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Dualities and intersection multiplicities

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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.
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
)
iZ
of modules equipped
with a differential (
X
i
)
iZ
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
)
iZ
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
)
iZ
of maps s
i
: X
i
Y
i+1
such that
σ
i
ρ
i
=
Y
i+1
s
i
+ s
i1
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
in
, and the differen-
tials are
Σ
n
X
i
= (1)
n
X
in
. 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 nite 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 RSbi-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 R1
· · · 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 Rcomplexes 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 Qlinear 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
n1
, 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→∞
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