phd defense, may 24, 2007 intersection multiplicities and...

PhD defense, May 24, 2007

Intersection multiplicities andGrothendieck spacesEsben Bistrup Halvorsen

Department of Mathematical Sciences

University of Copenhagen

– p.1/15

INTERSECTION MULTIPLICITIES

Throughout,R denotes a commutative, Noetherian, localring with maximal idealm.

– p.2/15


Throughout,R denotes a commutative, Noetherian, localring with maximal idealm. Define

D(R) = the derived category ofR.

– p.2/15




Df2(R) = the full subcategory of finite complexes.

– p.2/15





Pf(R) = the full subcategory of finite complexes

isomorphic to a bounded complex of projectives.

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Let X, Y ∈ Df2(R) with Supp X ∩ Supp Y = {m}.

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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


The ringR satisfies vanishingif

χ(X, Y) = 0 whendim(Supp X) + dim(Supp Y) < dim R.

– p.3/15


The ringR satisfies vanishing if


This generalizes the vanishing conjecture by Serre.

– p.3/15




This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishingif this holds when bothX ∈ P

f(R) and Y ∈ Pf(R).

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This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishing if this holds when bothX ∈ P


Theorem (Dutta–Hochster–McLaughlin). Thevanishing conjecture does not hold, so not all ringssatisfy vanishing!

– p.3/15




This generalizes the vanishing conjecture by Serre. Thering R satisfies weak vanishing if this holds when bothX ∈ P


Theorem (Dutta–Hochster–McLaughlin). Thevanishing conjecture does not hold, so not all ringssatisfy vanishing!

So far, all rings satisfy weak vanishing.

– p.3/15


Assume thatR is complete of prime characteristicp andwith perfect residue field.

– p.4/15


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



f2(R) with


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



f2(R) with


dim(Supp X) + dim(Supp Y) ≤ dim R. The Duttamultiplicity of X andY is defined as


1

pe codim(Supp X)χ(LFe(X), Y),

whereLF is the left-derived Frobenius functor.

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f2(R) with




1


whereLF is the left-derived Frobenius functor, given by

X = · · · −→ Xi

∂Xi−→ Xi−1 −→ · · ·

– p.4/15



f2(R) with




1



X ≃ · · · −→ Rm(aij)−→ Rn −→ · · ·

– p.4/15



f2(R) with




1



LF(X) ≃ · · · −→ Rm(a

pij)

−→ Rn −→ · · · .

– p.4/15



f2(R) with




1



LF(X) ≃ · · · −→ Rm(a

pij)

−→ Rn −→ · · · .

The Dutta multiplicity satisfies vanishing. – p.4/15

GROTHENDIECK GROUPS

Let C be a full subcategory of the category of finitecomplexes.

– p.5/15

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

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



Sinceχ(−, Y) is zero on exact complexes and additive onshort exact sequences, it factors through a Grothendieckgroup.

– p.5/15

GROTHENDIECK GROUPS




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

GROTHENDIECK GROUPS





χ(X, Y) = χ([X], Y)– p.5/15

GROTHENDIECK GROUPS





χ(X, Y) = χ([X], Y) = χ([X′], Y)– p.5/15

GROTHENDIECK GROUPS





χ(X, Y) = χ([X], Y) = χ([X′], Y) = χ(X′, Y).– p.5/15

GROTHENDIECK GROUPS

We can also compute

χ∞(X, Y)

– p.6/15

GROTHENDIECK GROUPS

We can also compute


1

pe codim(Supp X)χ([LFe(X)], Y)

where[LFe(X)] ∈ K0(C).

– p.6/15

GROTHENDIECK GROUPS

We can also compute


1


= lime→∞

χ(1

pe codim(Supp X)[LFe(X)], Y)

wherep−e codim(Supp X)[LFe(X)] ∈ K0(C)⊗Z Q.

– p.6/15

GROTHENDIECK GROUPS

We can also compute


1


= lime→∞

χ(1


= χ( lime→∞

1


where???

– p.6/15

GROTHENDIECK SPACES

Let X be a specialization-closed subset ofSpec R.

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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

GROTHENDIECK SPACES



Define

Df2(X) = the full subcategory ofDf

2(R) of complexes

with support contained inX.

– p.7/15

GROTHENDIECK SPACES



Define


2(R) of complexes


Pf(X) = the full subcategory ofPf(R) of complexes


– p.7/15

GROTHENDIECK SPACES



Define


2(R) of complexes


Pf(X) = the full subcategory ofPf(R) of complexes


Let Xc denote the maximal subset ofSpec R such that

X∩Xc = {m} and dim X + dim Xc ≤ dim R.

– p.7/15

GROTHENDIECK SPACES

Let X be a specialization-closed subset ofSpec R.

– p.8/15

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

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

GROTHENDIECK SPACES

Now we can calculate . . .

– p.9/15

GROTHENDIECK SPACES

Now we can calculate

χ∞(X, Y)

– p.9/15

GROTHENDIECK SPACES



1

pe codim Xχ([LFe(X)], Y)

whereX = Supp X.– p.9/15

GROTHENDIECK SPACES



1


= lime→∞

χ(1

pe codim X[LFe(X)], Y)

whereX = Supp X.– p.9/15

GROTHENDIECK SPACES



1


= lime→∞

χ(1


= lime→∞

χ(1

pe codim XFeX([X]), Y)

whereX = Supp X andFeX([X]) = [LFe(X)].

– p.9/15

GROTHENDIECK SPACES



1


= lime→∞

χ(1


= lime→∞

χ(1


= lime→∞

χ(ΦeX([X]), Y)

whereX = Supp X andΦeX = p−e codim XFe

X.– p.9/15

GROTHENDIECK SPACES



1


= lime→∞

χ(1


= lime→∞

χ(1


= lime→∞

χ(ΦeX([X]), Y)

= χ( lime→∞

ΦeX([X]), Y)

whereX = Supp X andlime→∞ ΦeX([X]) ∈ GP

f(X).– p.9/15

GROTHENDIECK SPACES

Theorem 1. Let X be a specialization-closed subset of

Spec R and let α ∈ GPf(X).

– p.10/15

GROTHENDIECK SPACES


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

GROTHENDIECK SPACES


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

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

GROTHENDIECK SPACES


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

GROTHENDIECK SPACES


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

GROTHENDIECK SPACES

Translating the theorem to complexes, the Duttamultiplicity χ∞(X, Y) can be computed.

– p.12/15

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

NUMERICAL PROPERTIES

Assume thatR is complete of prime characteristicp andwith perfect residue field.

– p.13/15


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



f(X). If

χ(α, Y) = χ(α(0), Y) for all Y ∈ Pf(Xc), thenα satisfies

numerical vanishing.

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f(X). If


numerical vanishing. If this holds for all elements in allGrothendieck spaces, thenR satisfiesnumerical vanishing.

– p.13/15



f(X). If


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



f(X). If


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


The duality functorRHomR(−, R) onPf(X) induces an

automorphism(−)∗ on GPf(X).

– p.14/15




An elementα ∈ GPf(X) is self-dualif it satisfies

α = (−1)codimXα∗.

– p.14/15




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





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





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

RING PROPERTIES

vanishing

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RING PROPERTIES

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phd defense, may 24, 2007 intersection multiplicities and...

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