Following Tom Bridgeland. An introduction to motivic Hall algebras. arXiv:1002.4372, 2010.

1. SEMINAR ON MOTIVIC HALL ALGEBRAS
DANIEL HUYBRECHTS, HEINRICH HARTMANN
1. Introduction
In this seminar we want to go through the recent paper T. Bridgeland - An
introduction to motivic Hall algebras [Bri10b]. Hall algebras were introduced
by C. Ringel in [Rin90] and are used in current research e.g. to prove identi-ties
between Donaldson–Thomas invariants [KS08] [JS10] [Bri10a]. However,
we do not intend to touch this applications during the seminar. Instead we
will focus on the methods used in the definition of the motivic Hall algebra:
Motives and Stacks.
The seminar is aimed at master-level students. We assume familiarity
with the basic theory of schemes as presented in Hartshorne’s [Har77] book,
or in the lecture courses Algebraic Geometry I and II at this university.
This is still a preliminary version of the program. More detailed explana-tions
for the talks 5 will be provided during the semester.
2. Program Outline
2.1. Grothendieck rings of varieties. This talk shall cover section 2 of
Bridgeland’s paper [Bri10b].
We define Grothendieck rings of varieties K(Var) as the free abelian group
generated by isomorphism classes of varieties modulo the scissors relation,
Z(V ar)/([X] − [Z] − [U])
where X Z is a closes subvariety with complement U. Note that varieties
are not assumed to be irreducible.
We will refer to the class [X] of a variety X as it’s motive.
Next introduce the notion of a Zariski fibration and of a geometric bi-jection.
Explain how to rewrite the scissors relation in terms of geometric
bijections.
Explain how the motive of Gln decomposes as
[Gln] =
Yn
k=1
Lk(Lk − 1),
where L = [A1] is the Lefschetz motive.
Give a proof of Lemma 2.12 in the context of schemes:
K(V ar) = K(Shm).
Note: We will ignore everything related to algebraic spaces in this talk.
Finnaly explain the following result of F. Bittner [Bit04] as explained in
C. Peters’ lecture notes [Pet06].
Date: April 8, 2011.
1

2. 2 DANIEL HUYBRECHTS, HEINRICH HARTMANN
The group K(V ar) is generated by the classes of smooth projective vari-eties
subject to the relations
• [;] = 0 and
• the blowup relation [X] − [Y ] = [Z] − [E], where BlY X = Z is the
blowup of a smooth projective variety along a smooth subvariety Y
with exceptional divisor E.
Note: Skip everything related to Euler characteristcs and Betti polyno-mials
(Lemma 2.3). The next talk will explain them in detail.
2.2. Motivic invariants. A motivic invariant is an map I : Ob(V ar) −!
where is an abelian group, which is additive with respect to the scissors
relation: I(X) = I(Z) + I(U). Therefore I descends to an additive map:
I : K(V ar) −! .
This talk shall introduce several topological motivic invariants following
the lectures of C. Peters [Pet06].
The first important example is the Euler characteristic:
e(X) =
X
i
c(Xan,Q) 2 Z.
(−1)idimQHi
Sketch the proof, which involves the following statements:
(1) Finiteness of dimQHi
c(X,Q),
(2) Vanishing of dimQHi
c(X,Q) if i 0,
(3) Additivity e(X) = e(U) + e(Z).
A proof of 1 and 2 can be found e.g. in Appendix B of Peters–Steenbrink
[PS06]. The proof of 3 involves the existence of an exact sequence
c(U) −! Hi
c(X) −! Hi
c(Z) −! Hi+1
c (U),
Hi
which depends on existence of a Whitney stratification.
Illustrate the additivity of the Euler characteristic at the example [P1] =
[C] + [pt] = [C] + 2[pt].
Generalizing the example of the Euler characteristic, there are more re-fined
motivic invariants.
For a smooth projective variety over C we have the Hodge polynomial:
h(x, y)(X) =
X
p+q=n
(−1)p+qhp,q(X)xpyq
and the Betti polynomial b(t)(X) = h(t, t)(X). These polynomials can be
extended to functions on the Grothendieck ring:
h : K(V ar) −! Z[x, y]
There are two proofs of this statement: One uses the Deligne’s theory of
mixed Hodge structures on non-projective or non-smooth varieties. The
other one, which shall be presented, uses Bittner’s description of K(V ar).
In fact, we need to define a motivic invariant I only for smooth projective
varietes, and check the well-definedness only for the blowup relation. Note
however, that the values of I on non-projective or non-smooth varieties are
very difficult to compute.

3. SEMINAR ON MOTIVIC HALL ALGEBRAS 3
In this the same way we get an even more refined invariant, called Hodge
characteristic
h(X) =
X
i
(−1)iHi(X,Q) 2 K(Hodge structures)
with values in the Grothendieck group of Hodge structures. The talk can
directly present a proof of this theorem and get the existence of Hodge- and
Betti-polynomials as a corollary.
As an application prove Lemma 2.3 in [Bri10b].
2.3. Algebraic spaces. This talk is a quick introduction of Atrin’s notion
of algebraic space. We follow the presentation in Knutson’s book [Knu70].
The following topics shall be discussed:
• Grothendieck topologies and sheaves (I.1-I.3)
• Etale equivalence relations (I.5)
• The definition of algebraic spaces (II.1)
• Properties of algebraic spaces and morphisms between them: sepa-reted,
reduced, of finite type (II.3).
• Theorem: Every algebraic space X contains a dense open subspace
U X which is a scheme. (II.6.6)
• Grothendieck groups of algebraic spaces: K(Sp) = K(V ar).
([Bri10b], Lemma 2.12.)
Algebraic spaces are a generalization of schemes which occure naturally
in moduli theory. A moduli problem (following Grothendieck) is a functor
M: (Shm) ! (Set)op. A (fine) moduli space forMis a scheme M together
with an isomorphism of functors:
:M−! M := Hom(_,M).
However many interesting moduli problems do not have fine moduli spaces.
Functors of the form M have two important properties:
(1) They are sheaves in a Grothendieck topology on (Shm).
(2) The admit a covering by affine schemes.
Ad1: Recall, that morphisms f : T ! M can be glued from fi : Ui ! T,
where S
Ui = X. This can be rephrased as saying: M is a sheaf for the
Zariski topology on (Shm). Grothendick observed that morphisms can be
glued not only be done anlog Zariski covers, but also along etale maps U !
M (or even fppf morphisms).
Ad2: A scheme X can be glued from affine schemes by an equivalence
relation
R = `
Ui Uj //
/
`
Ui/ X
where the maps i : R ! U are local isomorphism in the Zariski topology.
Note that, if X is quasi-compact and separated, then R is again affine.
Algebraic spaces are defined as functors X : (Shm) ! (Set)op, which are
(1) sheaves in the etale Gorthendieck topology on (Shm),
(2) can be glued from affine schemes via an etale equivalence relation.

4. 4 DANIEL HUYBRECHTS, HEINRICH HARTMANN
The condition 2 means, that every algebraic space can be represented as
a quotient in the category of sheaves on (Shm):
R
1
/
2
/ U/ X
where R and U are schemes and i are etale maps, such that R ! U ×U is a
“categorical equivalence relation”. Finite group actions give rise to examples
of etale equivalence relations, which are not Zariski-trivial.
If (P) is a property of schemes, which is stable in under etale maps (e.g.
reduced, non-singular), then we say X has (P) if there exists a presentation
(A) such that U has (P).
A morphism between algebraic spaces f : X1 ! X2 can always be lifted
to a suitable presentations of Xi:
R1/
/
fR
U1/
fU
X
f
R2 //
/ U2/ X2
This allows us to define properties (e.g. closed, open immersion) of f by
requiring fU to have the corresponding property.
Now that we know what a closed subspace of an algebraic space is, we
define the Grothendieck group of algebraic spaces K(Sp) in the same way
as for schemes.
The main theorem of the talk is the following isomorphy
K(V ar) = K(Sp).
The main step is to show, that every algebraic space X contains a dense
open subset U X which is a scheme. We want to see the details of the
proof.
2.4. Stacks. This is a basic introduction into stacks following [Góm01]. See
also [Vis05], [BCE+12], [LMB00]. The following topics shall be covered:
• Stacks as groupoid valued sheaves
• Example: moduli-stack of coherent sheavesM, quotient stacks X/G
• Stacks as categories fibered in groupoids
• Deligne–Mumford and Artin stacks
• Basic properties: separated, finite-type, . . .
• Fiber products of stacks (Appendix in [Bri10b])
Roughly speaking, a Deligne–Mumford stack X is a generalization of the
notion of an algebraic space (and therefore of a scheme), which allows points
x 2 X to have finite automorphism groups AutX(x).
In this seminar we will mainly work with algebraic stacks (=Artin stacks).
Here, the automorphism groups are allowed to algebraic groups of positive
dimension.

5. SEMINAR ON MOTIVIC HALL ALGEBRAS 5
The relation between schemes, algebraic spaces and stacks is summarized
in the following diagram:
(shemes)/ (algebraic spaces)/
(etale sheaves)/
Fun((Shm), (Set)op)
(DM-stacks)/
(stacks)/ 2-Fun((Shm), (Grp)op)
jjj4jjjj jjjj
(algebraic-stacks)
In this diagrams all arrows represent fully-faithful functors.
Many interesting moduli functors can be represented in the category of
algebraic stacks. The focus of our seminar lies on the moduli functor of
coherent sheaves on a smooth projective scheme X:
MX : (Shm) −! (Grp)op, T7! {E 2 Coh(T × X) | E flat/T } .
WhereMX(T) is endowed with the structure of a category with morphisms:
isomorphisms of sheaves.
Note that MX is not a “strict” functor, since pull back of sheaves is not
functorial. But there is a natural isomorphism fgE = (g f)E, which
makes MX a 2-functor.
There is an equivalent way of describing 2-functors F : (Shm) ! (Grp)op
as categories fibered in groupoids (CFG). The CFG associated to MX, is
the category of pairs (T,E), where E 2MX(T) with morphisms (f : T0 !
T, : fE
!
E0), together with the forgetful functor (T,E)7! T.
A 2-functor F : (Shm) ! (Grp)op is called stack if it satisfies axioms of
a sheaf in the etale (or fppf) topology.
Examples of stacks are given by quotient stacks X/G, 1 and the stack of
coherent sheaves MX.
In analogy to algebraic spaces we define algebraic- (resp. DM-) stacks to
be stacks X which can be represented as a quotient of an algebraic space U
by an action of a groupoid R:
R
1
/
2
/ U/ X,
such that i are smooth (resp. etale), and R ! U × U is separated and
quasi-compact. If R is a groupoid with only identity morphisms, then we
recover the notion of an equivalence relation.
A morphism of stacks f : X ! Y is said to be representable if for all
schemes U and maps U ! Y , the fibered product U ×Y X is an algebraic
space. If (P) is a property of algebraic spaces, that is local in the etale
topology, then we say that f has (P) if f is representable and U ×Y X ! U
has (P).
2.5. Moduli of sheaves. The goal of this talk is to show, that the 2-functor
MX is an algebraic stack locally of finite type over C.
This involves the following steps:
1In this seminar, the symbol [X/G] is reserved for the class of the quotient stack in the
Grothendieck group K(St).

6. 6 DANIEL HUYBRECHTS, HEINRICH HARTMANN
• Verify the stack axioms: This uses descend for coherent sheaves in
fppf topology [Vis05].
• Construct an atlas for MX. The stategy is to write MX as a quo-tient
stack of a Quot-scheme.
Sketch the construction of the Quot-sheme, as quotient of a Grassmanian
[HL97].
Moreover we want to compare the moduli stackMX to the moduli spaces
of semi-stable coherent shevaves Mss
X(v, h), with v 2 N(X), h 2 Amp(X).
2.6. Grothendieck rings of stacks. This talk shall cover section 3 of
[Bri10b]: Grothendieck rings of stacks K(St).
The main goal is to prove the comparison theorem:
K(V ar/C)[L−1, (Li − 1)−1|i 1] = K(St/C).
This involves the following steps:
• In the definition of K(St) involves only stacks with affine automor-phism
groups.
• Theorem (Kresh): Every algebraic stack X with affine stabilizers
can be stratified by stacks of the form Y/Gln.
• This uses a famous theorem of Chevalley that characterizes affine
groups as those which do not allow a non-trivial homomorphism
onto an abelian varity.
Finally introduce the relative Grothendieck group K(St/S) for a stack S
and explain the basic properties.
2.7. The motivic Hall algebra. We introduce the stacksM(n) parametriz-ing
n-step filtrations of coherent sheaves E on X and the fundamental maps:
ai, b :M(n) −!MX =M(1).
Show that M(n) is an algebraic stack using the corresponding property of
MX and the relative Quot-scheme.
The motivic Hall algebra is defined as
H(X) = K(St/MX)
endowed with the convolution product x y = b (a1 × a2)(x).
The main result of this talk ist the associativity of the convolution prod-uct.
2.8. Integration map. Introduce the grading of H(X) indexed by the nu-merical
Grothendieck group = N(Coh(X)). Recall the definition of a
Poisson algebra and introduce the quantum torus Z[].
Define regular elements in H(X) and the semi-classical version Hsc(X) of
the Hall algebra. State the main theorems of this seminar:
5.1 Commutativity of Hsc(X).
5.2 Existence of the integration map
I : Hsc(X) −! Z[]
which is a homomorphism of Poisson algebras.

7. SEMINAR ON MOTIVIC HALL ALGEBRAS 7
The integration map depends on the choice of a constructible function
on the moduli stack MX. Recall the definition of such a function. The
coice = 1 already leads to interesting invariants. For Donaldson–Thomas
invariants one has to use the, so called, Behrend function.
The brave speaker might have a look at Theorem 5.3. proved in [JS10].
2.9. Fibers of the convolution map. This talk covers section 6 of
[Bri10b]: Introduce the extension functor k(E1,E2) and show repre-sentability
by the relative Ext-sheaf ExtOS (E1,E2).
This is used to prove Proposition 6.2. which states that the pullback of
the convolution map (a1, a2) to a sheme S
M(2)(f, g)
/M(2)
a1,a2
S
f,g
/MX ×MX
can be represented by stacks of the form
Ext1
OS (E2,E1)/HomOS (E2,E1)
after suitable stratification of S.
2.10. Proofs of 5.1 and 5.2. This talk is the highlight of the seminar:
Explain the proofs of Theorem 5.1 and 5.2. as presented in section 7. Include
Prop. 1 of [Mac74].
References
[BCE+12] Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi,
Lothar Göttsche, and Andrew Kresch. Algebraic Stacks. in preparation, avail-able
online., 2012.
[Bit04] F. Bittner. The universal euler characteristic for varieties of characteristic zero.
Compos. Math. 140, 2004.
[Bri10a] Tom Bridgeland. Hall algebras and curve counting invariants. arXiv:1002.4374,
2010.
[Bri10b] Tom Bridgeland. An introduction to motivic Hall algebras. arXiv:1002.4372,
2010.
[Góm01] Tomás L. Gómez. Algebraic stacks. Proc. Indian Acad. Sci. Math. Sci.,
111(1):1–31, 2001.
[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Grad-uate
Texts in Mathematics, No. 52.
[HL97] Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of
sheaves. Aspects of Mathematics, E31. Friedr. Vieweg Sohn, Braunschweig,
1997.
[JS10] Dominic Joyce and Y. Song. A theory of generalized Donaldson–Thomas in-variants.
arXiv:0810.5645, 2010.
[Knu70] D. Knutson. Algebraic Spaces. Lecture Notes in Mathematics. Springer, 1970.
[KS08] Maxim Kontsevich and Yan Soibelman. Stability structures, motivic
Donaldson–Thomas invariants and cluster ransformations. arXiv:0811.2435,
2008.
[LMB00] G. Laumon and L. Moret-Bailly. Champs algebriques. Ergebnisse der
Matehmatik. Springer, 2000.
[Mac74] R. D. MacPherson. Chern classes for singular algebraic varieties. Ann. of Math.
(2), 100:423–432, 1974.
[Pet06] C. Peters. Tata lectures on motivic hodge theory. available on his homepage,
2006.

8. 8 DANIEL HUYBRECHTS, HEINRICH HARTMANN
[PS06] C. Peters and J. Steenbrink. Mixed Hodge structures. Ergebnisse der
Matehmatik. Springer, 2006.
[Rin90] Claus Michael Ringel. Hall algebras and quantum groups. Invent. Math.,
101(3):583–591, 1990.
[Vis05] Angelo Vistoli. Grothendieck topologies, fibered categories and descent theory.
In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr.,
pages 1–104. Amer. Math. Soc., Providence, RI, 2005.