Schur functors and motives [PhD thesis]

57 Pages·0.343 MB·English

by  Carlo Mazza

SCHUR FUNCTORS AND MOTIVES
BY CARLO MAZZA
A dissertation submitted to the
Graduate School|New Brunswick
Rutgers, The State University of New Jersey
in partial ful(cid:12)llment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Mathematics
Written under the direction of
Charles Weibel
and approved by
New Brunswick, New Jersey
January, 2004
ABSTRACT OF THE DISSERTATION
Schur functors and motives
by Carlo Mazza
Dissertation Director: Charles Weibel
In [Kim], Kimura introduced the notion of a \(cid:12)nite dimensional" motive
(which we will refer to as \Kimura-(cid:12)nite" motive) and he conjectured that all
Q-linear motives modulo rational equivalence are Kimura-(cid:12)nite. The same no-
tion was introduced independently in a di(cid:11)erent context by O’Sullivan. Kimura-
(cid:12)niteness has been the subject of several articles recently ([GP02], [GP], [AK02]).
In [GP], Guletski(cid:21)(cid:16) and Pedrini proved that if X is a smooth projective surface
withp = 0,thenthemotiveofX isKimura-(cid:12)niteifandonlyifBloch’sconjecture
g
holds for X, i.e., the kernel of the Albanese map vanishes.
InthisdissertationweintroducethenotionofSchur-(cid:12)nitemotives, thatis,mo-
tives which are annihilated by a Schur functor. We study its relation to Kimura-
(cid:12)niteness and in particular we show that this new notion is more (cid:13)exible than
Kimura’s. Moreover, we show that the motive of any curve is Kimura-(cid:12)nite.
Inthe(cid:12)rstchapterwe(cid:12)rstintroducesomebasicnotionscomingfromrepresen-
tation theory, such as Schur functors and Tannakian categories. Then, we recall
the constructions of the categories of classical motives, and also of Voevodsky’s
triangulated category DM.
ii
In the second chapter, we de(cid:12)ne the central notion of Schur-(cid:12)niteness. We
study its basic properties and its relations with Kimura-(cid:12)niteness in the most
general setting. We then proceed to study more particular examples, i.e., how
Schur-(cid:12)niteness behaves with respect to short exact sequences in abelian cate-
gories and triangles in derived categories.
The third and last chapter analyzes the class of Schur-(cid:12)nite objects in the
categories of classical motives and in the category DM. We prove that Schur-
(cid:12)niteness has the two out of three property for triangles in DM, and this allows
us to prove that the motive of every curve is Kimura-(cid:12)nite. (This last result has
also been obtained by Guletski(cid:21)(cid:16).) We close with an example due to O’Sullivan of
a Schur-(cid:12)nite motive which is not Kimura-(cid:12)nite.
iii
Acknowledgements
I would like to thank Chuck Weibel for reasons too numerous to list.
I am deeply indebted to both Luca Barbieri Viale and Filippo De Mari for
being my mentors during the early stages of my career.
I am grateful to Wolmer Vasconcelos, Siddhartha Sahi and Claudio Pedrini
fortheir valuable comments andforserving as members of the defense committee.
I would also like to thank Friedrich Knop and Bruno Kahn for the exchanges of
ideas.
On a more personal note, I thank Marco, Jooyoun, Manuela, Tony, Andrea,
Daniela, Davide and Alina for helping me survive grad school.
This dissertation is dedicated to my parents.
iv
Table of Contents
Abstract : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii
Acknowledgements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iv
1. Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.1. Representation theory and Schur functors . . . . . . . . . . . . . . 2
The Littlewood-Richardson rule . . . . . . . . . . . . . . . 7
The Kronecker coe(cid:14)cients . . . . . . . . . . . . . . . . . . 8
1.2. Tannakian categories . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3. Classical motives . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4. Voevodsky’s categories . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Schur-(cid:12)niteness : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21
2.1. De(cid:12)nitions and basic properties . . . . . . . . . . . . . . . . . . . 21
2.1.1. Combinatorial dimension . . . . . . . . . . . . . . . . . . . 25
2.2. Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3. Extensions and derived categories . . . . . . . . . . . . . . . . . . 30
3. Motives : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37
3.1. Classical motives . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2. The category DM . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3. A non Kimura-(cid:12)nite motive . . . . . . . . . . . . . . . . . . . . . 47
References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50
Vita : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52
v
1
Chapter 1
Preliminaries
In this chapter we will deal with the basic de(cid:12)nitions and properties that we will
need in the later chapters.
Let us (cid:12)rst establish a few basic notations and terminology.
De(cid:12)nition 1.1. Let C be a category. We say that C is symmetric monoidal
if there exists a bifunctor (cid:10) : C (cid:2) C ! C, which we will call tensor product,
and an identity object (cid:0) (or (cid:0) , if the category is understood) which satisfy the
C
associativity and the commutativity constraints (see [ML98, p. 161 and p.184]).
Let A and B be two symmetric monoidal categories. A monoidal functor
F : A ! B is a functor which respects the tensor structures (see [ML98, p. 164]).
Let k be a (cid:12)eld. An additive category is k-linear if every hom is endowed with
the structure of a k-vector space in such a way that composition is k-bilinear.
A k-linear symmetric monoidal category is a category which is k-linear and
symmetric monoidal such that the tensor product is k-bilinear.
Let A be any category. Recall that p 2 End(A) is an idempotent if p(cid:14)p = p.
A category A is idempotent complete if each idempotent splits (see [ML98, p.
20]). If A is any category, we can construct the idempotent completion A#
which is universal for functors into categories in which every idempotent splits.
The objects of A# are the pairs (A;p) where A is an object of A and p is an
idempotent. If A is a symmetric monoidal, or additive, category, then so is A#.
De(cid:12)nition 1.2. A tensor category is a symmetric monoidal category which is ad-
ditive, idempotent complete, and where the tensor product is bilinear. A k-linear
2
tensor category is a k-linear symmetric monoidal category which is idempotent
complete and where the tensor product is k-bilinear.
Consider a functor F between two tensor categories. We say that F is a
tensor functor it is a symmetric monoidal functor and it respects the idempotent
splittings. If moreover the two categories are k-linear, then we say that F is a
k-linear tensor functor if it is a tensor functor and it is k-linear.
We say that a k-linear tensor category C is k-tensorielle if it is abelian and
(cid:24)
End((cid:0) ) = k.
C
Vocabulary Warning. The terminology used for symmetric monoidal cate-
gories is varied. In [Del90], Deligne writes \cat(cid:19)egorie tensorielle sur k" for our
k-tensorielle category. In [Del02], \cat(cid:19)egorie k-tensorielle" has the same mean-
ing as our k-tensorielle category. Moreover, a category satisfying the axioms of
[Del02, 1.2], is a k-linear tensor category. In [DM82], Deligne and Mumford write
\tensor category" for our symmetric monoidal category (see also [DM82, 1.4]).
Example 1.3. LetGbeagroup. Then thecategory Rep(G)of(cid:12)nite dimensional
representations of G is a k-tensorielle category. This is the subject of the next
section.
Exercise 1.4. Let C be a Q-linear tensor category. Then we write C(cid:6) for the
super-category, i.e., the objects of C(cid:6) are the pairs (V;W) of objects. We will
write (cid:22)(cid:0) for the object (0;(cid:0) ) of C(cid:6). The tensor product is de(cid:12)ned as usual, and
C C
recall that the switch acts as ((cid:0)1) times the product of the degrees. With this
de(cid:12)nition, C(cid:6) is a Q-linear tensor category. Note that (cid:22)(cid:0) (cid:10) (cid:22)(cid:0) (cid:24)= (cid:0) .
C C C
1.1 Representation theory and Schur functors
The main reference for this introductory part on representation theory will be
[FH91]. In this section, k will be a (cid:12)eld of characteristic zero.
3
Let G be a (cid:12)nite group. A (left) representation of G on a (cid:12)nite dimensional
k-vector space V is a homomorphism (cid:26) : G ! End(V). We will often confuse
g 2 G with its image (cid:26)(g) and if v 2 V we will often write g (cid:1) v for ((cid:26)(g))(v).
If V and W are two representations of G, then both V (cid:8) W and V (cid:10) W are
representations of G (G acts on V (cid:10)W by g(v(cid:10)w) = g(v)(cid:10)g(w)).
It will be convenient to introduce the character of a representation in order to
formulate Frobenius reciprocity. Let (cid:26) be a representation of G on V. We de(cid:12)ne
the character of (cid:26) by
(cid:31) (g) = trace((cid:26)(g)):
V
This is a complex valued function. The character is a class function and respects
both the direct sum and the tensor product of representations.
Let V be a representation of G and let us suppose that V decomposes as
Va1 (cid:8):::(cid:8)Van, where the V ’s are distinct and irreducible representations of G.
1 n i
Then each a is the inner product ((cid:31) ;(cid:31) ). (If (cid:11) and (cid:12) are two class functions
i V Vi
on G, we de(cid:12)ne the inner product ((cid:11);(cid:12)) to be 1 (cid:11)(g)(cid:12)(g).)
jGj g2G
Let V and W be two representations of G, wPith W irreducible. Then we
de(cid:12)ne [V : W] to be the multiplicity ((cid:31) ;(cid:31) ) of W inside V. In particular,
V W
V = (cid:8) [V : W ]W , where W runs over all irreducible representations of G.
i i i i
Let H be a subgroup of G. Let V be a representation of G and W a repre-
sentation of H. We will write res(V) for V considered as a representation for H.
Moreover, we de(cid:12)ne IndGW = Z[G](cid:10) W.
H H
Lemma 1.5. (Frobenius reciprocity) Consider a representation W of H and V
of G. Then
((cid:31) ;(cid:31) ) = ((cid:31) ;(cid:31) ) :
IndGW V G W res(V) H
H
In particular, if both V and W are irreducible representations, then we have that
[IndGW : V] = [resV : W].
H
4
In particular, we are interested in studying the representations of the sym-
metric group on n elements, (cid:6) .
n
Example 1.6. Let V be a 1-dimensional vector space. There are only two repre-
sentations of (cid:6) on V: the trivial representation (where g 2 G acts as g (cid:1)v = v)
n
and the alternating one (where g(cid:1)v = sgn(g)v, where sgn(g) is the sign of the per-
mutation g). Now let V be an n-dimensional vector space, with a basis e ;:::;e .
1 n
Let (cid:6) act on V by g (cid:1)e = e . The subspace V0 generated by e +:::+e is
n i g(i) 1 n
invariant; let U be the complement. We say that U is the standard representation
for (cid:6) . It is always an irreducible representation of (cid:6) .
n n
Let us (cid:12)x an integer n. Then a partition (cid:21) = (n ;:::;n ) of n is a sequence on
1 r
numbers n (cid:21) n (cid:21) ::: (cid:21) n such that n +:::+n = n and we will write j(cid:21)j = n.
1 2 r 1 r
We will represent such a partition with a particular kind of boxes diagram which
is called Young diagram. For example the three partitions of the number 3 are
represented as the following diagrams.
(3) : (2;1) : (1;1;1) :
We will often write (pq) for the partition (p;:::;p) of pq, which is represented by
the rectangle with q rows and p columns. For example, the partitions of n = 4
are: (4);(3;1);(22);(2;12);(14). If (cid:21) is a partition of n, then we write (cid:21)t for the
transpose of (cid:21), i.e., the partition whose Young diagram is the transpose of the
Young diagram of (cid:21) (e.g., (n)t = (1n);(p;1q)t = (q;1p);(3;2)t = (22;1)).
Let (cid:21) be a partition of n. Then a tableau of the Young diagram corresponding
to the partition (cid:21) is a numbering of the boxes by the integers 1;:::;j(cid:21)j = n. For
example if (cid:21) = (4;2;1), then some tableau are the following.
1 2 3 4 7 6 5 4 5 1 7 2
5 6 3 2 6 4
7 1 3
Let us (cid:12)x a tableau and consider the subgroups of (cid:6) called P, which preserves
n
5
each row, and Q, which preserves each column. In the group ring Z[(cid:6) ] de(cid:12)ne
n
a = e and b = sgn(g)e :
(cid:21) g (cid:21) g
g2P g2Q
X X
Still in Z[(cid:6) ] de(cid:12)ne c = a (cid:1)b .
n (cid:21) (cid:21) (cid:21)
Lemma 1.7. (See [FH91, 4.3].) In the group ring Z[(cid:6) ], there exists an integer
n
n such that c2 = n c . Then the image V = Q[(cid:6) ]c of c is an irreducible
(cid:21) (cid:21) (cid:21) (cid:21) (cid:21) n (cid:21) (cid:21)
representation of S and n = n!=dimV .
n (cid:21) (cid:21)
This construction yields a one-to-one correspondence between the partitions of
n and the irreducible representations of (cid:6) .
n
Choosing a di(cid:11)erent tableau for (cid:21) gives a di(cid:11)erent, but isomorphic, represen-
tation.
Example 1.8. From [FH91], we can list the correspondences between partitions
and irreducible representations for low n. For n = 2:
trivial alternating
For n = 3:
trivial standard alternating
Let n be integers so that n +:::+n = n and consider (cid:6) (cid:2):::(cid:2)(cid:6) (cid:18)
i 1 r n1 nr
(cid:6) . Let (cid:22) be a partition of n , and let V be the corresponding irreducible
n i i (cid:22)i
representation. Note that V (cid:10):::(cid:10)V is an irreducible representation of (cid:6) (cid:2)
(cid:22)1 (cid:22)r n1
:::(cid:2)(cid:6) . For every (cid:21) partition of n, let V be the corresponding representation.
nr (cid:21)
We de(cid:12)ne
[(cid:21) : (cid:22) ;:::;(cid:22) ] = [res(V ) : V (cid:10):::(cid:10)V ] = [Ind(V (cid:10):::(cid:10)V ) : V ];
1 r (cid:21) (cid:22)1 (cid:22)r (cid:22)1 (cid:22)r (cid:21)
wherethelastequality holdsbyFrobeniusreciprocity. Notethat[(cid:21) : (cid:22) ;:::;(cid:22) ] =
1 r
0 whenever j(cid:21)j 6= j(cid:22) j.
i i
P