ofPf(X) is theQ-vector spaceGP
f(X)presented by generators[X], one for each isomorphismclass of a complexX in P
f(X), and relations
[X] = [X′] when χ(X,−) = χ(X′,−)
as metafunctionsDf2(Xc) → Q.
The spaceGPf(X) is equipped with the initial topology of
the family ofQ-linear maps
χ(−, Y) : GPf(X) → Q for Y ∈ D
f2(Xc).
– p.8/15
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GROTHENDIECK SPACES
Now we can calculate . . .
– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y)
– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y) = lime→∞
1
pe codim Xχ([LFe(X)], Y)
whereX = Supp X.– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y) = lime→∞
1
pe codim Xχ([LFe(X)], Y)
= lime→∞
χ(1
pe codim X[LFe(X)], Y)
whereX = Supp X.– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y) = lime→∞
1
pe codim Xχ([LFe(X)], Y)
= lime→∞
χ(1
pe codim X[LFe(X)], Y)
= lime→∞
χ(1
pe codim XFeX([X]), Y)
whereX = Supp X andFeX([X]) = [LFe(X)].
– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y) = lime→∞
1
pe codim Xχ([LFe(X)], Y)
= lime→∞
χ(1
pe codim X[LFe(X)], Y)
= lime→∞
χ(1
pe codim XFeX([X]), Y)
= lime→∞
χ(ΦeX([X]), Y)
whereX = Supp X andΦeX = p−e codim XFe
X.– p.9/15
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GROTHENDIECK SPACES
Now we can calculate
χ∞(X, Y) = lime→∞
1
pe codim Xχ([LFe(X)], Y)
= lime→∞
χ(1
pe codim X[LFe(X)], Y)
= lime→∞
χ(1
pe codim XFeX([X]), Y)
= lime→∞
χ(ΦeX([X]), Y)
= χ( lime→∞
ΦeX([X]), Y)
whereX = Supp X andlime→∞ ΦeX([X]) ∈ GP
f(X).– p.9/15
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GROTHENDIECK SPACES
Theorem 1. Let X be a specialization-closed subset of
Spec R and let α ∈ GPf(X).
– p.10/15
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GROTHENDIECK SPACES
Theorem 1. Let X be a specialization-closed subset of
Spec R and let α ∈ GPf(X). Then there is a unique
decomposition
α = α(0) + · · ·+ α(u)
in which each α(i) is either zero or an eigenvector for
ΦX with eigenvalue p−i.
– p.10/15
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GROTHENDIECK SPACES
Theorem 1. Let X be a specialization-closed subset of
Spec R and let α ∈ GPf(X). Then there is a unique
decomposition
α = α(0) + · · ·+ α(u)
in which each α(i) is either zero or an eigenvector for
ΦX with eigenvalue p−i. The components
α(0), . . . , α(u) are recursively defined by
α(0) = lime→∞
ΦeX(α) and
α(i) = lime→∞
pieΦeX(α − (α(0) + · · ·+ α(i−1))),
– p.10/15
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GROTHENDIECK SPACES
. . . and there is a formula
α(0)
...
α(u)
=
1 1 · · · 1
1 p−1 · · · p−u
...... . . . ...
1 p−u · · · p−u2
−1
α
ΦX(α)...
ΦuX(α)
.
– p.11/15
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GROTHENDIECK SPACES
. . . and there is a formula
α(0)
...
α(u)
=
1 1 · · · 1
1 p−1 · · · p−u
...... . . . ...
1 p−u · · · p−u2
−1
α
ΦX(α)...
ΦuX(α)
.
The numberu is thevanishing dimensionof α; it measures,in a sense, how farα is from satisfying vanishing. Inparticular,α satisfies vanishing if and only ifα = α(0).
– p.11/15
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GROTHENDIECK SPACES
. . . and there is a formula
α(0)
...
α(u)
=
1 1 · · · 1
1 p−1 · · · p−u
...... . . . ...
1 p−u · · · p−u2
−1
α
ΦX(α)...
ΦuX(α)
.
The numberu is the vanishing dimension ofα; it measures,in a sense, how farα is from satisfying vanishing. Inparticular,α satisfies vanishing if and only ifα = α(0).
We haveu ≤ max(codim X− 2, 0).
– p.11/15
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GROTHENDIECK SPACES
Translating the theorem to complexes, the Duttamultiplicity χ∞(X, Y) can be computed.
– p.12/15
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GROTHENDIECK SPACES
Translating the theorem to complexes, the Duttamultiplicity χ∞(X, Y) is the first entry in
1 1 · · · 1
pt pt−1 · · · pt−u
...... . . . ...
put pu(t−1) · · · pu(t−u)
−1
χ(X, Y)
χ(LF(X), Y)...
χ(LFu(X), Y)
,
wheret = codim(Supp X).
– p.12/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field.
– p.13/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX be a specialization-closedsubset ofSpec R and letα ∈ GP
f(X).
– p.13/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX be a specialization-closedsubset ofSpec R and letα ∈ GP
f(X). If
χ(α, Y) = χ(α(0), Y) for all Y ∈ Pf(Xc), thenα satisfies
numerical vanishing.
– p.13/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX be a specialization-closedsubset ofSpec R and letα ∈ GP
f(X). If
χ(α, Y) = χ(α(0), Y) for all Y ∈ Pf(Xc), thenα satisfies
numerical vanishing. If this holds for all elements in allGrothendieck spaces, thenR satisfiesnumerical vanishing.
– p.13/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX be a specialization-closedsubset ofSpec R and letα ∈ GP
f(X). If
χ(α, Y) = χ(α(0), Y) for all Y ∈ Pf(Xc), thenα satisfies
numerical vanishing. If this holds for all elements in allGrothendieck spaces, thenR satisfies numerical vanishing.
Theorem 2. The ring R satisfies numerical vanishing
if and only if all elements of GPf({m}) do.
– p.13/15
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NUMERICAL PROPERTIES
Assume thatR is complete of prime characteristicp andwith perfect residue field. LetX be a specialization-closedsubset ofSpec R and letα ∈ GP
f(X). If
χ(α, Y) = χ(α(0), Y) for all Y ∈ Pf(Xc), thenα satisfies
numerical vanishing. If this holds for all elements in allGrothendieck spaces, thenR satisfies numerical vanishing.
Theorem 2. The ring R satisfies numerical vanishing
if and only if all elements of GPf({m}) do. In
particular, this holds if and only if
χ(LF(Z)) = pdim Rχ(Z)
for all complexes Z in Pf({m}).
– p.13/15
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NUMERICAL PROPERTIES
The duality functorRHomR(−, R) onPf(X) induces an
automorphism(−)∗ on GPf(X).
– p.14/15
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NUMERICAL PROPERTIES
The duality functorRHomR(−, R) onPf(X) induces an
automorphism(−)∗ on GPf(X).
An elementα ∈ GPf(X) is self-dualif it satisfies
α = (−1)codimXα∗.
– p.14/15
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NUMERICAL PROPERTIES
The duality functorRHomR(−, R) onPf(X) induces an
automorphism(−)∗ on GPf(X).
An elementα ∈ GPf(X) is self-dual if it satisfies
α = (−1)codimXα∗, andα is numerically self-dualif itsatisfiesχ(α, Y) = (−1)codimXχ(α∗, Y) for allY ∈ P
f(Xc).
– p.14/15
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NUMERICAL PROPERTIES
The duality functorRHomR(−, R) onPf(X) induces an
automorphism(−)∗ on GPf(X).
An elementα ∈ GPf(X) is self-dual if it satisfies
α = (−1)codimXα∗, andα is numerically self-dual if itsatisfiesχ(α, Y) = (−1)codimXχ(α∗, Y) for allY ∈ P
f(Xc). If this holds for all elements in allGrothendieck spaces, thenR satisfiesself-dualityornumerical self-duality, respectively.
– p.14/15
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NUMERICAL PROPERTIES
The duality functorRHomR(−, R) onPf(X) induces an
automorphism(−)∗ on GPf(X).
An elementα ∈ GPf(X) is self-dual if it satisfies
α = (−1)codimXα∗, andα is numerically self-dual if itsatisfiesχ(α, Y) = (−1)codimXχ(α∗, Y) for allY ∈ P
f(Xc). If this holds for all elements in allGrothendieck spaces, thenR satisfies self-duality ornumerical self-duality, respectively.
Theorem 3 (w. Frankild). When R is complete ofprime characteristic p and with perfect residue field,
(−1)codimXα∗ = α(0) − α(1) + · · ·+ (−1)uα(u). – p.14/15
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RING PROPERTIES
vanishing
weak vanishing – p.15/15
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RING PROPERTIES
vanishing
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��vanishing
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��vanishing
��numericalself-duality
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��regular +3 vanishing
��numericalself-duality
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��regular +3 vanishing
��
dim ≤ 2ks
numericalself-duality
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
numericalself-duality
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
�%C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
numericalself-duality
��weak vanishing – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
�%C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
numericalself-duality
��weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
�%C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
C
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
numericalself-duality
��
Gorensteinof dim ≤ 5
s{ nnnnnnnnnnn
nnnnnnnnnnn
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
$,PPPPPPPPPPPP
PPPPPPPPPPPP
numericalself-duality
��
Gorensteinof dim ≤ 5
s{ nnnnnnnnnnn
nnnnnnnnnnn
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
��
+3 vanishing
��
dim ≤ 2ks
completeintersection
$,PPPPPPPPPPPP
PPPPPPPPPPPP
numericalself-duality
��
Gorensteinof dim ≤ 5
ks
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
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dim ≤ 2ks
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Gorensteinof dim ≤ 5
ks
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
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+3 vanishing
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dim ≤ 2ks
completeintersection
+3 numericalvanishing
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Gorensteinof dim ≤ 5
ks
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
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+3 vanishing
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dim ≤ 2ks
completeintersection
+3 numericalvanishing
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Gorensteinof dim ≤ 3
ks
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Gorensteinof dim ≤ 5
ks
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
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+3 vanishing
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dim ≤ 2ks
completeintersection
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numericalvanishing
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Gorensteinof dim ≤ 3
ks
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Gorensteinof dim ≤ 5
ks
weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
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+3 vanishing
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dim ≤ 2ks
completeintersection
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numericalvanishing
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Gorensteinof dim ≤ 3
ks
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numericalself-duality
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Gorensteinof dim ≤ 5
ks
Cohen–Macaulay weak vanishing dim ≤ 4ks – p.15/15
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RING PROPERTIES
self-dualityKS
��regular
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+3 vanishing
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dim ≤ 2ks
completeintersection
+3
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numericalvanishing
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Gorensteinof dim ≤ 3
ks
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ks
Cohen–Macaulay weak vanishing dim ≤ 4ks – p.15/15