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arXiv:0705.1253v1 [math.AC] 9 May 2007

DUALITIES AND INTERSECTION MULTIPLICITIES

ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

Abstract. Let R be a commutative, noetherian, local ring. Topological Q–

vector spaces modelled on full subcategories of the derived category of R are

constructed in order to study intersection multiplicities.

1. Introduction

Let R be a commutative, noetherian, local ring and let X and Y be homolog-

ically bounded complexes over R with ﬁnitely genera ted homology and supports

intersecting at the ma ximal ideal. When the projective dimension of X or Y is

ﬁnite, their intersection multiplicity is deﬁned as

χ(X, Y ) = χ(X ⊗

L

R

Y ),

where χ(−) denotes the Euler characteristic deﬁned as the alternating sum of the

lengths of the homology modules. When X and Y are modules, this deﬁnition

agrees with the intersection multiplicity deﬁned by Serre [22].

The ring R is said to satisfy vanishing when

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

If the above holds under the restriction that both complexes have ﬁnite projective

dimension, R is said to satisfy weak vanishing.

Assume, in addition, that dim(Supp X) + dim(Supp Y ) 6 dim R and tha t R has

prime characteristic p. The Dut ta multiplicity of X and Y is deﬁned when X has

ﬁnite projective dimension as the limit

χ

∞

(X, Y ) = lim

e→∞

1

p

e codim(Supp X)

χ(LF

e

(X), Y ),

where LF

e

denotes the e-fold composition of the left-derived Frobenius functor; the

Frobenius functor F was systematically used in the classical work by Peskine and

Szpiro [18]. When X and Y are modules, χ

∞

(X, Y ) is the usual Dutta multiplicity;

see Dutta [6].

Let X be a specialization-closed subset of Spec R and let D

f

(X) denote the full

subc ategory of the derived category of R comprising the homologically bounded

complexes with ﬁnitely generated homology and support contained in X. The

symbols P

f

(X) and I

f

(X) denote the full subcategories of D

f

(X) comprising the

complexes that are isomorphic to a complex of projective or injective modules,

2000 Mathematics Subject Classiﬁcation. Primary 13A35, 13D22, 13H15, 14F17.

Key words and phrases. Frobenius endomorphism, Frobenius functors, Dutta multiplicity, i n-

tersection multiplicity.

Preliminary version, February 5, 2008.

The ﬁrst author is a Steno stipendiat supported by FNU, the Danish Research Council.

The second author is partially supported by FNU, the Danish Research Council .

1

2 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

respectively. The Grothendieck spaces GD

f

(X), GP

f

(X) and GI

f

(X) are topolog-

ical Q–vector spaces modelled on these categories. The ﬁrst two of these spaces

were introduced in [11] but were there modelled on ordinary non-derived cate-

gories of complexes. The construction of Gro thendieck spaces is similar to that o f

Grothendieck groups but targ e ted at the study of intersection multiplicities.

The main re sult of [11] is a diagonalization theorem in prime characteristic p for

the automorphism on GP

f

(X) induced by the Frobe nius functor. A conse quence of

this theorem is that every element α ∈ GP

f

(X) can be decomposed as

α = α

(0)

+ α

(1)

+ · · · + α

(u)

,

where the component of degree zero describes the Dutta multiplicity, whereas the

components of higher degree describe the extent to which vanishing fa ils to hold for

the the intersection multiplicity. This paper presents (see Theorem 6.2) a similar

diagonaliza tion theore m for a functor that is analogous to the Frobenius functor and

has been studied by Herzog [13]. A consequence is that every element β ∈ GI

f

(X)

can be decomposed as

β = β

(0)

+ β

(1)

+ · · · + β

(v)

,

where the component of degree zero desc ribes an analog of the Dutta multiplicity,

whereas the co mponents of higher degree describ e the extent to which vanishing

fails to hold for the Euler form, introduced by Mori and Smith [16]. Another

consequence (see Theorem 6.12) is that R satisﬁes weak vanishing if only the Eu-

ler characteristic of homologically bounded complexes with ﬁnite-length homology

changes by a factor p

dim R

when the analogous Frobenius functor is applied.

The duality functor (−)

∗

= RHom

R

(−, R) on P

f

(X) induces an automorphism

on GP

f

(X) which in prime characteristic p is given by (see Theorem 7.5)

(−1)

codim X

α

∗

= α

(0)

− α

(1)

+ · · · + (−1)

u

α

(u)

.

Even in arbitrary characteristic, R satisﬁes vanishing if and only if all elements

α ∈ GP

f

(X) are self-dual in the sense that α = (−1)

codim X

α

∗

; and R satisﬁes

weak vanishing if all elements α ∈ GP

f

(X) ar e numerically self-dual, meaning that

α−(−1)

codim X

α

∗

is in the kernel of the homo morphism GP

f

(X) → GD

f

(X) induced

by the inclusion of the underlying categories (see Theorem 7.4). Rings for which

all elements of the Grothendieck spa c es GP

f

(X) are numerically self-dual include

Gorenstein rings of dimension less than or equal to ﬁve (see Proposition 7.11) and

complete intersections (se e Proposition 7.7 together with [11, Example 3 3]).

Notation

Throughout, R denotes a c ommutative, noetherian, local ring with unique max-

imal ideal m and residue ﬁeld k = R/m. Unless otherwise stated, modules and

complexes are assumed to be R–modules and R–complexes, respectively.

2. Derived categories and functors

In this section we review notation and results from the theory of derived cate-

gories, and we introduce a new star duality and derived versions of the Frobenius

functor and its natural analog. For details on the derived catego ry and derived

functors, consult [9, 12, 23].

DUALITIES AND INTERSECTION MULTIPLICITIES 3

2.1. Derived categories. A complex X is a sequence (X

i

)

i∈Z

of modules equipped

with a diﬀerential (∂

X

i

)

i∈Z

lowering the homological degr ee by one. The homology

complex H(X) of X is the c omplex whose modules ar e

H(X)

i

= H

i

(X) = K e r ∂

X

i

/ Im ∂

i+1

and whose diﬀerentials are trivial.

A morphism of complexes σ : X → Y is a family (σ

i

)

i∈Z

of homomorphisms

commuting with the diﬀerentials in X and Y . The morphism of complexes σ is

a quasi-isomorphism if the induced map o n homology H

i

(σ): H

i

(X) → H

i

(Y ) is

an iso morphism in every degree. Two morphisms of complexes σ, ρ: X → Y a re

homotopic if there exists a family (s

i

)

i∈Z

of maps s

i

: X

i

→ Y

i+1

such that

σ

i

− ρ

i

= ∂

Y

i+1

s

i

+ s

i−1

∂

X

i

.

Homotopy yields an equivalence relation in the group Hom

R

(X, Y ) o f morphisms

of complexes, and the homotopy category K (R) is obtained from the category of

complexes C(R) by decla ring

Hom

K(R)

(X, Y ) = Hom

C(R)

(X, Y )/ homotopy.

The collection S of quasi-isomorphisms in the triangulated category K(R) form

a multiplicative system of morphisms. The derived category D(R) is obtained by

(categorica lly ) localizing K(R) with respect to S. Thus, quasi-isomorphisms become

isomorphisms in D (R); in the sequel, they ar e denoted ≃.

Let n be an integer. The symbol Σ

n

X denotes the co mplex X shifted (or trans-

lated or suspended) n degrees to the left; that is, against the direction of the

diﬀerential. The modules in Σ

n

X are given by (Σ

n

X)

i

= X

i−n

, and the diﬀeren-

tials are ∂

Σ

n

X

i

= (−1)

n

∂

X

i−n

. The symbol ∼ denotes isomorphisms up to a shift in

the derived category.

The full subcategory of D(R) consisting of c omplexes with bounded, ﬁnitely

generated homology is denoted D

f

(R). Complexes from D

f

(R) are called ﬁnite

complexes. The symbols P

f

(R) and I

f

(R) denote the full subcategories of D

f

(R)

consisting of complexes that a re isomorphic in the derived categor y to a bounded

complex of projective modules and isomorphic to a bounded complex of injective

modules, respectively. Note that P

f

(R) coincides with the full subca tegory F

f

(R)

of D

f

(R) consisting of complexes isomorphic to a complex of ﬂat modules.

2.2. Support. The spectrum of R, denoted Spec R, is the set of prime ideals of R.

A subset X of Spec R is specialization-closed if it has the property

p ∈ X and p ⊆ q =⇒ q ∈ X

for all pr ime ideals p and q. A subset that is closed in the Zariski topology is, in

particular, specialization-closed.

The support of a complex X is the set

Supp X =

n

p ∈ Spec R

H(X

p

) 6= 0

o

.

A ﬁnite complex is a complex with b ounded homology and ﬁnitely generated ho-

mology modules; the support of such a complex is a closed and hence specializatio n-

closed subset o f Spec R.

For a specialization-closed subset X of Spec R, the dimension of X, denoted

dim X, is the usual Krull dimension of X. When dim R is ﬁnite, the co-dimension

of X, denoted codim X, is the number dim R − dim X. For a ﬁnitely generated

4 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

module M , the dimension and co-dimension of M, denoted dim M and codim M,

are the dimension and co-dimension of the support of M .

For a specialization-closed subset X of Spec R, the symbols D

f

(X), P

f

(X), and

I

f

(X) denote the full sub c ategories of D

f

(R), P

f

(R), and I

f

(R), respectively, con-

sisting of complexes whose support is contained in X. In the case where X equals

{m}, we simply write D

f

(m), P

f

(m) and I

f

(m), respectively.

2.3. Derived functors. A complex P is said to be semi-projective if the functor

Hom

R

(P, −) sends surjective quasi-isomorphisms to surjective quasi-isomorphisms.

If a complex is bounded to the right and co ns ists of projective modules, it is semi-

projective. A semi-projective resolution of M is a quasi-isomorphism π : P → X

where P is semi-projective.

Dually, a complex I is said to be semi-injective if the functor Hom

R

(−, I) sends

injective quasi-isomorphisms to surjective quasi-isomorphisms. If a complex is

bounded to the le ft and consists of injective modules, it is semi-injective. A semi-

injective resolution of Y is a quasi-isomorphism ι: Y → I where I is semi-injective.

For e xistence of semi-projective and semi-injective resolutions se e [2].

Let X and Y be complexes. The left-derived tensor product X ⊗

L

R

Y in D(R) of

X and Y is deﬁned by

P ⊗

R

Y ≃ X ⊗

L

R

Y ≃ X ⊗

R

Q,

where P

≃

−→ X is a semi-projective resolution of X and Q

≃

−→ Y is a semi-projective

resolution of Y . The right-derived homomorphism complex RHom

R

(X, Y ) in D(R)

of X and Y is deﬁned by

Hom

R

(P, Y ) ≃ RHo m

R

(X, Y ) ≃ Hom

R

(X, I),

where P

≃

−→ X is a semi-pr ojective resolution of X and Y

≃

−→ I is a semi-injective

resolution of Y . When M and N are modules,

H

n

(M ⊗

L

R

N)

∼

=

Tor

R

n

(M, N ) a nd H

−n

(RHom

R

(M, N ))

∼

=

Ext

n

R

(M, N )

for all integers n.

2.4. Stability. Let X and Y be specialization-c losed subsets of Spec R and le t X

be a complex in D

f

(X) and Y be a complex in D

f

(Y). Then

X ⊗

L

R

Y ∈ D

f

(X ∩ Y) if X ∈ P

f

(X) or Y ∈ P

f

(Y),

X ⊗

L

R

Y ∈ P

f

(X ∩ Y) if X ∈ P

f

(X) and Y ∈ P

f

(Y),

X ⊗

L

R

Y ∈ I

f

(X ∩ Y) if X ∈ P

f

(X) and Y ∈ I

f

(Y),

X ⊗

L

R

Y ∈ I

f

(X ∩ Y) if X ∈ I

f

(X) and Y ∈ P

f

(Y),

RHom

R

(X, Y ) ∈ D

f

(X ∩ Y) if X ∈ P

f

(X) or Y ∈ I

f

(Y),

RHom

R

(X, Y ) ∈ P

f

(X ∩ Y) if X ∈ P

f

(X) and Y ∈ P

f

(Y),

RHom

R

(X, Y ) ∈ I

f

(X ∩ Y) if X ∈ P

f

(X) and Y ∈ I

f

(Y) and

RHom

R

(X, Y ) ∈ P

f

(X ∩ Y) if X ∈ I

f

(X) and Y ∈ I

f

(Y).

(2.4.1)

DUALITIES AND INTERSECTION MULTIPLICITIES 5

2.5. Functorial isomorphisms. Throughout, we will make use of the functorial

isomorphisms stated below. As we will not need them in the most general setting,

the reader should bear in mind that not all the boundedness c o nditions imposed

on the complexes are strictly necessary. For details the re ader is referred e.g., to [5,

A.4] and the reference s therein.

Let S be another commutative, noetherian, local r ing. Let K, L, M ∈ D(R), let

P ∈ D(S) and let N ∈ D(R, S), the derived category of R–S–bi-mo dules . There

are the next functorial isomorphisms in D(R, S).

M ⊗

L

R

N

≃

−→ N ⊗

L

R

M.(Comm)

(M ⊗

L

R

N) ⊗

L

S

P

≃

−→ M ⊗

L

R

(N ⊗

L

S

P ).(Assoc)

RHom

S

(M ⊗

L

R

N, P )

≃

−→ RHom

R

(M, RHom

S

(N, P )).(Adjoint)

RHom

R

(M, RHom

S

(P, N ))

≃

−→ RHom

S

(P, R Hom

R

(M, N )).(Swap)

Moreover, there a re the following evaluation morphisms.

σ

KLP

: RHom

R

(K, L) ⊗

L

S

P → RHom

R

(K, L ⊗

L

S

P ).(Tensor- e val)

ρ

P LM

: P ⊗

L

S

RHom

R

(L, M) → RHom

S

(RHom

R

(P, L), M).(Hom-eval)

In addition,

• the morphism σ

KLP

is invertible if K is ﬁnite, H(L) is bounded, and either

P ∈ P(S) or K ∈ P(R); and

• the morphism ρ

P LM

is invertible if P is ﬁnite, H(L) is bounded, and either

P ∈ P(R) or M ∈ I(R).

2.6. Dualizing complexes. A ﬁnite complex D is a dualizing complex for R if

D ∈ I

f

(R) and R

≃

−→ RHom

R

(D, D).

Dualizing complexes are essentially unique: if D and D

′

are dualizing complexes

for R, then D ∼ D

′

. To check whether a ﬁnite complex D is dualizing is equivalent

to checking whether

k ∼ RHom

R

(k, D).

A dualizing complex D is said to be normalized when k ≃ RHom

R

(k, D). If R is

a Cohen–Mac aulay ring of dimension d and D is a normalized dualizing complex,

then H(D) is concentrated in degree d, and the module H

d

(D) is the (so-called)

canonical module; see [3, Chapter 3]. O bserve that Supp D = Spec R.

If D is a no rmalized dualizing complex for R, then it is isomorphic to a c omplex

0 → D

dim R

→ D

dim R−1

→ · · · → D

1

→ D

0

→ 0

consisting of injective modules, where

D

i

=

M

dim R/p=i

E

R

(R/p)

and E

R

(R/p) is the injective hull (or envelope) of R/p for a prime ideal p; in

particular, it follows that D

0

= E

R

(k).

When R is a homomorphic image of a loca l Gorenstein ring Q, then the R–

complex Σ

n

RHom

Q

(R, Q), where n = dim Q − dim R, is a normalized dualizing

complex over R. In particular, it follows from Cohen’s structure theorem for com-

plete local rings that any complete ring admits a dualizing complex. Conversely, if

6 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

a local ring admits a dualizing complex, then it must be a homomor phic image of a

Gorenstein ring; this follows from Kawasaki’s proof of Sharp’s conjecture; see [14].

2.7. Dagger duality. Assume that R admits a normalized dualizing complex D

and consider the duality morphism of functors

id

D(R)

→ RHom

R

(RHom

R

(−, D), D).

It follows essentially from (Hom-eval) that the contravariant functor

(−)

†

= RHom

R

(−, D)

provides a duality on the category D

f

(R) which restricts to a duality between

P

f

(R) and I

f

(R). This duality is sometimes referred to as dagger duality. According

to (2.4.1), if X is a specialization-closed subset of Spec R, then dagger duality gives a

duality on D

f

(X) which re stricts to a duality between P

f

(X) and I

f

(X) as described

by the following commutative diagram.

D

f

(X)

(−)

†

//

D

f

(X)

(−)

†

oo

P

f

(X)

(−)

†

//

OO

I

f

(X).

(−)

†

oo

OO

Here the vertical arrows are full embeddings of categories. For more details on

dagger duality, see [12].

2.8. Foxby equivalence. Assume that R admits a normalized dualizing complex

D and consider the two contravariant adjoint functors

D ⊗

L

R

− and RHom

R

(D, −),

which come naturally equipped the unit and co-unit morphisms

η : id

D(R)

→ RHom

R

(D, D ⊗

L

R

−) and ε: D ⊗

L

R

RHom

R

(D, −) → id

D(R)

.

It follows essentially from an application of (Tensor-eval) and (Hom-eval) that the

categories P(R) and I(R) are na tur ally equivalent via the above two functors. This

equivalence is usually known as Foxby equivalence and was introduced in [1], to

which the r e ader is referred for further details.

According to (2.4.1), for a specialization-closed subset X of Spec R, Foxby equiv-

alence restricts to an equivalence between P

f

(X) and I

f

(X) as described by the

following diagram.

P

f

(X)

D⊗

L

R

−

//

I

f

(X).

RHom

R

(D,−)

oo

DUALITIES AND INTERSECTION MULTIPLICITIES 7

2.9. Star duality. C onsider the duality morphism of functors

id

D(R)

→ RHom

R

(RHom

R

(−, R), R).

From an application of (Hom-eval) it is readily seen that the functor

(−)

∗

= RHom

R

(−, R)

provides a duality on the category P

f

(R). According to (2.4.1), for a specialization-

closed subset X of Spec R, star duality restricts to a duality on P

f

(X) as described

by following diagram.

P

f

(X)

(−)

∗

//

P

f

(X).

(−)

∗

oo

When R admits a dualizing complex D, the star functor can also be described in

terms of the dagger and Foxby functors. Indeed, it is straig htforward to show that

the following three contravariant endofunctors on P

f

(R) are isomorphic.

(−)

∗

, RHom

R

(D, −

†

), and (D ⊗

L

R

−)

†

.

It is equally s traightforward to show that the following four contrava riant endo-

functors on I

f

(R) are isomorphic.

(−)

† ∗ †

, RHom

R

(D, −)

†

, D ⊗

L

R

(RHom

R

(D, −)

∗

) and D ⊗

L

R

(−)

†

.

They provide a duality on I

f

(R). In the sequel, the four isomorphic functors are

denoted (−)

⋆

. According to (2.4.1), for a specialization-closed subset X of Spec R,

this new kind of star duality restricts to a duality on I

f

(X) as described by the

following diagram.

I

f

(X)

(−)

⋆

//

I

f

(X).

(−)

⋆

oo

The dagger duality, Foxby equivalence and star duality functors ﬁt together in

the following diagram.

D

f

(X)

(−)

†

//

D

f

(X)

(−)

†

oo

(−)

∗

##

P

f

(X)

OO

(−)

†

//

D⊗

L

R

−

))

I

f

(X)

OO

(−)

†

oo

RHom

R

(D,−)

ii

(−)

⋆

{{

(2.9.1)

In the lower part o f the diagr am, the three types of functors, dagger, Foxby and

star, always commute pairwise, and the composition of two of the three types yields

a functor of the third type. For example, star duality and dagger duality always

commute and compose to give Foxby equivalence, since we have

(−)

∗†

≃ (−)

†⋆

≃ D ⊗

L

R

− and (−)

⋆†

≃ (−)

†∗

≃ RHom

R

(D, −).

8 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

2.10. Frobenius endofunctors. Assume that R is complete of prime character-

istic p and with perfect residue ﬁeld k. The endomorphism

f : R → R deﬁned by f(r) = r

p

for r ∈ R is c alled the Frobenius endomorphism on R. The n-fold c ompos itio n of

f, denoted f

n

, oper ates on a generic element r ∈ R by f

n

(r) = r

p

n

. We let

f

n

R

denote the R–algebra which, as a ring, is identical to R but, as a module, is viewed

through f

n

. Thus , the R–module structure on

f

n

R is given by

r · x = r

p

n

x for r ∈ R and x ∈

f

n

R.

Under the present assumptions on R, the R–module

f

n

R is ﬁnitely generated (see ,

for example, Roberts [21, Section 7.3]).

We deﬁne two functors from the category of R–modules to the category of

f

n

R–

modules by

F

n

(−) = − ⊗

R

f

n

R and G

n

(−) = Hom

R

(

f

n

R, −),

where the resulting modules are ﬁnitely generated modules with R–structure ob-

tained from the ring

f

n

R = R. The functor F

n

is called the Frobenius functor and

has been studied by Peskine and Szpiro [18]. The functor G

n

has been studied by

Herzog [13] and is analogous to F

n

in a sense that will be des c ribed below. We ca ll

this the analogous Frobenius functor. The R–structure on F

n

(M) is given by

r · (m ⊗ x) = m ⊗ rx

for r ∈ R, m ∈ M and x ∈

f

n

R, and the R–structure on G

n

(N) is given by

(r · ϕ)(x) = ϕ(rx)

for r ∈ R, ϕ ∈ Hom

R

(

f

n

R, N ) and x ∈

f

n

R. Note tha t here we also have

(rm) ⊗ x = m ⊗ (r · x) = m ⊗ r

p

x and rϕ(x) = ϕ(r · x) = ϕ(r

p

x).

Peskine and Szpiro [18, Th´eor`eme (1.7)] have proven that, if M has ﬁnite projective

dimension, then so does F (M), and Herzog [13, Satz 5.2] has prove n that, if N has

ﬁnite injective dimension, then so does G(N).

It fo llows by deﬁnition that the functor F

n

is right-exact while the functor G

n

is left-exact. We deno te by LF

n

(−) the left-derived of F

n

(−) and by RG

n

(−) the

right-derived of G

n

(−). When X and Y are R–complexes with semi-projective and

semi-injective resolutions

P

≃

−→ X and Y

≃

−→ I,

respectively, these derived functors are obtained as

LF

n

(X) = P ⊗

R

f

n

R and RG

n

(Y ) = Hom

R

(

f

n

R, I),

where the re sulting complexes are viewed through their

f

n

R–structure, which makes

them R–complexes since

f

n

R as a ring is just R. Observe that we may identify these

functors with

LF

n

(X) = X ⊗

L

R

f

n

R and RG

n

(Y ) = R Hom

R

(

f

n

R, Y ).

DUALITIES AND INTERSECTION MULTIPLICITIES 9

2.11. Lemma. Let R be a complete ring of prime characteristic and with perfect

residue ﬁeld, and let X be a specialization-closed subset of Spe c R. Then the Frobe-

nius functors commute with dagger and star duality in the sense that

LF

n

(−)

†

≃ RG

n

(−

†

), RG

n

(−)

†

≃ LF

n

(−

†

),

LF

n

(−)

∗

≃ LF

n

(−

∗

) and RG

n

(−)

⋆

≃ RG

n

(−

⋆

).

Here the ﬁrst row contains isomorphisms of functors between P

f

(X) and I

f

(X), while

the second row contains isomorphisms of endofunctors on P

f

(X) and I

f

(X), respec-

tively. Finally, the Frobenius functors commute with Foxby equivalence in the sense

that

D ⊗

L

R

LF

n

(−) ≃ RG

n

(D ⊗

L

R

−) and

RHom

R

(D, RG

n

(−)) ≃ LF

n

(RHom

R

(D, −))

as functors from P

f

(X) to I

f

(X) and from I

f

(X) to P

f

(X), respectively.

Proof. Let ϕ: R → S be a local homomorphism making S into a ﬁnitely generated

R–module, and let D

R

denote a normalized dualizing complex for R. Then D

S

=

RHom

R

(S, D

R

) is a normalized dualizing complex for S. Pick an R–complex X

and consider the next string of natural isomorphisms.

RHom

S

(X ⊗

L

R

S, D

S

) = RHom

S

(X ⊗

L

R

S, RHom

R

(S, D

R

))

≃

←− RHom

R

(X ⊗

L

R

S, D

R

)

≃

−→ RHom

R

(S, RHom

R

(X, D

R

)).

Here, the two isomor phism follow from (Adjoint). The computation shows that

(− ⊗

L

R

S)

†

S

≃ RHom

R

(S, −

†

R

)

in D(S). A similar computation using the natural isomorphisms (Adjoint) and

(Hom-eval) shows that

(−)

†

R

⊗

L

R

S ≃ RHom

R

(S, −)

†

S

.

Under the present assumptions, the n-fold composition of the Frobenius endomor-

phism f

n

: R → R is module-ﬁnite map. Therefore, the above isomorphisms of

functors yield

LF

n

(−)

†

≃ RG

n

(−

†

) and LF

n

(−

†

) ≃ RG

n

(−)

†

.

Similar considerations establish the remaining isomorphisms of functors.

2.12. Corollary. Let R be a complete ring of prime characteristic and with per-

fect residue ﬁeld, and let X be a specialization-closed subset of Sp e c R. Then the

Frobenius functor RG

n

is an endofunctor on I

f

(X).

Proof. From the above lemma, we learn that

RG

n

(−) ≃ (−)

†

◦ LF

n

◦ (−)

†

and since LF

n

is an endofunctor on P

f

(X) the conclusion is immediate.

10 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

2.13. Lemma. Let R be a complete ring of prime characteristic and with perfect

residue ﬁeld. For complexes X, X

′

∈ P

f

(R) and Y, Y

′

∈ I

f

(R) there are isomor-

phisms

LF

n

(X ⊗

L

R

X

′

) ≃ LF

n

(X) ⊗

L

R

LF

n

(X

′

),

RG

n

(X ⊗

L

R

Y ) ≃ LF

n

(X) ⊗

L

R

RG

n

(Y ),

RG

n

(RHom

R

(X, Y )) ≃ RHom

R

(LF

n

(X), RG

n

(Y ))

LF

n

(RHom

R

(X, X

′

)) ≃ RHom

R

(LF

n

(X), LF

n

(X

′

)) and

LF

n

(RHom

R

(Y, Y

′

)) ≃ RHom

R

(RG

n

(Y ), RG

n

(Y

′

)).

Proof. We prove the ﬁrst and the third isomorphism. The rest are obtained in a

similar manner using Lemma 2.11 and the functorial isomorphisms.

Let F

≃

−→ X and F

′

≃

−→ X

′

be ﬁnite free reso lutions . Then it follows

LF

n

(X ⊗

L

R

X

′

) ≃ F

n

(F ⊗

R

F

′

)

≃ F

n

(F ) ⊗

R

F

n

(F

′

)

≃ LF

n

(X) ⊗

L

R

LF

n

(X

′

).

Here the ﬁrst isomorphism follows as F ⊗

R

F

′

is isomo rphic to X ⊗

L

R

X

′

; the second

isomorphism follows from e.g., [11, Proposition 12(vi)].

From Corollary 2.12 we learn that

RG

n

(Y ) ≃ (LF

n

(Y

†

))

†

,

and therefore we may compute as follows.

RHom

R

(LF

n

(X), RG

n

(Y )) ≃ RHom

R

(LF

n

(X), (LF

n

(Y

†

))

†

)

≃ RHom

R

(LF

n

(X) ⊗

L

R

LF

n

(Y

†

), D)

≃ RHom

R

(LF

n

(X ⊗

L

R

Y

†

), D)

≃ LF

n

(X ⊗

L

R

Y

†

)

†

≃ (LF

n

(RHom

R

(X, Y )

†

)

†

≃ RG

n

(RHom

R

(X, Y )).

Here the second isomorphism follows by (Adjoint); the third from the ﬁrst statement

in the Lemma; the fourth from deﬁnition; the ﬁfth isomorphism follows from (Hom-

eval); and the last isomorphism follows from Corollary 2.12.

2.14. Remark. Any complex in P

f

(R) is isomorphic to a bounded complex of

ﬁnitely g enerated, free modules, and it is well-known that the Frobenius functor

acts on such a complex by simply raising the entries in the matrices representing

the diﬀerentials to the p

n

’th power. To b e precise, if X is a complex in the form

X = · · · −→ R

m

(a

ij

)

−→ R

n

−→ · · · −→ 0,

then LF

n

(X) = F

n

(X) is a complex in the form

LF

n

(X) = · · · −→ R

m

(a

p

n

ij

)

−→ R

n

−→ · · · −→ 0.

DUALITIES AND INTERSECTION MULTIPLICITIES 11

If R is Cohen–Macaulay with canonical module ω, then it follows from dagger

duality that any complex in I

f

(R) is isomorphic to a complex Y in the form

Y = 0 −→ · · · −→ ω

n

(a

ji

)

−→ ω

m

−→ · · · ,

and RG

n

acts on Y by raising the entries in the matrices representing the diﬀer-

entials to the p

n

’th power, s o that RG

n

(Y ) = G

n

(Y ) is a complex in the form

RG

n

(Y ) = 0 −→ · · · −→ ω

n

(a

p

n

ji

)

−→ ω

m

−→ · · · .

3. Intersection multiplicities

3.1. Serre’s intersection multiplicity. If Z is a co mplex in D

f

(m), then its

ﬁnitely many homology modules all have ﬁnite length, and the Euler characteristic

of Z is deﬁned by

χ(Z) =

X

i

(−1)

i

length H

i

(Z).

Let X and Y be ﬁnite complexes with Supp X ∩ Supp Y = {m}. The intersection

multiplicity of X and Y is deﬁned by

χ(X, Y ) = χ (X ⊗

L

R

Y ) when either X ∈ P

f

(R) or Y ∈ P

f

(R).

In the case where X and Y a re ﬁnitely generated modules, χ(X, Y ) coincides with

Serre’s intersection multiplicity; see [22].

Serre’s vanishing conjecture can be generalized to the statement that

(3.1.1) χ(X, Y ) = 0 if dim(Supp X) + dim(Supp Y ) < dim R

when either X ∈ P

f

(R) or Y ∈ P

f

(R). We will say tha t R satisﬁes vanishing when

the above holds; note that this, in general, is a stronger condition than Serre’s

vanishing conjecture for modules. It is k nown that R satisﬁes vanishing in certain

cases, for example when R is regular. However, it does not hold in general, as

demonstrated by Dutta, Hochster and McLaughlin [8].

If we require that both X ∈ P

f

(R) and Y ∈ P

f

(R), condition (3.1.1) becomes

weaker. When this weaker condition is satisﬁes, we say that R satisﬁes weak van-

ishing. It is known that R satisﬁes weak vanishing in many cases, for example if R

is a complete intersection; see Roberts [19] or Gillet and Soul´e [10]. There are, so

far, no counterexamples preventing it from holding in full generality.

3.2. Euler form. Let X and Y be ﬁnite complexes with Supp X ∩ Supp Y = {m}.

The Euler form of X and Y is deﬁned by

ξ(X, Y ) = χ(RHom

R

(X, Y )) when either X ∈ P

f

(R) or Y ∈ I

f

(R).

In the case where X and Y a re ﬁnitely generated modules, χ(X, Y ) coincides with

the Euler form introduced by Mori and Smith [1 6].

If R admits a dualizing c omplex, then from Mori [17, Lemma 4.3(1) and (2)] and

the deﬁnition of (−)

⋆

, we obtain

ξ(X, Y ) = χ(X , Y

†

) whenever X ∈ P

f

(R) or Y ∈ I

f

(R),

χ(X

∗

, Y ) = χ(X , Y

†

) whenever X ∈ P

f

(R), and

ξ(X, Y

⋆

) = ξ(X

†

, Y ) whenever Y ∈ I

f

(R).

(3.2.1)

12 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

Since the dagger functor does not change supports of complexes, the ﬁrs t formula

in (3.2.1) shows that R satisﬁes vanishing exactly when

(3.2.2) ξ(X, Y ) = 0 if dim(Supp X) + dim(Supp Y ) < dim R

when either X ∈ P

f

(R) or Y ∈ I

f

(R), and that R s atisﬁes weak vanishing exactly

when (3.2.2) holds when we requir e both X ∈ P

f

(R) and Y ∈ I

f

(R).

3.3. Dutta multiplicity. Assume that R is complete of prime characteristic p and

with perfect residue ﬁeld. L e t X and Y be ﬁnite complexes with

Supp X ∩ Supp Y = {m} and dim(Supp X) + dim(Supp Y ) 6 dim R.

The Dutta multiplicity of X and Y is deﬁned by

χ

∞

(X, Y ) = lim

e→∞

1

p

e co dim(Supp X)

χ(LF

e

(X), Y ) when X ∈ P

f

(R).

When X and Y are ﬁnitely generated modules, χ

∞

(X, Y ) coincides with the Dutta

multiplicity deﬁned in [6].

The Euler form prompts to two natural a nalogs of the Dutta multiplicity. We

deﬁne

ξ

∞

(X, Y ) = lim

e→∞

1

p

e co dim(Supp Y )

ξ(X, RG

e

(Y )) when Y ∈ I

f

(R), and

ξ

∞

(X, Y ) = lim

e→∞

1

p

e co dim(Supp X)

ξ(LF

e

(X), Y ) when X ∈ P

f

(R).

We immediately note, using (3.2 .1) together with Lemma 2.11, that

ξ

∞

(X, Y ) = χ

∞

(Y

†

, X) whenever Y ∈ I

f

(Y), and

ξ

∞

(X, Y ) = χ

∞

(X

∗

, Y ) whenever X ∈ P

f

(X).

4. Grothendieck spaces

In this sectio n we present the deﬁnition and basic prope rties of Grothendieck

spaces. We will introduce three types of Gr othendieck s paces, two of which were

introduced in [11]. The cons tructions in loc. cit. are diﬀerent from the ones her e

but yield the same spac e s.

4.1. Complement. For any specialization-closed subset X of Spec R, a new subset

is deﬁned by

X

c

=

n

p ∈ Spec R

X ∩ V (p) = {m} and dim V (p) 6 codim X

o

.

This set is engineered to be the largest subset of Spec R such that

X ∩ X

c

= {m} and dim X + dim X

c

6 dim R.

In fact, when X is closed,

dim X + dim X

c

= dim R.

Note that X

c

is specialization-closed and that X ⊆ X

cc

.

DUALITIES AND INTERSECTION MULTIPLICITIES 13

4.2. Grothendieck space. Let X be a specialization- closed subset of Spec R. The

Grothendieck space of the category P

f

(X) is the Q–vector space GP

f

(X) presented

by elements [X]

P

f

(X)

, one for each isomorphism class of a complex X ∈ P

f

(X), and

relations

[X]

P

f

(X)

= [

e

X]

P

f

(X)

whenever χ(X, −) = χ(

e

X, −)

as metafunctions (“functions” from a category to a set) D

f

(X

c

) → Q.

Similarly, the Grothendieck space of the category I

f

(X) is the Q–vector space

GI

f

(X) presented by elements [Y ]

I

f

(X)

, o ne for each isomorphism class of a complex

Y ∈ I

f

(X), and relations

[Y ]

I

f

(X)

= [

e

Y ]

I

f

(X)

whenever ξ(−, Y ) = ξ(−,

e

Y )

as metafunctions D

f

(X

c

) → Q.

Finally, the Grothendieck space of the category D

f

(X) is the Q–vector space

GD

f

(X) presented by elements [Z]

D

f

(X)

, one for each isomorphism class of a com-

plex Z ∈ D

f

(X), and relations

[Z]

D

f

(X)

= [

e

Z]

D

f

(X)

whenever χ(−, Z) = χ(−,

e

Z)

as meta functions P

f

(X

c

) → Q. Because of (3.2.1), these relations are exactly the

same as the relations

[Z]

D

f

(X)

= [

e

Z]

D

f

(X)

whenever ξ(Z, −) = ξ(

e

Z, −)

as metafunctions I

f

(X

c

) → Q.

By deﬁnition of the Grothendieck space GP

f

(X) there is, for each complex Z in

D

f

(X

c

), a well-deﬁned Q–linear map

χ(−, Z) : GP

f

(X) → Q given by [X]

P

f

(X)

7→ χ(X, Z).

We equip GP

f

(X) with the initial topology induced by the family of maps in the

above form. This topology is the coarsest topology on GP

f

(X) making the above

map continuous for all Z in D

f

(X

c

). Likewise, for each complex Z in D

f

(X

c

), there

is a well-deﬁned Q–linear map

ξ(Z, −): GI

f

(X) → Q given by [Y ]

I

f

(X)

7→ ξ(Z, Y ),

and we equip GI

f

(X) with the initial topology induced by the family of maps in the

above form. Finally, for each c omplex X in P

f

(X

c

), there is a well-deﬁned Q–linear

map

χ(X, −): GD

f

(X) → Q given by [Z]

D

f

(X)

7→ χ(X, Z),

and we equip GD

f

(X) with the initial topology induced by the family of ma ps in

the above form. By (3.2.1), this topology is the same a s the initial topology induced

by the family of (well-deﬁned, Q–linear ) maps in the form

ξ(−, Y ): GD

f

(X) → Q given by [Z]

D

f

(X)

7→ ξ(Z, Y ),

for complexes Y in I

f

(X

c

).

It is straightforward to see that addition and scalar multiplication are continuo us

operations on Grothendieck space s, making GP

f

(X), GD

f

(X) and GI

f

(X) topolog-

ical Q–vector spaces. We shall always consider Grothendieck spaces as topological

Q–vector spac e s, so that, for example, a “homomorphism” between Grothendieck

14 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

spaces means a homomorphism of topological Q–vector spaces: that is, a continu-

ous, Q–linear map.

The following proposition is an improved version of [11, Proposition 2(iv) and (v)].

4.3. Proposition. Let X be a specialization-closed subset of Spec R.

(i) Any element in GP

f

(X) can be written in the form r[X]

P

f

(X)

for some

r ∈ Q and some X ∈ P

f

(X), any element in GI

f

(X) can be written in the

form s[Y ]

I

f

(X)

for some s ∈ Q and some Y ∈ I

f

(X), and any element in

GD

f

(X) can be written in the form t[Z]

D

f

(X)

for some t ∈ Q and some

Z ∈ D

f

(X). Moreover, X, Y and Z may be chosen so that

codim(Supp X) = codim(Supp Y ) = codim(Supp Z) = codim X.

(ii) For any complex Z ∈ D

f

(X), we have the identity

[Z]

D

f

(X)

= [H(Z)]

D

f

(X)

.

In particular, the Q–vector space GD

f

(X) is generated by elements in the

form [R/p]

D

f

(X)

for prime ideals p in X.

Proof. (i ) By cons truction, any element α in GP

f

(X) is a Q–linear combination

α = r

1

[X

1

]

P

f

(X)

+ · · · + r

n

[X

n

]

P

f

(X)

where r

i

∈ Q and X

i

∈ P

f

(X). Since a shift of a complex changes the sign of the

corresponding element in the Grothendieck space, we can assume that r

i

> 0 for

all i. Choo sing a greatest common denominator for the r

i

’s, we can ﬁnd r ∈ Q such

that

α = r(m

1

[X

1

]

P

f

(X)

+ · · · + m

n

[X

n

]

P

f

(X)

) = r[X]

P

f

(X)

,

where the m

i

’s are natural numbers and X is the direct s um over i of m

i

copies of

X

i

.

In order to prove the last statement of (i), choose a prime ideal p = (a

1

, . . . , a

t

)

in X which is ﬁrst in a chain p = p

0

( p

1

( · · · ( p

t

= m of prime ideals in X of

maximal length t = codim X. Note that X ⊇ V (p) and that the Koszul complex

K = K(a

1

, . . . , a

t

) has support exac tly equal to V (p). It follows that

α = α + 0 = r[X]

P

f

(X)

+ r[K]

P

f

(X)

− r[K]

P

f

(X)

= r[X ⊕ K ⊕ ΣK]

P

f

(X)

,

where codim(Supp(X ⊕ K ⊕ ΣK)) = codim X. The same argument applies to

elements of GI

f

(X) and GD

f

(X).

(ii) Any complex in D

f

(X) is isomo rphic to a bounded c omplex. After an

appropriate shift, we may assume that Z is a complex in D

f

(X) in the form

0 → Z

n

→ · · · → Z

1

→ Z

0

→ 0

for some natural number n. Since H

n

(Z) is the kernel of the map Z

n

→ Z

n−1

, we

can construct a short exact sequence of complexes

0 → Σ

n

H

n

(Z) → Z → Z

′

→ 0,

where Z

′

is a complex in D

f

(X) concentrated in the same degrees as Z. The

complex Z

′

is exact in degree n , and H

i

(Z

′

) = H

i

(Z) for i = n − 1, . . . , 0. In the

Grothendieck space GD

f

(X), we then have

[Z]

D

f

(X)

= [Σ

n

H

n

(Z)]

D

f

(X)

+ [Z

′

]

D

f

(X)

.

DUALITIES AND INTERSECTION MULTIPLICITIES 15

Again, Z

′

is isomorphic to a c omplex concentrated in degree n − 1, · · · , 0, so we

can repeat the process a ﬁnite number o f times and achieve that

[Z]

D

f

(X)

= [Σ

n

H

n

(Z)]

D

f

(X)

+ · · · + [Σ H

1

(Z)]

D

f

(X)

+ [H

0

(Z)]

D

f

(X)

= [Σ

n

H

n

(Z) ⊕ · · · ⊕ Σ H

1

(Z) ⊕ H

0

(Z)]

D

f

(X)

= [H(Z)]

D

f

(X)

.

The above analysis shows tha t any element of GD

f

(X) can be written in the form

r[Z]

D

f

(X)

= r

X

i

(−1)

i

[H

i

(Z)]

D

f

(X)

,

which means that GD

f

(X) is generated by modules. Taking a ﬁltration of a module

establishes that GD

f

(X) must be generated by elements of the form [R/p]

D

f

(X)

for

prime ideals p in X.

4.4. Induced Euler characteristic. The Euler characteristic χ : D

f

(m) → Q

induces an isomorphism

1

(4.4.1) GD

f

(m)

∼

=

−→ Q given by [Z]

D

f

(m)

7→ χ(Z).

See [11] for more deta ils . We also denote this isomorphism by χ. The iso morphism

means that we can identify the intersection multiplicity χ(X, Y ) and the Euler form

ξ(X, Y ) of c omplexes X and Y with e lements in GD

f

(m) of the form

[X ⊗

L

R

Y ]

D

f

(m)

and [RHom

R

(X, Y )]

D

f

(m)

,

respectively.

4.5. Induced inclusion. Let X be a spec ialization-closed subset of Spec R. It is

straightforward to verify that the full embeddings of P

f

(X) and I

f

(X) into D

f

(X)

induce homomo rphisms

2

GP

f

(X) → GD

f

(X) given by [X]

P

f

(X)

7→ [X]

D

f

(X)

, and

GI

f

(X) → GD

f

(X) given by [Y ]

I

f

(X)

7→ [Y ]

D

f

(X)

.

If X and X

′

are specialization-closed subsets of Spec R s uch that that X ⊆ X

′

, then

it is straightforward to verify that the full embeddings of P

f

(X) into P

f

(X

′

), I

f

(X)

into I

f

(X

′

) and D

f

(X) into D

f

(X

′

) induce homomorphisms

GP

f

(X) → GP

f

(X

′

) given by [X]

P

f

(X)

7→ [X]

P

f

(X

′

)

,

GI

f

(X) → GI

f

(X

′

) given by [Y ]

I

f

(X)

7→ [Y ]

I

f

(X

′

)

, and

GD

f

(X) → GD

f

(X

′

) given by [Z]

D

f

(X)

7→ [Z]

D

f

(X

′

)

.

The maps obtained in this way ar e called inclusion homomorphisms, and we shall

often denote them by an overline: if σ is an element in a Grothendieck space , then

σ denotes the image of σ after an application of an inclusion homomo rphisms.

1

That is, a Q–linear homeomorphism.

2

That is, continuous, Q–linear maps.

16 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

4.6. Induced tensor product and Hom. Proposition 4.7 b e low shows that the

left-derived tensor product functor and the right-derived Ho m-functor induce bi-

homomorphisms

3

on Grothendieck s paces. To clarify the contents of the proposi-

tion, let X and Y be specialization-closed subsets of Spec R such that X ∩ Y = {m}

and dim X + dim Y 6 dim R. Proposition 4.7 states, for example, that the right-

derived Hom-functor induces a bi-homomorphism

Hom: GP

f

(X) × GI

f

(Y) → GI

f

(m).

Given elements σ ∈ GP

f

(X) and τ ∈ GI

f

(Y), we can, by P rop osition 4 .3, write

σ = r[X]

P

f

(X)

and τ = s[Y ]

I

f

(Y)

,

where r and s are rational numbers, X is a complex in P

f

(X) and Y is a complex

in I

f

(Y). The bi-homomorphism above is then given by

(4.6.1) (σ, τ) 7→ Hom(σ, τ) = r s [RHom

R

(X, Y )]

D

f

(m)

.

We shall use the symbol “⊗” to denote any bi-homomorphism on Grothendieck

spaces induced by the left-derived tensor product and the symbol “Hom” to denote

any bi-homomorphism induced by right-derived Hom-functor. Together with the

isomorphism in (4.4.1) it follows that the intersection multiplicity χ(X, Y ) and

Euler form ξ(X, Y ) can be identiﬁed with elements in GD

f

(m) of the form

[X]

P

f

(X)

⊗ [Y ]

D

f

(Y)

, [X]

D

f

(X)

⊗ [Y ]

P

f

(Y)

,

Hom([X]

D

f

(X)

, [Y ]

I

f

(Y)

) and Hom([X]

P

f

(X)

, [Y ]

D

f

(Y)

).

4.7. Proposition. Let X and Y be specialization-closed subsets of Spec R such that

X ∩ Y = {m} and dim X + dim Y 6 dim R. The left-derived tensor product induces

bi-homomorphisms as in the ﬁrst column below, and the right-derived Hom-functor

induces bi-homomorphisms as in the second column below.

GP

f

(X) × GD

f

(Y) → GD

f

(m), GP

f

(X) × GD

f

(Y) → GD

f

(m),

GD

f

(X) × GP

f

(Y) → GD

f

(m), GD

f

(X) × GI

f

(Y) → GD

f

(m),

GP

f

(X) × GP

f

(Y) → GP

f

(m), GP

f

(X) × GI

f

(Y) → GI

f

(m),

GP

f

(X) × GI

f

(Y) → GI

f

(m), GP

f

(X) × GP

f

(Y) → GP

f

(m),

GI

f

(X) × GP

f

(Y) → GI

f

(m) and GI

f

(X) × GI

f

(Y) → GP

f

(m).

Proof. We verify that the map

Hom: GP

f

(X) × GI

f

(Y) → GI

f

(m)

given as in (4.6.1) is a well-deﬁned bi-homomorphism, leaving the same veriﬁcations

for the remaining maps as an e asy exercise for the reader.

Therefore, assume that X and

e

X a re complexes from P

f

(X) and that Y and

e

Y

are complexes from I

f

(Y) such that

σ = [X]

P

f

(X)

= [

e

X]

P

f

(X)

and τ = [Y ]

I

f

(Y)

= [

e

Y ]

I

f

(Y)

.

In order to show that the map is a well-deﬁned Q–bi-linear map, we are required

to demonstrate that

[RHom

R

(X, Y )]

I

f

(m)

= [RHom

R

(

e

X,

e

Y )]

I

f

(m)

.

3

That is, maps that are continuous and Q–linear in each variable.

DUALITIES AND INTERSECTION MULTIPLICITIES 17

To this end, let Z be an arbitrary complex in D

f

({m}

c

) = D

f

(R). We want to

show that

ξ(Z, RHom

R

(X, Y )) = ξ(Z, RHom

R

(

e

X,

e

Y )).

Without loss of generality, we may assume that R is complete; in particular, we

may assume that R admits a normalized dualizing complex. Observe that

Z ⊗

R

X ∈ D

f

(X) ⊆ D

f

(Y

c

) and Z ⊗

L

R

Y

†

∈ D

f

(Y) ⊆ D

f

(X

c

).

Applying (3.2.1), (Hom-eval) and (Assoc), we learn that

ξ(Z, RHom

R

(X, Y )) = χ(Z, RHom

R

(X, Y )

†

)

= χ(Z, X ⊗

L

R

Y

†

)

= χ(X, Z ⊗

L

R

Y

†

).

(4.7.1)

A similar computation shows that ξ(Z, RHom

R

(

e

X, Y )) = χ(

e

X, Z ⊗

L

R

Y

†

), and since

[X]

P

f

(X)

= [

e

X]

P

f

(X)

, we conclude that

ξ(Z, RHom

R

(X, Y )) = ξ(Z, RHom

R

(

e

X, Y )).

An application of (Adjoint) yields that

ξ(Z, RHom

R

(

e

X, Y )) = ξ(Z ⊗

L

R

e

X, Y ),

and similarly ξ(Z, RHom

R

(

e

X,

e

Y )) = ξ(Z ⊗

L

R

e

X,

e

Y ). Since [Y ]

I

f

(Y)

= [

e

Y ]

I

f

(Y)

, we

conclude that

ξ(Z, RHom

R

(

e

X, Y )) = ξ(Z, R Hom

R

(

e

X,

e

Y )).

Thus, we have that

ξ(Z, RHom

R

(X, Y )) = ξ(Z, RHom

R

(

e

X,

e

Y )),

which establishes well-deﬁnedness.

By deﬁnition, the induced Hom-map is Q–linear. To establish that it is con-

tinuous in, say, the ﬁrst variable it suﬃces for ﬁxed τ ∈ GI

f

(Y) to show that, to

every ε > 0 and every complex Z ∈ D

f

({m}

c

) = D

f

(R), there exists a δ > 0 and

a complex Z

′

∈ D

f

(X

c

) such that

|χ(σ, Z

′

)| < δ =⇒ |ξ(Z, Hom(σ, τ))| < ε.

We can wr ite τ = r[Y ]

I

f

(Y)

for an Y ∈ I

f

(Y) and a r ational number r > 0. According

to (4.7.1), the implication above is then achieved with Z

′

= Z ⊗

L

R

Y

†

and δ = ε/r.

Continuity in the second variable is shown by similar arguments.

In P roposition 4.8 below, we will show that the dagger, Foxby and s tar functors

from diagra m (2.9.1) induce isomorphisms of Grothendieck spaces. We shall denote

the isomorphisms induced by the star and dagger duality functors by the same

symbol as the original functor, whereas the isomorphisms induced by the Foxby

functors will be denoted according to Proposition 4.7 by D ⊗ − and Hom(D, −).

In this way, for example,

[X]

†

P

f

(X)

= [X

†

]

I

f

(X)

, [X]

∗

P

f

(X)

= [X

∗

]

P

f

(X)

and D ⊗ [X]

P

f

(X)

= [D ⊗

L

R

X]

I

f

(X)

.

18 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

4.8. Propositi on. Let X be a specialization-closed subset of Spec R, and assume

that R admits a dualizing complex. The functors from diagram (2.9.1) induce iso-

morphisms of Grothendieck spaces as described by the horizontal and circular arrows

in the following commutative diagram.

GD

f

(X)

(−)

†

//

GD

f

(X)

(−)

†

oo

(−)

∗

##

GP

f

(X)

OO

(−)

†

//

D⊗

L

R

−

))

GI

f

(X)

OO

(−)

†

oo

RHom

R

(D,−)

ii

(−)

⋆

{{

Proof. T he fact that the dagger, star and Foxby functors induce homomorphisms

on Grothendieck spaces follows immediately from Proposition 4.7. The fact that

the induced homomorphisms are isomorphisms follows immediately from 2.7, 2.8

and 2.9, since the underlying functors deﬁne dualities or equivalences of categories.

4.9. Proposition. Let X be a specialization-closed subset of Spec R and consider

the following elements of Grothendieck spaces.

α ∈ GP

f

(X), β ∈ GI

f

(X), γ ∈ GD

f

(X

c

) and σ ∈ GD

f

(m).

Then σ

†

= σ holds in GD

f

(m), and so do the fol lowing identities.

α ⊗ γ = Hom(γ, α

†

) = Hom(α, γ

†

) = Hom(α

∗

, γ)

Hom(α, γ) = α ⊗ γ

†

= Hom(γ, D ⊗ α) = α

∗

⊗ γ

Hom(γ, β) = β

†

⊗ γ = Hom(Hom(D, β), γ)

Hom(β

†

, γ) = Hom(γ

†

, β) = Hom(D , β) ⊗ γ = Hom(γ, β

⋆

)

Proof. Rec all from 2.9 that the Foxby functors can be written as the composition of

a star and a dagger functor. All identities follow from the formulas in (3.2.1). The

formula for σ is a consequence of the ﬁrst formula in (3.2 .1) in the case X = R.

4.10. Frobenius endomorphism. Assume that R is complete of prime charac-

teristic p and with perfect re sidue ﬁeld. Let X be a specialization-closed subset of

Spec R, and let n be a non-negative integer. The der ived Frobenius endofunctor

LF

n

on P

f

(X) induces an endomorphism

4

on GP

f

(X), which will be denoted F

n

X

;

see [11] for further details. It is given for a complex X ∈ P

f

(X) by

F

n

X

([X]

P

f

(X)

) = [LF

n

(X)]

P

f

(X)

.

Let

Φ

n

X

=

1

p

n codim X

F

n

X

: GP

f

(X) → GP

f

(X).

According to [11, Theorem 19], the endomorphism Φ

n

X

is diagonalizable.

4

That is, a continuous, Q–linear operator.

DUALITIES AND INTERSECTION MULTIPLICITIES 19

In Lemma 2.11, we established that the functor RG

n

is an endofunctor on I

f

(X)

which can be written as

RG

n

(−) = (−)

†

◦ LF

n

◦ (−)

†

.

Thus, RG

n

is composed of functors that induce homomorphisms on Grothendieck

spaces, and hence it too induces a homomorphism GI

f

(X) → GI

f

(X). We denote

this endomorphism on GI

f

(X) by G

n

X

. It is given for a complex Y ∈ I

f

(X) by

G

n

X

([Y ]

I

f

(X)

) = [RG

n

(Y )]

I

f

(X)

.

Let

Ψ

n

X

=

1

p

n codim X

G

n

X

: GI

f

(X) → GI

f

(X).

Theorem 6.2 shows that Ψ

n

X

also is a diagonalizable automorphism.

For complexes X ∈ P

f

(X) and Y ∈ I

f

(X) we shall write Φ

n

X

(X) and Ψ

n

X

(Y )

instead of Φ

n

X

([X]

P

f

(X)

) and Ψ

n

X

([Y ]

I

f

(X)

), respectively. The isomorphism in (4.4.1)

together with Proposition 4.7 shows that the Dutta multiplicity χ

∞

(X, Y ) and

its two analogs ξ

∞

(X, Y ) and ξ

∞

(X, Y ) from Section 3.3 can be identiﬁed with

elements in GD

f

(m) of the form

lim

e→∞

(Φ

e

X

(X) ⊗ [Y ]

D

f

(Y)

), lim

e→∞

Hom([X]

D

f

(X)

, Ψ

e

Y

(Y )) and

lim

e→∞

Hom(Φ

e

X

(X), [Y ]

D

f

(Y)

).

5. Vanishing

5.1. Vanishing. Let X be a specialization-closed subset of Spec R a nd co ns ider an

element α in GP

f

(X), an element β in GI

f

(X) and an element γ in GD

f

(X). The

dimensions of α, β and γ are deﬁned as

dim α = inf

n

dim(Supp X)

α = r[X]

P

f

(X)

for some r ∈ Q and X ∈ P

f

(X)

o

,

dim β = inf

n

dim(Supp Y )

α = s[Y ]

P

f

(X)

for some s ∈ Q and Y ∈ I

f

(X)

o

and

dim γ = inf

n

dim(Supp Z)

γ = t[Z]

D

f

(X)

for some t ∈ Q and Z ∈ D

f

(X)

o

.

In particular, the dimension of an element in a Grothendieck spa c e is −∞ if and

only if the element is trivial. We say that α satisﬁes vanishing if

α ⊗ σ = 0 in GD

f

(m) for all σ ∈ GD

f

(X

c

) with dim σ < codim X,

and that α satisﬁes weak vanishing if

α ⊗ τ = 0 in GD

f

(m) for all τ ∈ GP

f

(X

c

) with dim τ < codim X.

Similarly, we say that β satisﬁes vanishing if

Hom(σ, β) = 0 in GD

f

(m) for all σ ∈ GD

f

(X

c

) with dim σ < codim X,

and that β satisﬁes weak vanishing if

Hom(τ, β) = 0 in GD

f

(m) for all τ ∈ GP

f

(X

c

) with dim τ < codim X.

20 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

The vanishing dimension of α and β is deﬁned as the numbers

vdim α = inf

n

u ∈ Z

α ⊗ σ = 0 for all σ ∈ GD

f

(X

c

)

with dim σ < codim X − u

o

and

vdim β = inf

n

v ∈ Z

Hom(σ, β) = 0 for all σ ∈ GD

f

(X

c

)

with dim σ < codim X − v

o

.

In par ticula r, the vanishing dimension of an element in a Grothendieck space is

−∞ if and only if the ele ment is trivial, and the vanishing dimension is less than

or equal to 0 if and only if the element satisﬁes vanishing.

5.2. Remark. If X is a complex in P

f

(R) with X = Supp X, then the element

α = [X]

P

f

(X)

in GP

f

(X) satisﬁes vanishing exactly when

χ(X, Y ) = 0 for all co mplexes Y ∈ D

f

(X

c

) with dim(Supp Y ) < codim X,

and α s atisﬁes weak vanishing exactly when

χ(X, Y ) = 0 for all co mplexes Y ∈ P

f

(X

c

) with dim(Supp Y ) < codim X.

The va nis hing dimension of α measures the extent to which vanishing fails to hold:

the vanishing dimension of α is the inﬁmum of integers u such that

χ(X, Y ) = 0 for all complexes Y ∈ D

f

(X

c

) with dim(Supp Y ) < codim X − u.

It follows that the ring R satisﬁes vanishing (or weak vanishing, respectively) as

deﬁned in 3.1, if and only if all elements of GP

f

(X) for all specialization-clo sed

subsets X of Spec R satisfy vanishing (or weak vanishing, respectively).

If Y is a complex in I

f

(R) with X = Supp Y , then the element β = [Y ]

I

f

(X)

in

GI

f

(X) satisﬁes vanishing exactly when

ξ(X, Y ) = 0 for all complexes X ∈ D

f

(X

c

) with dim(Supp X) < codim X.

and β satisﬁes weak vanishing exactly when

ξ(X, Y ) = 0 for all complexes X ∈ P

f

(X

c

) with dim(Supp X) < codim X.

The vanishing dimension of β measures the extent to which vanishing of the Euler

form fails to hold: the vanishing dimension of β is the inﬁmum of integers v such

that

ξ(X, Y ) = 0 for all complexes X ∈ D

f

(X

c

) with dim(Supp X) < codim X − v.

Because of the formulas in (3.2.1), it follows that the ring R satisﬁes va nis hing (or

weak vanishing, respectively) if a nd only all e le ments of GI

f

(X) for all specialization-

closed subsets X of Sp ec R satisfy vanishing (or weak vanishing, respectively).

5.3. Remark. For a specialization clos ed subset X of Spec R and elements α ∈

GP

f

(X), β ∈ GI

f

(X) and γ ∈ GD

f

(X), we have the following formulas for dimension.

dim γ = dim γ

†

,

dim α = dim α

†

= dim α

∗

= dim(D ⊗ α) and

dim β = dim β

†

= dim β

⋆

= dim Hom(D, β).

These follow immediately from the fact that the dagger, star and Foxby functors

do no t change supports of complexes. Further, we have the following formulas for

DUALITIES AND INTERSECTION MULTIPLICITIES 21

vanishing dimension.

vdim α = vdim α

†

= vdim α

∗

= vdim(D ⊗ α) and

vdim β = vdim β

†

= vdim β

⋆

= vdim Hom(D, β).

These follow immediately from the above together with (3.2.1).

5.4. Propo sition. Let X be a specialization-closed subset of Spec R , let α ∈ GP

f

(X)

and let β ∈ GI

f

(X). Then the following hold.

(i) If codim X 6 2 then vanishing holds for all elements in GP

f

(X) and GI

f

(X).

In particular, we always have

vdim α, vdim β 6 max(0, codim X − 2).

(ii) Let X

′

be a specialization-closed subset of Spec R with X ⊆ X

′

. Then

vdim

α 6 vdim α − (codim X − codim X

′

) and

for

α ∈ GP

f

(X

′

). For any given s in the range 0 6 s 6 vdim α, we

can always ﬁnd an X

′

with s = codim X − codim X

′

such that t he above

inequality becomes an equality. Likewise,

vdim

β 6 vdim β − (codim X − codim X

′

)

for

β ∈ GI

f

(X

′

), and for any given s in the range 0 6 s 6 vdim β, we

can always ﬁnd an X

′

with s = codim X − codim X

′

such that t he above

inequality becomes an equality.

(iii) The element α satisﬁes weak vanishing if and only if, for all specialization-

closed subsets X

′

with X ⊆ X

′

and codim X

′

= codim X − 1,

α = 0 as an element of GD

f

(X

′

).

Similarly, the element β satisﬁes weak vanishing if and only if, for all

specialization-closed subsets X

′

with X ⊆ X

′

and codim X

′

= codim X − 1,

β = 0 as an element of GD

f

(X

′

).

Proof. Because of Proposition 4.9 and the formulas in Remark 5.3, it suﬃces to

consider the statements for α and GP

f

(X). But the in this case, (i ) and (ii) are

already contained in [11, Example 6 and Remark 7], and (iii) follows by consider-

ations similar to those pr oving (ii) in [11, Remark 7].

The following two propositions present conditions that are equivalent to hav-

ing a certain vanishing dimension for elements of the Grothendieck space GI

f

(X).

There are similar results fo r elements of the Grothendieck space GP

f

(X); see [11,

Proposition 23 and 24].

5.5. Propositi on. Let X be a specialization-closed subset of Spec R, and let β ∈

GI

f

(X). Then the following conditions are equivalent.

(i) vdim β 6 0.

(ii) Hom(γ, β) = 0 for all γ ∈ GD

f

(X

c

) with dim γ < codim X.

(iii)

β = 0 in GI

f

(X

′

) for any specialization-closed subset X

′

of Spec R with

X ⊆ X

′

and codim X

′

< codim X.

(iv) β = 0 in GI

f

(X

′

) for any specialization-closed subset X

′

of Spec R with

X ⊆ X

′

and codim X

′

= codim X − 1.

22 ANDERS J. FRANKILD AND ESBEN BISTRUP HALVORSEN

Proof. By deﬁnition (i) is equivalent to (ii), and Proposition 4.3 in conjunction

with Remark 5.2 shows that (i) implies (iii ). Clearly (iii) is stronger than (iv ),

and (iv ) in conjunction with Proposition 5.4 implies (ii).

5.6. Proposition. Let X be a specialization-closed subset of Spec R, let β ∈ GI

f

(X),

and let u be a non-negative int eger. Then the following conditions are equivalent.

(i) vdim β 6 v.

(ii) Hom(γ, β) = 0 for all γ ∈ GD

f

(X

c

) with dim γ < codim X − v.

(iii)

β = 0 in of GI

f

(X

′

) for any specialization-closed subset X

′

of Spec R with

X ⊆ X

′

and codim X

′

< codim X − u.

(iv)

β = 0 in GI

f

(X

′

) for any specialization-closed subset X

′

of Spec R with

X ⊆ X

′

and codim X

′

= codim X − v − 1 .

Proof. T he structure of the proof is similar to that of Propositio n (5.5).

6. Grothendieck spaces in prime characteristic

According to [11, Theorem 19] the endomorphism Φ

X

on GP

f

(X) is diagonaliz-

able; the precise statement is recalled in the next theorem. This section establishes

that the