The KH-Theory of Complete Simplicial Toric Varieties and the Algebraic K-Theory of Weighted Projective Spaces

0.31 MB·English

by  Adam Massey

THE KH-THEORY OF COMPLETE SIMPLICIAL TORIC
VARIETIES AND THE ALGEBRAIC K-THEORY OF
WEIGHTED PROJECTIVE SPACES
ADAMMASSEY
2
1 Abstract. We show that, for a complete simplicial toric variety X,
0
we can determine its homotopy K-theory (denoted KH-theory) entirely
2 in terms of the torus pieces of open sets forming an open cover of X.

l WeaccomplishthisbyconstructingasimplicialschemeBOT andcon-
u X
structing a relationship between the spectrum KH(X) and a certain
J
3 spectrum determined by BOTX. We then construct conditions under
which, given two complete simplicial toric varieties with the same sim-

plicialstructure,thetwospectraKH(X)⊗QandKH(Y)⊗Qareweakly
]
T equivalent. WeapplythisresulttodeterminetherationalKH-theoryof
K weighted projective spaces. WenextexamineK-regularityfor complete
toric surfaces; in particular, we show that complete toric surfaces (and
.
h therefore 2-dimensional weighted projective spaces) are K -regular. We
0
at then determine conditions under which our approach for dimension 2
m worksinarbitrarydimensions,beforedemonstratingthatweightedpro-
jective spaces are not K -regular, and for dimensions bigger than 2 are
[ 1
also not in general K0-regular.
2
v
3
2
1
0
1. Introduction
.
7
0 Inthis paper, weexamine thebasicproperties of complete simplicial toric
2 varieties, and use these properties to compute their K-theory. Following the
1
ideas of [CHWW, Proposition 5.6], the K-theory of any toric variety (in
:
v particular, any weighted projective space) over a field of characteristic 0 is
i
X obtained as a direct sum of the KH-theory and another invariant of that
r toric variety called its FK. We begin by focusing on the KH-theory; the FK
a
groups turn out to be much more difficult and are handled separately.
We conjecture that if X and Y are two complete simplicial toric varieties
withthe samesimplicial structure(see Definition 3.2), then theKH-theories
of X and Y are isomorphic; however, we are currently unable to prove a
relationship between KH(X) and KH(Y) exists under these conditions. To
remedy this, we add additional conditions to force a relationship between X
and Y, and in turn a relationship between KH(X) and KH(Y). We do this
via the construction of simplicial schemes BOT and BOT (see Definition
X Y
4.3). Itturnsoutthatthesetwosimplicialschemeswillhavearelationshipif
weassumecertainrelationshipsbetweenthefansofX andY. Themaingoal
of the first half of this paper is to construct and examine this relationship.
Date: July 4, 2012.
Key words and phrases. Toric Varieties, Algebraic K-Theory, Weighted Projective
Spaces, K-regularity.
1
2 ADAMMASSEY
Using this relationship, we can construct a relationship between the KH-
theory of ordinary projective space and the KH-theory of any weighted pro-
jective space. In particular, if P(q ,...,q ) is a weighted projective space of
0 d
dimension d, then the rank of KH (P(q ,...,q )) is d+1.
0 0 d
Using these ideas, our goal throughout this paper will be to prove the
following theorem:
Theorem 1.1. Let P(q ,...,q ) be any weighted projective space of dimen-
0 d
sion d over a field k. Then:
(a) If the characteristic of k does not divide the product q ·q ···q , then
0 1 d
for every n, we have KH (P(q ,...,q ))⊗Q ∼= KH (Pd)⊗Q.
n 0 d n
If the characteristic of k is 0, then we can conclude the following additional
results:
(b) Any 2-dimensional weighted projective space P(a,b,c) is K -regular.
0
(c) Any d-dimensional weighted projective space P(q ,...,q ) whose sin-
0 d
gular set consists of only isolated singular points is K -regular.
0
(d) If our weighted projective space is of the form P(1,1,...,1,a),
then for n ≤ 0 we can conclude the stronger statement
K (P(1,1,...,1,a)) ∼= K (Pd).
n n
Theorem1.1isproveninseveralstages. Mostofourworktowardsproving
Theorem 1.1 is done by proving Theorem 3.3; as such, much of this paper
will focus on the proof of (and applications of) Theorem 3.3.
We begin in Section 2 by establishing the basic notation and terminology
that we will use throughout the paper. In Section 3, we introduce the
basic notion of the simplicial structure of a complete simplicial toric variety
before moving on to construct the simplicial scheme BOT in Section 4.
X
Using BOT , we determine conditions under which BOT and BOT are
X X Y
related in Section 5, and use that relationship to construct a relationship
between the KH-theories of X and Y in Section 6. We apply these ideas to
weighted projective spaces in Section 7.
Finally, we examine the K-regularity of complete toric varieties and
weighted projective spaces. Focusing primarily on the case where the di-
mension of the toric variety is 2 and the underlying field has characteristic
0, we begin by proving the F Decomposition Theorem for complete toric
K
surfaces in Section 8, and use it to prove that any complete toric surface is
K -regular. In Section 9, we examine the failure of the F Decomposition
0 K
Theorem to hold in higher dimensions, and determine additional conditions
on the toric variety that allow a variant of the F Decomposition The-
K
orem to hold. Using this variant, we can again show that, in this case,
higher dimensional complete toric varieties satisfying this extra condition
are K -regular. We conclude in Section 10 by demonstrating that weighted
0
projective spaces are not K -regular in general, and that if the dimension
1
of our weighted projective space is bigger than 2 and the singular set has
dimension bigger than 0, then our weighted projective space need not be
K -regular. Thus, weighted projective spaces are not in general K -regular
0 0
as well.
KH-THEORY OF COMPLETE SIMPLICIAL TORIC VARIETIES 3
2. Toric Varieties: Notations and Terminology
In this section we establish the notation and basic definitions/results
about toric varieties that we will assume throughout the paper. Much of
what we discuss can be found in [Ful], which is the standard reference on
the subject, and [Cox]. One can also find many of the basics that we as-
sume for this paper discussed in the early sections of [CHWW]. Note that
throughout this paper, unless otherwise mentioned, we make no assumption
on the characteristic of k.
We call X a toric variety if it is a normal variety along with a split
algebraic torus T ∼= Gn embedded as a Zariski dense open subset and an
m
action of T on X extending the obvious action of T on itself. The case in
which the torus is not split will not be considered in this paper.
Let N bea lattice of finite rank, and M = Hom(N,Z) be the dual lattice.
We let N = N ⊗ R and M = M ⊗ R = Hom(N ,Z). For m ∈ M and
R Z R Z R R
n∈ N , we let hm,ni denote the value of m applied to n.
R
A rational, strongly convex polyhedral cone σ ⊂ N is a subset of the
R
form σ = R v +···+R v for some v ,...,v ∈ N, and if both u and −u
≥0 1 ≥0 k 1 k
are in σ, then u = 0. Throughout this paper, when we say cone, we mean
a rational, strongly convex polyhedral cone. We define the dimension of σ,
denoted dim(σ), to be the dimension of the subspace Span{v ,...,v } ⊂ N .
1 k R
Given a cone σ, we define the dual cone to be
σ∨ = {m ∈ M |hm,ni ≥ 0 for n ∈ σ}. We define the affine toric
R
variety associated to σ, which we denote U , to be U = Spec(k[σ∨∩M]),
σ σ
where we write σ∨∩M multiplicatively. When we want to indicate that U
σ
is an open subset of the variety X, we will do so by writing it as UX.
σ
Given a cone σ, we say that τ is a face of σ, and write τ ≺ σ, if
τ = {n ∈ σ|hm,ni = 0} for some m ∈ σ∨. Notice that any face of a
strongly convex polyhedral cone is again a strongly convex polyhedral cone.
We define a facet of σ to be a face of codimension 1. A fan ∆ in N is a
R
collection of cones such that if τ ≺ σ and σ ∈ ∆, then τ ∈ ∆ and such that
if σ ,σ ∈ ∆ then σ ∩σ ≺ σ and σ ∩σ ≺ σ .
1 2 1 2 1 1 2 2
If τ ≺ σ ⊂ N , then we get an inclusion U −→ U , and in fact U is a
R τ σ τ
principal open subset of U . Therefore, given any fan ∆, we can construct
σ
a variety X(∆) by taking affine opens U for all σ ∈ ∆ and then, for all
σ
σ ,σ ∈ ∆, gluing U and U along U . X(∆) is in fact a toric variety,
1 2 σ1 σ2 σ1∩σ2
with U as its torus (here 0 denotes the cone consisting only of the origin in
0
N ). Thereis a converse, dueto Sumihiro,which says that any toric variety
R
is given by a fan inside some lattice; see [Sum]. When referring to the toric
variety associated to a fan ∆, we write X(∆); conversely, when given a toric
variety X, we denote by ∆ its associated fan and we denote by NX the
X
lattice in which ∆ lives.
X
SupposeX andY aretwotoricvarieties,and∆ and∆ aretheirrespec-
X Y
tive fans; if we have a lattice morphism that induces a map ϕ: ∆ −→ ∆
X Y
such that for any cone σ ∈ ∆ , ϕ(σ) is contained in a cone inside ∆ , then
X Y
ϕ induces a morphism of toric varieties X −→ Y. To construct the map,
noticethat becauseϕ(σ) iscontained inaconeinside∆ (whichwewillcall
Y
σ′ for convenience), we induce a map Uσ −→ Uσ′ for every cone σ ∈ ∆X.
Then we glue them together. Throughout this paper, we will use the notion
of toric morphism and lattice/fan morphism interchangeably.
4 ADAMMASSEY
We define the support of a fan ∆, denoted |∆|, to be the union of all
cones in ∆ as viewed inside N . We say a toric variety X is complete if
R
|∆ |= N .
X R
We say that a cone σ is simplicial if its minimal set of generators
{v ,...,v } is linearly independent over R. We say that a fan ∆ is sim-
1 k
plicial if every cone σ ∈ ∆ is simplicial, and we say that a toric variety X is
simplicial if its associated fan ∆ is simplicial. Similarly, we say that a cone
X
σ is smooth if its minimal set of generators {v ,...,v } is part of a Z-basis
1 k
of N. We say that a fan ∆ is smooth if every cone σ ∈ ∆ is smooth, and
we say that a toric variety X is smooth if its associated fan ∆ is smooth.
X
Notice that any smooth cone/fan/toric variety must also automatically be
simplicial.
Let N be a lattice, and σ ⊂ N be a cone. We define
R
N = (σ∩N)+(−σ∩N)tobethesublatticeofN generated byσ. Similarly,
σ
we define N = N/N .
σ σ
Oneimportantfanthatwecancreatefromagivenfan∆(insidethelattice
e
N) is the star of a given cone τ, denoted Star(τ). The star of the cone τ
is the fan consisting of all cones σ ∈ ∆ such that τ ≺ σ, but considered
inside the lattice N . The toric variety X(Star(τ)) is a closed subvariety of
τ
X(∆), and the dimension of the cone τ is the codimension of X(Star(τ))
e
insideX(∆); inparticular,ifτ isamaximalcone, thenX(Star(τ))isapoint
inside X(∆).
Proposition 2.1. Let σ be a p-dimensional cone in a lattice N with
dimNR = n. Then Uσ ∼= Uσ′ × T, where T ∼= Gmn−p is a split algebraic
torus of rank n−p.
Proof. The key idea is to take a splitting N ∼= N ⊕N′′; in this splitting
σ
we have σ = σ′ ⊕ {0}, where σ′ is a maximal cone in N . Taking duals
σ
gives us the splitting M ∼= M′ ⊕ M′′, and following the construction of
U given above yields Spec(k[σ∨ ∩M] = k[(σ′∨∩M′)⊕M′′]), which gives
σ
Uσ ∼= Uσ′ ×T. See [Ful, Page 29] for the full details.
(cid:3)
Definition 2.2. We define the “torus piece of U ” to be
σ
Spec(k[Hom(N ,Z)]), and we denote this torus piece by T . When
σ σ
we want to indicate that T is the torus part of an open subset U of the
σ σ
e
variety X, we will do so by writing it as TX.
σ
Remark 2.3. The torus T constructed in Proposition 2.1 is the torus piece
T of U in the sense of Definition 2.2. Indeed, observe that in the splitting
σ σ
of N given in the proof of Proposition 2.1, N′′ ∼= N ; hence N could have
σ σ
instead been used to construct the torus piece in Proposition 2.1. These
e e
ideas will be used interchangeably throughout the paper.
One particularly important class of simplicial toric varieties that we will
be interested in is the class of weighted projective spaces. Let q ,...,q be
0 d
positiveintegerswithgcd(q ,...,q ) = 1. Weightedprojectivespace,denoted
0 d
P(q ,...,q ), is constructed by
0 d
P(q ,...,q ) = Ad+1\{0}/ ∼ (2.1)
0 d
where (a ,...,a ) ∼ (b ,...,b ) if and only if there is a λ ∈ k such that
0 d 0 d
ai = λqi · bi for all i = 0,...,d. Notice that ordinary projective space
KH-THEORY OF COMPLETE SIMPLICIAL TORIC VARIETIES 5
is the weighted projective space where all the weights are 1; namely,
Pd = P(1,...,1). To find the fan, consider the lattice Zd+1 with basis
{f ,...,f }; then we consider the lattice N = Zd+1/hq f +···+q f i. Con-
0 d 0 0 d d
sider the set {y ,...,y } of residues of {f ,...,f } in N. Then ∆ is
0 d 0 d P(q0,...,qd)
the fan consisting of all cones generated by proper subsets of {y ,...,y }.
0 d
Observe that if all the weights are equal to 1 and we follow the method
described above for the construction of X(∆ ), we recover the usual con-
Pd
struction for Pd by gluing together copies of Ad.
3. The Simplicial Structure of a Complete Simplicial Toric
Variety
From [CHWW, Proposition 5.6], the algebraic K-theory of any toric va-
riety X is determined completely by its KH-theory and its F groups. We
K
begin by examining the KH-theory of complete simplicial toric varieties.
The initial impulse the reader might have is to use the fact that toric
varietieshaveaveryniceresolutionofsingularities(see[Cox,Theorem5.1]),
and then combine that idea with the fact that KH satisfies cdh-descent, as
proven in [Hae]. For very simple examples, such as P(1,1,...,1,a), this is a
reasonable approach.
However, this method is not effective as a general approach to calculating
the KH-theory of even just weighted projective spaces. The problem is that
even in dimension 2, the number of steps needed to resolve the singularities
can be quite large, and therefore we can be confronted with uncontrollable
exceptional varieties. Consider, for instance, the weighted projective space
P(1,5,7). To completely resolve all singularities, we must add in five addi-
tional one-dimensional cones. The exceptional variety will, in this case, be
a disjoint union of a chain of three copies of P1 and a chain of two copies of
P1.
Instead, we examine the simplicial structure of toric varieties, and how
thesimplicialstructureaffectstheKH-theory. Webeginwithcalculatingthe
KH-theory of U for any cone σ. Recall from Proposition 2.1 and Definition
σ
2.2 that, given any p-dimensional cone σ in N, where dimN = m, we get
R
Uσ ∼= Uσ′ ×Tσ, where Tσ is a split torus of rank m−p.
Proposition 3.1. Let σ be any p-dimensional cone. Then
K(T )−→ KH(T ) −→ KH(U ) (3.1)
σ σ σ
are weak equivalences as spectra. In other words, the KH groups of an open
set corresponding to a cone are just the K groups of its associated torus
piece.
Proof. The first map is a weak equivalence since algebraic tori are smooth.
The second map is seen to be a weak equivalence by combining the defini-
tion of U with [Wei1, Theorem 1.2, Part (a)]. See the proof of [CHWW,
σ
Proposition 5.6].
(cid:3)
It is the result of Proposition 3.1 that provides the intuition for the ap-
proach we use. Our goal will be to build some relationship between a com-
plete simplicial toric variety and a scheme that is built from all the torus
pieces of open sets associated to maximal cones covering that toric variety,
6 ADAMMASSEY
and then use this scheme of torus pieces to develop a relationship between
the KH-theories of two complete simplicial toric varieties with the same
simplicial structure.
As we saw above, the KH-theory of an open set associated to a cone
depended only on the torus piece. This leads us to consider ways in which
we might use the simplicial structure of a toric variety to determine its KH-
theory. In order to do this, we must first make our definition of simplicial
structure clear.
Definition 3.2. Let X be a complete simplicial toric variety, and ∆ (1)
X
be the set of 1-dimensional rays in the fan of X. Let S(∆ ) denote the set
X
of all sets of rays in ∆ (1) that form a cone in the fan ∆ . Since X is
X X
assumed simplicial, any subset of a set of rays forming a cone also forms a
cone; therefore, S(∆ ) forms a simplicial complex. We define the“simplicial
X
structure” of X to be the simplicial complex S(∆ ). We say that two simpli-
X
cial toric varieties X and Y have the “same simplicial structure” if S(∆ )
X
and S(∆ ) are isomorphic as simplicial complexes.
Y
Now that we have established our basic definitions, the next few sections
will be dedicated to proving and applying the following theorem, which is
our main technical result.
Theorem3.3. LetX andY be twocomplete simplicial toricvarietiesoverk
with the same simplicial structure; i.e. ϕ : S(∆ ) −→ S(∆ ) is an isomor-
X Y
phism of simplicial complexes, where ∆ is the fan for X and ∆ is the fan
X Y
for Y. Let ∆ live in the lattice NX and ∆ live in the lattice NY. Suppose
X Y
wehavealatticemorphism F : NX −→ NY whichisinjectivewithfinitecok-
ernel such that the restriction maps F| :NX −→ NY are also injective
NσX σ ϕ(σ)
with finite cokernel for any cone σ ∈ ∆ . Suppose further that the charac-
X
teristic of k does not divide |coker(F)|. Then KH(X)⊗Q and KH(Y)⊗Q
are weakly equivalent as spectra; in particular, KH (X)⊗Q ∼= KH (Y)⊗Q
n n
for all n.
There are several stages that go into the proof of Theorem 3.3. We begin
by focusing on a single complete simplicial toric variety and, in the spirit
of Proposition 3.1, seek to relate its KH theory to the KH theory of certain
torus pieces that are related to X. Then, once such a relationship is estab-
lished, we look to use the lattice conditions to build a map between these
tori that let us (after considering the above relationship with the complete
simplicial toric variety X) derive the relationship between the KH theories
of X and Y. We begin this construction in Section 4.
4. The Construction of the Simplicial Scheme BOT
X
As it turns out, simply knowing that X and Y have the same simplicial
structure is not enough to determine that they have the same KH-theory.
However, since KH satisfies Zariski descent, we can approach this problem
from the perspective of simplicial schemes. This is our next goal.
Construction 4.1. Let X be a complete simplicial toric variety. Then
X gives rise to a simplicial scheme, which we call U . To construct the
X
simplicial scheme structure, we simply take the Zariski cover given by open
sets associated to maximal cones in the fan of X, and then we take the Cˇech
KH-THEORY OF COMPLETE SIMPLICIAL TORIC VARIETIES 7
nerve of that cover. On the level of lattices, the face and degeneracy maps
are easy to describe. As in the statement of Theorem 3.3, let NX denote the
lattice in which the fan ∆ lives. For the face maps, letting τ = σ ∩···∩σ
X 0 n
and letting τ =σ ∩···∩σ ∩···∩σ , this map becomes d : U −→ U by
j 0 j n j τ τj
inclusion. Notice that, since τ ≺ τ , this face map can be obtained by taking
j
NX −→ NX to be the idebntity and taking τ −→ τ by inclusion. For the
j
degeneracy maps, if we again let τ = σ ∩···∩σ , then s : U −→ U by
0 n j τ τ
the identity, which is obtained by taking NX −→ NX to be the identity and
taking τ −→ τ to be the identity as well. Note that we can see very easily
that the maps d and s above satisfy the usual simplicial relations. For the
j j
remainder of this paper, when we say “simplicial scheme associated to X”,
we are referring to the simplicial scheme U .
X
OurgoalistorelatetheKHtheoryofacompletesimplicialtoricvarietyto
the KHtheory of its torus pieces, extending the intuition of Proposition 3.1.
Fortunately, our work has provided us a way to do this. Indeed, notice that
the definition of the simplicial scheme structure associated to a complete
simplicial toric variety is given completely in terms of open sets associated
to cones; since open sets associated to cones have algebraic tori as direct
factors, it will allow us to relate these two concepts. We begin with some
new definitions.
Definition 4.2. Let X be a complete simplicial toric variety with associated
simplicial scheme U as shown in Construction 4.1. Again, let NX denote
X
the lattice in which the fan ∆ lives, and let NX = NX/NX as defined at
X τ τ
the top of page 4. We define two lattice maps, which we call d and s , by the
e j j
following construction. Let τ = σ ∩···∩σ and τ = σ ∩···∩σ ∩···∩σ .
0 n j 0 j n
Then construct d :NX −→ NX by first lifting NX up to NXe, thenemapping
j τ τj τ b
NX to itself via the identity (with τ −→ τ via inclusion), and finally taking
e e e j e
canonical surjection onto NX. Similarly, construct s : NX −→ NX by
τj j τ τ
first lifting NX up to NX, then mapping NX to itself via the identity (with
τ e e e e
τ −→ τ via the identity), and finally taking canonical surjection onto NX.
e τ
Observe that, with this construction, s is just the identity map.
j
e
Definition 4.3. Let X be a complete simplicial toric variety with associ-
e
ated simplicial scheme U as shown in Construction 4.1. We define a new
X
simplicial scheme, which we call BOT , by the following properties:
X
(1) We define (BOT ) = T , where T is the asso-
X n α(σ0,···,σn) α(σ0,···,σn)
ciated torus piece for th`e open set Uσ0 ∩···∩Uσn (in the sense of
Definition 2.2) and α(σ ,··· ,σ ) is its rank. As before, the σ ’s are
0 n i
all maximal cones.
(2) We define the face and degeneracy maps, denoted dBOTX and sBOTX
j j
respectively, to component-wise be the morphisms of toric varieties
that are induced by the lattice maps d and s of Definition 4.2.
j j
Theorem 4.4. BOTX, as defined in Definiteion 4.3,eis a simplicial scheme.
Proof. Definition 4.3 has already given us our objects (BOT ) and our
X n
face and degeneracy maps, and it is an easy exercise to check that dBOTX
j
and sBOTX are well-defined and satisfy the usual simplicial identities.
j
(cid:3)
8 ADAMMASSEY
Now that we have shown BOT to be a simplicial scheme, we want
X
to relate BOT to the simplicial scheme U . Let qUX be the mor-
X X n
phism of schemes obtained by projecting each component of (U ) onto
X n
its torus piece. By it’s very definition, this gives us a morphism
qUX : (U ) −→ (BOT ) and we have:
n X n X n
Theorem 4.5. The morphism qUX : U −→ BOT given in degree n ≥ 0
X X
by qUX :(U ) −→ (BOT ) is a morphism of simplicial schemes.
n X n X n
Proof. This is straightforward by the construction of the maps dBOTX and
j
sBOTX.
j
(cid:3)
Theorem 4.6. The morphisms KH(qUX) : KH(BOT ) −→ KH(U ) and
X X
KH(qUX)⊗id : KH(BOT )⊗Q −→ KH(U )⊗Q of cosimplicial spectra
Q X X
induced by the map qUX of Theorem 4.5 are weak equivalences.
Proof. This follows by applying Proposition 3.1 componentwise.
(cid:3)
Corollary 4.7. The morphisms holim(KH(BOT )) −→ holim(KH(U ))
X X
and holim(KH(BOT )⊗Q)−→ holim(KH(U )⊗Q) are weak equivalences
X X
of Spectra.
Proof. This is immediate from Theorem 4.6 and the construction of the
holim functor given in [BK]. For our purposes, holim(−) = Tot(Π∗(R(−))),
where R denotes fibrant replacement, Π∗ denotes cosimplicial replacement,
and Tot is the total object functor.
(cid:3)
This allows us to conclude the following important consequence for
BOT .
X
Theorem 4.8. Let X be a complete simplicial toric variety. Then the two
spectra holim(KH(BOT )) and KH(X) are weakly equivalent.
X
Proof. By Corollary 4.7 and the fact that KH satisfies Zariski descent, we
get a zig-zag
holim(KH(BOT )) −→ holim(KH(U )) ←−KH(X) (4.1)
X X
where each map is a weak equivalence; thus the spectra holim(KH(BOT ))
X
and KH(X) are weakly equivalent as claimed.
(cid:3)
Theorem 4.8 completes our generalization of the intuition we presented
at the end of Section 3. Since holim(KH(BOT )) and KH(X) are weakly
X
equivalent, the nth stable homotopy group of holim(KH(BOT )) is isomor-
X
phictothegroupKH (X), showingthatKH (X) isindeeddeterminedonly
n n
by the torus pieces of open sets associated to maximal cones covering X as
claimed.
5. The Relationship Between BOT and BOT
X Y
With the work of Sections 3 and 4, we are now ready to prove Theorem
3.3. Section 4 focused on the construction of BOT for a given complete
X
KH-THEORY OF COMPLETE SIMPLICIAL TORIC VARIETIES 9
simplicialtoricvarietyX. Similarly,wecouldconstructBOT foradifferent
Y
complete simplicial toric variety Y. The question is: when are these two
simplicial schemes related? In general this is not known; however, if we
impose the conditions that X and Y have the same simplicial structure and
that we have a lattice morphism F : NX −→ NY which is injective with
finite cokernel such that the restriction maps F| : NX −→ NY are also
NσX σ ϕ(σ)
injective with finite cokernel for any cone σ ∈ ∆ , then we get a very useful
X
relationship.
Lemma 5.1. Let X and Y be two complete simplicial toric varieties with
the same simplicial structure, and suppose we have a lattice morphism
F : NX −→ NY which is injective with finite cokernel such that the re-
striction maps F| : NX −→ NY are also injective with finite cokernel
NσX σ ϕ(σ)
for any cone σ ∈ ∆ . Then the induced map F : NX −→ NY is also
X σ σ ϕ(σ)
injective with finite cokernel. Furthermore, we haeve theat |coker(eFσ)| divides
|coker(F)| for any cone σ.
e
Proof. Since both NX and NY are themselves lattices we have the follow-
σ ϕ(σ)
ing diagram of lattices
e e
0 // NX(cid:31)(cid:127) // NX πNσX // NX // 0
σ σ
F|NσX F e Feσ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // NY (cid:31)(cid:127) // NY // NY // 0
ϕ(σ) πNY ϕ(σ)
ϕ(σ)
e
where the rows are short exact sequences. By the Snake Lemma, we get the
exact sequence
0 // ker F // coker F| // coker(F) // coker F // 0
σ NX σ
(cid:16) (cid:17) (cid:16) σ (cid:17) (cid:16) (cid:17)
e e
where we get exactness on the right because the bottom row of our diagram
of lattices is shortexact. Now we recall that coker F| is assumed to be
NX
(cid:16) σ (cid:17)
finite; therefore, as the above exact sequence forces ker F to inject into
σ
(cid:16) (cid:17)
coker F| , we conclude that ker F is finite as weell, and therefore
NX σ
(cid:16) σ (cid:17) (cid:16) (cid:17)
0 since it is free abelian. Therefore, thee map Fσ is injective. From here,
it is immediate that F has finite cokernel, and that |coker(F )| divides
σ e σ
|coker(F)| for any cone σ.
e e (cid:3)
The importance of Lemma 5.1 is that F : NX −→ NY gives rise to
σ σ ϕ(σ)
a morphism between the torus pieces of UX and UY , which will in turn
eσ e ϕ(σ) e
allow us to construct a morphism BOT −→ BOT of simplicial schemes.
X Y
Theorem 5.2. Let X and Y be two complete simplicial toric varieties
with the same simplicial structure, and suppose we have a lattice mor-
phism F : NX −→ NY which is injective with finite cokernel such that
the restriction maps F| : NX −→ NY are also injective with finite
NσX σ ϕ(σ)
10 ADAMMASSEY
cokernel for any cone σ ∈ ∆ . Then F induces a morphism of simpli-
X
cial schemes BOT −→ BOT . Furthermore, for every n, the morphism
X Y
(BOT ) −→ (BOT ) is, in each component, an isogeny.
X n Y n
Proof. As we’ve already seen, NX and NY determine the torus pieces of
σ ϕ(σ)
UσX and UϕY(σ), and so Fσ : NσXe−→ NϕY(σe) determines a morphism between
the tori TσX and TϕY(σ).eSincee Fσ is einjective with finite cokernel (Lemma
5.1), the induced map on the correspondingtori is an isogeny; therefore, the
e
morphism (BOT ) −→ (BOT ) is, in each component, an isogeny.
X n Y n
So we need only show that the maps (BOT ) −→ (BOT ) commute
X n Y n
withthe face anddegeneracy maps. Recall from Definitions 4.2 and 4.3 that
the face and degeneracy maps for BOT are constructed by considering
X
maps on the appropriate lattices. Recall that we called these maps d and
j
s ;here,todifferentiatebetweenthecasewhenthey’reconstructedfromNX
j e
or NY, we denote these maps by dNX and sNX for the complete simplicial
e j j
toricvariety X andby dNY andsNY for thecomplete simplicialtoric variety
j j e e
Y. Then, in the face map case, we get the following diagram:
e e
idNX
NX ++ NX
bb❊❊❊❊❊❊α❊❊❊ deNX π(cid:2)(cid:2)NτXj α DD
NX j // NX
τ τj
F Feτe e Feτj F
(cid:15)(cid:15) deNY (cid:15)(cid:15)
NY j // NY
πNϕY(τ) 88 eϕ(τ) eϕ(τj)bb❊❊❊❊πNϕY(τj)
❊
❊
(cid:25)(cid:25) α ❊ (cid:5)(cid:5)
ww
NY 33 NY
idNY
Here F is injective by assumption, while F and F are injective by Lemma
τ τj
5.1. Observe that, obviously, the outer square commutes. We want to
e e
show that the inner square commutes; then after applying all the necessary
functors, we get that these induced maps on the tori commute with the face
maps of BOT component-wise, and therefore commute overall.
X
Notice that, for an element [x] ∈ NX, F ([x]) = π ◦F(x). In other
τ τ NY
ϕ(τ)
words, we lift [x] back to NX, applyeF, aend then apply πNY . As Fτ is
ϕ(τ)
independent of our choice of lift, we can just take the lift α sending [x] to
e
x. We will abuse notation and use α to denote all such lifts of this type.
This means that we can write F = π ◦F ◦α and F = π ◦F ◦α.
τ NY τj NY
ϕ(τ) ϕ(τj)
Similarly, we can write dNX =eπ ◦id ◦α and dNYe= π ◦id ◦α.
j NX NX j NY NY
τj ϕ(τj)
e e