Homological Algebra and Divergent Series
We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits.
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| Цитувати: | Homological Algebra and Divergent Series / V. Gorbounov, V. Schechtman // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 36 назв. — англ. |
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| author | Gorbounov, V. Schechtman, V. |
| author_facet | Gorbounov, V. Schechtman, V. |
| citation_txt | Homological Algebra and Divergent Series / V. Gorbounov, V. Schechtman // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 36 назв. — англ. |
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| description | We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 034, 31 pages
Homological Algebra and Divergent Series?
Vassily GORBOUNOV † and Vadim SCHECHTMAN ‡
† Department of Mathematical Sciences, King’s College, University of Aberdeen,
Aberdeen, AB24 3UE, UK
E-mail: vgorb@maths.abdn.ac.uk
‡ Laboratoire de Mathématiques Emile Picard, Université Paul Sabatier, Toulouse, France
E-mail: schechtman@math.ups-tlse.fr
Received October 01, 2008, in final form March 04, 2009; Published online March 24, 2009
doi:10.3842/SIGMA.2009.034
Abstract. We study some features of infinite resolutions of Koszul algebras motivated by
the developments in the string theory initiated by Berkovits.
Key words: Koszul resolution; Koszul duality, divergent series
2000 Mathematics Subject Classification: 13D02; 14N99
To Friedrich Hirzebruch on his 80-th anniversary, with admiration
1 Introduction
1.1. This article consists of two parts. The first part is a simple exercise on Mellin transform.
The second one is a review on some numerical aspects of Koszul duality. An object which lies
behind the two parts is a Tate resolution of a commutative ring over a field of characteristic
zero.
Let us give some more details on the contents. In the first part we develop the elegant ideas,
due to physicists [5], which allow to define the numerical invariants of projective varieties, doing
a regularization of some divergent series connected with their homogeneous rings.
Let R0 = k[x0, . . . , xN ] be a polynomial algebra over a field k, f1, . . . , fp ∈ R0 homogeneous
elements of degrees di = deg fi > 0 which generate the ideal I = (f1, . . . , fp). Consider the
quotient algebra A = R0/I; it is graded
A =
∞
⊕
j=0
Aj .
Geometrically the projection R0 −→ A corresponds to a closed embedding i : X := ProjA ↪→
PN := Proj R0; the projective variety X is defined in PN by the equations f1 = 0, . . . , fp = 0.
The algebra A is called the homogeneous ring of X (it depends on the embedding into the
projective space).
Let us call a semi-free resolution of A the following data:
(i) An associative unital bi-graded k-algebra R =
∞
⊕
i,j=0
Rj
i (so Rj
i ·Rm
l ⊂ Rj+m
i+l ); the indexes i
and j will be called the homological degree and the polynomial degree respectively. We set
Ri := ⊕jR
j
i , Rj := ⊕iR
j
i . We introduce a structure of a superalgebra on R by defining the
parity to be equal to the parity of the homological degree.
The multiplication has to be super-commutative, i.e. for x ∈ Ri, y ∈ Rl we have xy =
(−1)ilyx.
?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection
is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html
mailto:vgorb@maths.abdn.ac.uk
mailto:schechtman@math.ups-tlse.fr
http://dx.doi.org/10.3842/SIGMA.2009.034
http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html
2 V. Gorbounov and V. Schechtman
R should be equipped with a differential d : R −→ R, d2 = 0, such that d(Rj
i ) ⊂ Rj
i−1 and
for x ∈ Ri, y ∈ R,
d(xy) = dx · y + (−1)ix · dy.
(ii) R0 should be equipped with a morphism of algebras ε : R0 −→ A such that ε(Rj
0) ⊂ Aj .
(iii) For each j ≥ 0, Rj should be a resolution of Aj , i.e. the complexes
· · · −→ Rj
2
d−→ Rj
1
d−→ Rj
0
ε−→ Aj −→ 0
should be exact.
(iv) If one forgets the differential d, R should be a polynomial (super)algebra in homogeneous
generators. We suppose that in each polynomial degree one has a finite number of generators.
Semi-free resolutions exist for each A but they are not unique (sometimes they are called
Tate resolutions).
Example 1.1. Let us consider the Koszul complex
K(R0; f) := k[x0, . . . , xN ]⊗ Λ〈ξ1 · · · ξM 〉,
where Λ denotes the exterior algebra; the differential is defined by the formula dξj = fj . The
requirements (i), (ii), (iv) are fulfilled (the homological degree of xi is 1). The requirement (iii)
holds iff the sequence f = {f1, . . . , fM} is regular.
If this is the case, the Krull dimension of A is given by a simple formula
dim A = N + 1−M, (1.1)
where N + 1 is the number of xi and M is the number of ξj ; one can say that N + 1 − M is
the “superdimension” of the polynomial ring k[x0, . . . , xN ] ⊗ Λ〈ξ1 · · · ξM 〉 (the number of even
generators minus the number of odd generators).
In general one can consider a semi-free resolution of A as a natural replacement of the Koszul
complex. One of the main goals of this paper is to propose an analog of the equation (1.1) valid
for not necessarily complete intersections.
If our algebra A is not a complete intersection then a semi-free resolution will be infinite.
However, one can associate with such a resolution a sequence of integer numbers and to show
that dim A can be written as a “sum” of these numbers; this will be the required generalization
of (1.1). One puts the sum inside of the quotation marks since we have here a divergent series,
the sum of which is calculated be means of a regularization, following a classical procedure of
Riemann.
In fact, our formula is nothing else but the Mellin transform of the classical theorem by
Hilbert which says that dim A is equal to the order of pole of the Hilbert series of A at 1.
Moreover, we give a similar interpretation for the other numerical invariants of A such as the
degree. For details, see Sections 2.1–2.4.
Except for the case of the complete intersection which is rather seldom, there exists a wider
class of algebras which admit a remarkable explicit semi-free resolution. Namely, if A is a Koszul
algebra then one can take for R the Chevalley cochain complex of the graded Lie algebra L Koszul
dual to A; one can call it the Koszul–Chevalley resolution, cf. Section 3.4 for details.
As has been noted by many the notion of the Koszul duality is of fundamental importance
in Physics. For us a motivating reference is the work of M. Movshev and A. Schwarz. It turns
out it is closely related to the “gauge fields-strings duality” dear to Polyakov, cf. [28].
In the second part of this article we discuss a (probably) simplest nontrivial case of a Koszul
algebra: we take for A the homogeneous ring of the Veronese curve X = P1 ↪→ PN . This part
Homological Algebra and Divergent Series 3
contains no new results; rather it is a review of some classical and modern theorems related to
the Koszul–Chevalley resolution of A. This topic turns out remarkably rich; it revolves around
the Gauss cyclotomic identity. We see here Euler products, Witt theorem, Polya theory and
a formula of Polyakov. A particular case of a deep theorem by Kempf and Bezrukavnikov says
that A is a Koszul algebra. We will see in Theorem 3.17 that the “numerical” manifestation of
this fact is precisely the Gauss cyclotomic identity.
1.2. At the end of this introduction let us say a few words about the algebras interesting to
the physicists; we hope to return to these questions later on. In his seminal papers N. Berkovits
considers the algebra of functions on the space of pure spinors of dimension 10, the quotient of
the polynomial ring in 16 variables R0 = C[λ1, . . . , λ16] by the ideal generated by 10 quadratic
elements∑
α,β
λαγm
αβλβ , m = 0, . . . , 9.
The corresponding projective variety i : X = Proj A ↪→ P15 is the hermitian symmetric space
X = SO(10, C)/U(5). The canonical bundle is ωX = OX(−8) where OX(1) = i∗OP15 , cf. [13].
The Hilbert series of A is equal to
H(A; t) =
1− 10t2 + 16t3 − 16t5 + 10t6 − t8
(1− t)16
=
(1 + t)(1 + 4t + t2)
(1− t)11
.
The algebra A is Koszul1.
Let L =
∞
⊕
i=1
Li be the graded Lie algebra Koszul dual to A and C∗(L) its Chevalley cochain
complex – the Koszul–Chevalley resolution of A. As a graded algebra it is the tensor product
C∗(L) = S(L[1]∗) = SL∗1 ⊗ ΛL∗2 ⊗ SL∗3 ⊗ · · ·
(more precisely the inductive limit of finite products; here S denotes the symmetric algebra and
the star denotes the dual space). Let us consider a graded commutative algebra
C1(L) = C∗(L)⊗ ΛL∗1. (1.2)
One can define a differential on this algebra that will make a dga whose cohomology will be
H∗(C1(L)) = TorR0
∗ (A, C) =: Q. (1.3)
The dga C1(L) is quasi-isomorphic to the Berkovits algebra studied by Movshev and Schwarz,
cf. [24].
The graded commutative algebra Q is called the algebra of syzygies. For i ≥ 1 denote by
L≥i ⊂ L the graded Lie subalgebra ⊕
j≥i
Lj . It follows from the above description that Q is
isomorphic to the algebra H∗(L≥2; C) – this is a result contained in [13].
Generalizing this construction, consider for each i ≥ 1 a graded commutative algebra
Ci(L) = C∗(L)⊗ ΛL∗1 ⊗ SL∗2 ⊗ · · · ⊗ FiL
∗
i ,
where Fi = S if i is even and Λ otherwise. One expects that it is possible to introduce
a differential on Ci(L) which provides a dga whose cohomology is H∗(L≥i). For example, an
algebra quasi-isomorphic to C3(L) has been studied in a very interesting paper [1].
1The Kempf–Bezrukavnikov theorem says that the coordinate rings of homogeneous spaces of the form G/P
where G is semisimple complex and P is a parabolic, are Koszul. Another manifestation of this is the Kapranov’s
description (à la Beilinson and Bernstein–Gelfand–Gelfand) of the derived category of coherent sheaves on G/P .
4 V. Gorbounov and V. Schechtman
The algebra of syzygies Q is a remarkable object. It is finite-dimensional over C, Q =
3
⊕
i=0
Qi
and admits, after [13] a scalar product Qi ⊗ Q3−i −→ C compatible with the multiplication.
One can imagine Q as the cohomology of a smooth compact oriented variety of dimension 3.
A few words about the aim of the paper and the novelty of the results. The physicists are
interested in quantum strings fluctuating on the singular space Spec(A). To pass to strings
one has to study the chiral analogs of the above algebras, cf. [1, 3]. The construction of the
chiral analog of such algebras, the chiralization, can be accomplished with the use of free infinite
resolutions of the above algebras. In this paper we put together the properties of Koszul algebras
and their infinite resolutions which contribute to the properties of the chiral algebras [1, 3]. We
introduce and study in Part I the regularization technique following the insight from [5] which
allows to define a part of the above chiral algebra structure. The Part II contains samples
of calculations with the equivariant Hilbert series which one needs to define the action of an
appropriate Kac–Moody on the chiralization of the above algebras. We are planning to return
to chiralization of these objects in a separate publication.
Otherwise, one can regard this topic from the point of view of the string topology of Chas and
Sullivan. The string theory is the study of spaces of loops. Let M be a closed oriented manifold
of dimension d, ΩM its free loop space. A fundamental theorem of Chas and Sullivan, cf. [8],
says that the homology of M shifted by d
H∗(ΩM) := H∗+d(ΩM)
admits a structure of a Batalin–Vilkovisky (BV) algebra. If M is simply connected then H∗(ΩM)
is isomorphic to the Hochschild cohomology
H∗(ΩM) = HH∗(C∗(M), C∗(M)),
where C∗(M) is the complex of singular cochains of M , cf. [7, 21].
An algebraic counterpart of the Chas–Sullivan theory is the following remarkable result (“the
cyclic Deligne conjecture”), cf. [18, 32]: let A be an associative algebra with an invariant scalar
product; then the complex of Hochschild cochains CH∗(A) admits a structure of a homotopy
BV algebra.
Returning to the Berkovits algebras C1(L), Q, cf. (1.2), (1.3), it would be very interesting to
study their Hochschild (as well as cyclic) cohomology. In view of the previous remarks, it should
be closely related to the state space of the Berkovits string2.
It is worth mentioning certain analogy between the “chiral” and “topological” points of
view. For example the Deligne conjecture: the Hochschild cochain complex CH∗(A) of an
associative algebra A is a homotopy Gerstenhaber algebra (or more precisely, an algebra over
the operad e2 of chains of little discs), resembles the Lian–Zuckerman conjecture [20]: the
space of a topological (i.e. N = 2 supersymmetric) vertex algebra is a homotopy Gerstenhaber
algebra. If an associative algebra is equipped with an invariant scalar product then CH∗(A)
becomes a homotopy BV algebra. Which complementary structure one needs on a topological
vertex algebra to become a homotopy BV algebra? In other words, does there exist a vertex
counterpart of the cyclic Deligne conjecture?
2It seems that the Polyakov’s gauge fields-strings correspondence translates in algebra into the assertion: the
Hochschild cohomologies of Koszul dual algebras are isomorphic
Homological Algebra and Divergent Series 5
First part
2 Numerical invariants and regularization
“Les séries divergentes sont en général quelque chose de bien fatal, et
c’est une honte qu’on ose y fonder aucune démonstration. On peut
démontrer tout ce qu’on veut en les employant. . . ”
N.-H. Abel, a letter to Holmboe, January 16, 1826
2.1 Semi-free resolutions and Hilbert series
2.1.1. Let
A = R0/(f1, . . . , fM ) =
∞
⊕
i=0
Ai,
and i : X := ProjA ↪→ PN := ProjR0 be as in the Introduction; denote by L = i∗OPN (1) the
corresponding very ample line bundle.
Set hi = dimk Ai and let
H(A; t) =
∞∑
i=0
hit
i
be the Hilbert series; we suppose X to be connected, so H(A; 0) = h0 = 1. Let
R : · · · −→ R2 −→ R1 −→ R0 −→ A −→ 0 (2.1)
be a semi-free resolution of A. For R1 one can take R1 =
M
⊕
l=1
R0ξl, where the variables ξl are
odd and dξl = fl, so ξl ∈ Rdi
1 .
Example 2.1. Suppose that f1, . . . , fM is a regular sequence, that is, X is a complete intersec-
tion. Then one can take for R the Koszul complex, Ri = R0 ⊗ ΛiW , where W =
M
⊕
l=1
kξl.
The exact sequence (2.1) immediately gives an expression of the Hilbert series:
H(A; t) =
M∏
l=1
(1− tdl)
(1− t)N+1
=
M∏
l=1
(1 + t + · · ·+ tdl−1)
(1− t)d+1
=
P (t)
(1− t)d+1
, (2.2)
where
d = N −M = dim X. (2.3)
One notes that
deg X =
M∏
l=1
dl = P (1). (2.4)
6 V. Gorbounov and V. Schechtman
2.1.2. Let us return to the general case. Recall that R as an algebra is a polynomial super-
algebra in homogeneous variables. Let us define integer numbers an = the number of even
generators of polynomial degree n minus the number of odd generators of polynomial degree n.
Then the exact sequence (2.1) gives a product expression
H(t) = H(A; t) =
∞∏
n=1
(1− tn)−an (2.5)
similar to (2.2).
This formula shows in particular that the numbers an depend only on A but not on the
resolution R.
Similarly to (2.3) and (2.4) we want to show that
dim X + 1 = “a1 + a2 + a3 + · · · ”, (2.6)
log(deg X) = −“(log 1) · a1 + (log 2) · a2 + (log 3) · a3 + · · · ”. (2.7)
It is natural also to consider the “higher moments”:
“
∞∑
n=1
nlan”, l ≥ 1. (2.8)
However, the series at the right hand side are divergent, so one puts the sums in the quotation
marks. Our aim will be to perform the summation of these series3. There are several ways of
doing the summation.
For example, one can write a Lambert series
f(t) =
∞∑
n=1
an ·
nte−nt
1− e−nt
,
cf. [14, App. IV, (1.1)].
This series diverges if |t| is small; but one can show that if |t| is sufficiently big then the series
absolutely converges and t−1f(t) is a rational function of y = e−t. Therefore one extend f(t) to
the complex plane; this function will be holomorphic at t = 0 (i.e. and y = 1). It is natural to
define
“
∑
an” := f(0) = res
t=0
f(t)
t
.
The identity (2.6) will hold true. Moreover, the higher moments (2.8) may be expressed in terms
of the coefficients of the Taylor series of f(t) and 0; this is noted in [5].
Another classical way of regularization is using the Mellin transform and working with the
Dirichlet series. This is what we are going to do.
2.2 Möbius inversion
2.2.1. Take the logarithm of (2.5):
log H(t) = −
∞∑
n=1
an log (1− tn) =
∞∑
n=1
∞∑
l=1
an
tln
k
=
∞∑
m=1
∑
l|m
al
m/l
· tm (2.9)
3The classical sources about divergent series are the books [14, 31].
Homological Algebra and Divergent Series 7
and then the derivative:
tH ′(t)
H(t)
=
∞∑
n=1
nantn
1− tn
=
∞∑
m=1
∑
l|m
lal
· tm. (2.10)
In other words, if one denotes
tH ′(t)
H(t)
=
∞∑
m=1
bmtm, (2.11)
then
bm =
∑
l|m
lal. (2.12)
2.2.2. Recall that the Möbius function µ : N+ = {1, 2, . . .} −→ {−1, 0, 1} is defined by:
µ(1) = 1, µ(n) = (−1)l if n = p1p2 · · · pl is a product of l distinct prime numbers, and µ(n) = 0
if n contains squares.
Another definition is by a generating series: if one defines the Riemann ζ function by the
Euler product
ζ(s) =
∞∑
n=1
n−s =
∏
p prime
(
1− p−s
)−1
,
then
ζ(s)−1 =
∏
p prime
(
1− p−s
)
=
∞∑
n=1
µ(n)n−s. (2.13)
The Möbius inversion formula says that if f : N+ −→ C is a function and if a function
g : N+ −→ C is defined by
g(m) =
∑
l|m
f(l),
then
f(m) =
∑
l|m
µ(m/l)g(l). (2.14)
Applying this formula to the function f(m) = mam, we get
mam =
∑
l|m
µ(m/l)bl. (2.15)
2.3 Dirichlet series
2.3.1. Let P (t) be a polynomial with complex coefficients such that P (0) = 1, so one can
write
P (t) =
p∏
l=1
(1− αlt). (2.16)
8 V. Gorbounov and V. Schechtman
Consider the product (2.5) for P (t):
P (t) =
∞∏
n=1
(1− tn)−an .
So if
tP ′(t)
P (t)
=
∞∑
m=1
bmtm,
then
mam =
∑
l|m
µ(m/l)bl.
On the other hand
tP ′(t)
P (t)
= −
p∑
r=1
αrt
1− αrt
= −
p∑
r=1
∞∑
m=1
αm
r tm, (2.17)
i.e.
bm = −
p∑
r=1
αm
r ,
wherefrom
am = −m−1
p∑
r=1
∑
l|m
µ(m/l)αl
r.
One puts
ρ(P ) = max
r
|αr|.
It follows that if ρ(P ) > 1, then |am| grows as fast as m−1ρ(P )m.
2.3.2. Let us consider the Dirichlet series
∞∑
n=1
ann−s.
We see that if ρ(P ) ≤ 1 then this series absolutely converges for Re(s) > 1, and if ρ(P ) > 1, it
diverges for all s; in this case (which in fact is interesting to us) one has to do something else.
Let us write formally with [5]:
−
∞∑
n=1
ann−s =
∞∑
n=1
p∑
r=1
∑
l|n
µ(n/l)αl
rn
−1−s =
p∑
r=1
∞∑
l,m=1
µ(m)αl
r(lm)−1−s
=
∞∑
m=1
µ(m)m−1−s ·
p∑
r=1
∞∑
l=1
αl
rl
−1−s = ζ(s + 1)−1
p∑
r=1
∞∑
l=1
αl
rl
−1−s.
Now if we rewrite, after Riemann,
l−1−s =
1
Γ(s + 1)
∫ ∞
0
e−lttsdt,
Homological Algebra and Divergent Series 9
wherefrom
∞∑
l=1
αl
rl
−1−s =
1
Γ(s + 1)
∫ ∞
0
∞∑
l=1
αl
re
−lttsdt =
1
Γ(s + 1)
∫ ∞
0
αre
−tts
1− αre−t
dt
(justified if |αr| < 1). Because
e−t
1− αre−t
=
1
et − αr
,
the integral∫ ∞
0
αre
−tts
1− αre−t
dt
absolutely converges for Re(s) > −1 if αr 6∈ [0, 1] (since P (1) = 1, one has αr 6= 0 automatically).
One notes that
p∑
r=1
αre
−t
1− αre−t
= −e−tP ′(e−t)
P (e−t)
. (2.18)
On the other hand, let us write the functional equation for ζ(s) (cf. [34, 13.151]):
21−sΓ(s)ζ(s) cos(πs/2) = πsζ(1− s)
and replace s by s + 1:
2−sΓ(s + 1)ζ(s + 1) cos(π(s + 1)/2) = πs+1ζ(−s),
i.e.
1
Γ(s + 1)ζ(s + 1)
= −2−sπ−s−1 sin(πs/2)ζ(−s)−1.
It follows:
∞∑
n=1
ann−s =
1
Γ(s + 1)ζ(s + 1)
∫ ∞
0
e−tP ′(e−t)
P (e−t)
tsdt
= −2−sπ−s−1 sin(πs/2)
ζ(−s)
∫ ∞
0
e−tP ′(e−t)
P (e−t)
tsdt.
2.3.3. Consider the last integral
IP (s) :=
∫ ∞
0
e−tP ′(e−t)
P (e−t)
tsdt.
It is well defined and represents a holomorphic function of s on the half-plane Re(s) > −1 as
soon as the condition (2.19) below is verified:
The roots of P (t) do not lie in the segment [0, 1]. (2.19)
As it was kindly pointed out by one of the referees the condition (2.19) holds. Indeed, since the
algebra A is affine, it has polynomial growth, so that the radius of convergence of the series
H(A, t) is 0 ≤ t < 1. Because the coefficients of the formal power series H(A, t) are nonnegative,
we have H(A, t) > 0 for all 0 ≤ t < 1. It follows that P (A, t) > 0 with H(A, t). Also, the is
function H(A, t) is rational (Hilbert–Serre), so that it has a pole at t = 1 of the order d + 1.
This means that the value P (A, 1) is nonzero as well.
10 V. Gorbounov and V. Schechtman
2.3.4. Following Riemann and using the notations of [34], consider the integral
JP (s) =
∫ (0+)
∞
e−tP ′(e−t)
P (e−t)
(−t)sdt, (2.20)
where
∫ (0+)
∞ denotes the integral along the contour
C = {t = a + εi, ∞ > a ≥ ε} ∪ {t =
√
2εeiθ, π/4 ≤ θ ≤ 7π/4}
∪ {t = a− εi, ε ≤ a < ∞}, (2.21)
where ε > 0 is sufficiently small. Here (−t)s = es log(−t), and we use the branch of the logarithm
π ≥ arg t ≥ −π on the small circle. Then
JP (s) = 2i sin(πs)IP (s),
cf. [34, 12.22]. But JP (s) is an entire function. Therefore, if one sets
zP (s) = −2−sπ−s−1 sin(πs/2)
ζ(−s)
· 1
2i sin(πs)
JP (s) =
i · 2−s−2π−s−1
cos(πs/2)ζ(−s)
JP (s), (2.22)
this defines the analytic continuation of (2.20) to a meromorphic function on the complex plane.
2.3.5. Let us return to the situation of Section 2.1; it is known that the Hilbert series is
a rational function of the form
H(A; t) =
P (A; t)
(1− t)d+1
,
where d = dim X = dim A− 1 and P (A; t) is a polynomial with integer coefficients (cf. a simple
Example 2.2). One has P (A; 0) = H(A; 0) = 1 since we suppose that X connected.
In view of the preceding discussion we define
z(X,L; s) =
1
Γ(s + 1)ζ(s + 1)
∫ ∞
0
e−tH ′(A; e−t)
H(A; e−t)
tsdt = d + 1 + zP (s) (2.23)
(because Γ(s + 1)ζ(s + 1) =
∫∞
0 (ts/(et − 1))dt). So if
H(A; t) =
∞∏
n=1
(1− tn)−An , (2.24)
then
z(X,L; s) = “
∞∑
n=1
Ann−s”,
the quotation marks mean that the sum is regularized. We define “the Dirichlet summation”:
“
∞∑
n=1
An” := z(X,L; 0),
and
“
∞∑
n=1
log n ·An” := −z′(X,L; 0),
Homological Algebra and Divergent Series 11
2.3.6. We have ζ(0) = −1/2 and ζ ′(0) = − log
√
2π 6= 0 (cf. [33, Ch. VII, § 9]). The integral
IP (0) is well defined (below we shall compute its value), so the formula (2.23) gives zP (0) = 0,
wherefrom
z(X,L; 0) = d + 1,
cf. (2.6). On the other hand, z′(X,L; s) = z′P (s), and a unique term giving a nontrivial contri-
bution into z′(X,L; 0), is
−
(
2−sπ−s−1 · (π/2) cos(πs/2)
ζ(−s)
)∣∣∣∣
s=0
· IP (0).
The first factor gives 1, whereas
IP (0) =
∫ ∞
0
e−tP ′(e−t)
P (e−t)
dt = −
∫ ∞
0
d log P (e−t)
dt
dt
= − log P (e−t)
∣∣∞
0
= − log P (0) + log P (1) = log P (1),
wherefrom
z′(X,L; 0) = log P (1).
But it is known that P (1) = deg X, which implies
deg X = ez′(X,L;0),
cf. (2.7).
Example 2.2 (Veronese curve and the Weil zeta function). Let X = P1, L = O(q + 1),
q ≥ 1. Then An = Γ(X,O((q + 1)n), which implies
H(A; t) =
∞∑
n=0
((q + 1)n + 1)tn =
1 + qt
(1− t)2
,
and
P (t) = 1 + qt =
∞∏
m=1
(1− tm)pm(−q),
where
pm(x) =
1
m
∑
l|m
µ(m/l)xl.
It is well known that if q is a prime power then pm(q) is equal to the number of monic irreducible
polynomials of degree m in Fq[T ]. In fact, we have
ZFq(A1, t) =
1
P (−t)
=
∞∏
m=1
(1− tm)−pm(q)
– this is the Euler product of the zeta function of the affine line over the finite field Fq.
(The idea that the Hilbert series of a variety is equal to the Weil zeta function of another
one seems very strange. Cf. [12] however4.)
4We are grateful to Yu.I. Manin who showed us this very interesting paper.
12 V. Gorbounov and V. Schechtman
2.4 Values at negative points
2.4.1. Let s = −m be a negative integer. In this case one can close the contour in the
integral (2.20):
JP (−m) = (−1)m
∫
|t|=ε
Q(t)t−mdt = 2πi · (−1)m res
t=0
Q(t)t−m,
where
Q(t) =
e−tP ′(e−t)
P (e−t)
= −d log P (e−t)
dt
.
So if Q(t) = q0 + q1t/1! + q2t
2/2! + · · · is the Taylor series at 0 then
JP (−m− 1) = 2πi · (−1)m+1 qm
m!
.
Here are some first values:
q0 =
P ′(1)
P (1)
, q1 = −q0 + q2
0 −
P ′′(1)
P (1)
,
q2 = q0 − 3q2
0 + 2q3
0 + 3(1− q0)
P ′′(1)
P (1)
+
P ′′′(1)
P (1)
.
2.4.2. One can say this in a different way: if
P (t) = 1 + c1t + · · ·+ cDtD,
and we imagine the numbers ci as “Chern classes” then qm will be the coefficients of the
logarithm of the Todd genus . . .
2.4.3. Now applying (2.22): for m = 1, the function ζ(s) has a simple pole at s = 1 with the
residue 1, whence
zP (−1) =
2P ′(1)
P (1)
. (2.25)
Remark 2.3. Suppose that P (t) is a “reciprocal” polynomial, i.e.
tdeg P P (1/t) = P (t),
which happens rather often. In this case it is easy to see that
2P ′(1)
P (1)
= deg P,
so zP (−1) = deg P .
2.4.4. For m > 1 one has to distinguish two cases.
(a) If m = 2l is even then cos(πl) = (−1)l, and after Euler,
ζ(2l) = (−1)l−1 (2π)2l
2(2l)!
b2l,
Homological Algebra and Divergent Series 13
where the Bernoulli numbers are defined by the generating series
S
eS − 1
= 1− S
2
+
∞∑
l=1
b2l
(2l)!
S2l.
Here are some first values:
b2 =
1
6
, b4 = − 1
30
, b6 =
1
42
, b8 = − 1
30
.
It follows:
zP (−2l) =
2lq2l−1
b2l
.
So if the numbers an are defined by (2.24) then
“
∞∑
n=1
n2lan” = z(X,L;−2l) = d + 1 +
2lq2l−1
b2l
(2.26)
– a formula found empirically in [5, (4.21)].
(b) If m = 2l + 1, l > 0 then zP (s) has a simple pole at s = −m, with the residue
res
s=−2l−1
zP (s) = (−1)l (2π)2lq2l
(2l + 1)!ζ(2l + 1)
.
2.4.5. Hurwitz zeta function.
(a) We have JP (s) =
p∑
r=1
Jαr(s) where
Jα(s) = −
∫ (0+)
∞
α
et − α
(−t)sdt.
Let us choose a value β = log α. The expression under the integral fα(t, s) = (α/et − α) · (−t)s
has the poles at points tn = β + 2πin, n ∈ Z. By the Cauchy formula,
Jα(s) + 2πi
∞∑
n=−∞
res
t=tn
fα(t, s) = 0
if Re(s) < −1, then
Jα(s) = 2πi
∞∑
n=−∞
(−β − 2πin)s, (2.27)
which gives JP (s) in the form of a series absolutely convergent for Re(s) < −1, expressible in
terms of the Hurwitz zeta functions.
14 V. Gorbounov and V. Schechtman
(b)
Example 2.4. Take P (t) = 1 + pt, where p ∈ Z>0, cf. Example 2.2. Then q0 = p/(p + 1),
q1 = −p/(p + 1)2 (cf. Section 2.4.1), so
JP (−2)
2πi
= − p
(p + 1)2
.
On the other hand, take β = log(−p) = log p + πi, and the above series will be written as
∞∑
n=−∞
(− log p− πi− 2πin)−2 =
∞∑
n=0
{
(− log p + (2n + 1)πi)−2 + (− log p− (2n + 1)πi)−2
}
,
so one came to an identity
∞∑
n=0
{
(− log p + (2n + 1)πi)−2 + (− log p− (2n + 1)πi)−2
}
= − p
(p + 1)2
. (2.28)
For example, for p = 3, −p/(p+1)2 = −0, 1875, whereas an approximate value given by MAPLE
is
1000∑
n=0
{
(− log p + (2n + 1)πi)−2 + (− log p− (2n + 1)πi)−2
}
= −0, 187449 . . . ;
we see that the convergence is very slow.
(c) If s = −1, the series (2.27) in our example is still “Eisenstein summable”, and on gets
a rational value:
∞∑
n=0
{
(− log p + (2n + 1)πi)−1 + (− log p− (2n + 1)πi)−1
}
= −2(p− 1)
p + 1
. (2.29)
This is an easy corollary of the decomposition of cot z into simple fractions5. But this value is
different from −q0 = −p/(p + 1): we have “an anomaly”.
2.4.6. Suppose that our variety X is smooth, H i(X,L⊗n) = 0 for all i > 0 and n ≥ 0. Then
An = Γ(X,L⊗n), and we can switch on the Riemann–Roch–Hirzebruch [15]:
hn = dim H0(X,L⊗n) =
∫
X
enc1(L)Td(TX),
where TX denotes the tangent bundle, Td the Todd genus, given for line bundle E by the
formula,
Td (E) =
c1(E)
1− e−c1(E)
,
so Td (TX) = 1 + Td1(TX) + Td2(TX) + · · · , where
Td1(TX) =
c1(TX)
2
, Td2(TX) =
c2
1(TX) + c2(TX)
12
.
5We thank Oleg Ogievietsky who has noted this.
Homological Algebra and Divergent Series 15
One writes: enc1(L) =
∞∑
i=0
c1(L)ini/i!, so hn = R(n), where
R(t) =
(∫
X
c1(L)d
)
· td
d!
+
(∫
X
c1(L)d−1c1(TX)
)
· td−1
2(d− 1)!
+
+
(∫
X
c1(L)d−2(c1(TX)2 + c2(TX))
)
· td−2
12(d− 2)!
+ · · · = rd
td
d!
+ rd−1
td−1
(d− 1)!
+ · · ·
is a polynomial of degree d = dim X, the Hilbert polynomial of the ring A; it is a polynomial
with rational coefficients which takes integer values at integer argument, whence
R(t) =
d∑
i=0
(−1)d−ied−i
(
t + i
i
)
,
where e0, e1, . . . , ed ∈ Z. For example,
e0 = rd =
∫
X
c1(L)d,
e1 =
d + 1
2
rd − rd−1 =
1
2
∫
X
(
(d + 1)c1(L)d − c1(L)d−1c1(TX)
)
,
e2 = rd−2 −
d
2
rd−1 +
(d + 1)(3d− 2)
24
rd
(we use
∑
1≤i<j≤d
ij = d(d2 − 1)(3d + 2)/24).
On the other hand it is known that
ei =
P (i)(1)
i!
.
It follows:
ez′(X,L;0) = P (1) = e0 =
∫
X
c1(L)d,
so one finds the degree of X. Next
zP (−1) = −2P ′(1)
P (1)
= −2e1
e0
= − 1
e0
∫
X
(
(d + 1)c1(L)d − c1(L)d−1c1(TX)
)
= −d− 1 +
1
e0
∫
X
c1(L)d−1c1(TX),
i.e.
z(X,L;−1) =
1
e0
∫
X
c1(L)d−1c1(TX),
etc.
One can put these calculations into a formal “generating function” for the moments
Ml = “
∞∑
n=1
nlan”.
Note that from the formulas above and using the expansion
log
(
1− et
)
= log(−t) +
t
2
+
∞∑
l=1
b2l
2l(2l)!
t2l
16 V. Gorbounov and V. Schechtman
one concludes the formal identity:
− log P (et) = “
∞∑
l=0
log(l)al + M1
t
2
+
∞∑
l=1
(M2l)
b2l
2l(2l)!
t2l”. (2.30)
The following elegant formula is due to F. Hirzebruch [16]: the polynomial P (t) is a charac-
teristic number defined as
P (t) =
∫
X
(1− t)e(1−t)c1Â((1− t)2p1, . . . , (1− t)2ip2i, . . . )
1− te(1−t)g
,
where c1, p1, p2, . . . are the first Chern class and the Pontryagin classes of the manifold X.
From this it is immediate to express the left hand side of (2.30) as a characteristic number.
Indeed
log P (et) = log
∫
X
(1− et)e(1−et)c1Â((1− et)2p1, . . . , (1− et)2ip2i, . . . )
1− et+(1−et)
.
In order to expand the right hand side into a series of t observe that the constant term of the
series under the logarithm is∫
X
1
1− g
= P (1).
Therefore
log
1∫
X
1
1−g
∫
X
(1− et)e(1−et)c1Â((1− et)2p1, . . . , (1− et)2ip2i, . . . )
1− et+(1−et)
is a series in t equal to log P (et)− log P (1). This gives the desired generating function.
Example 2.5. Let Xg be the moduli space of semi-stable vector bundles of rank 2 with tri-
vial determinant over a Riemann surface of genus g; it carries the canonical determinant line
bundle Lg, cf. [2]. Consider a graded algebra
Ag =
∞
⊕
i=0
H0
(
Xg,L⊗i
g
)
(we thank Peter Zograf who proposed to consider this example).
The coefficients of the Hilbert series Hg(t) = H(Ag; t) can be calculated using the Verlinde
formula [2]. Here are some first examples, cf. [36]:
H2(t) =
1
(1− t)4
, H3(t) =
1 + t + t2 + t3
(1− t)7
=
1− t4
(1− t)8
, (2.31)
H4(t) =
1 + 6t + 21t2 + 40t3 + 21t4 + 6t5 + t6
(1− t)10
. (2.32)
The highest coefficient rd/d! of the Hilbert polynomial has been calculated in [Wi], who found
the value 2ζ(2g − 2)/(2π2)g−1; here d = dim Xg = 3g − 3, whence
Pg(1) = (−1)g (3g − 3)!
(2g − 2)!
b2g−2 = (−1)g+1 (3g − 3)!
(2g − 1)!
ζ(−2g + 3) (2.33)
(one denotes Pg(t) = Hg(t)(1 − t)d+1). For example, for g = 4 one finds P4(1) = 96, which is
compatible with (2.32).
Homological Algebra and Divergent Series 17
The above discussion gives a strange expression of this number as a regularized infinite
product (starting from g = 4), of the form
∞∏
n=1
nan . For example, the beginning of the product
for P4(t) will be:
P4(t) = (1− t)−6
(
1− t3
)16(1− t4
)9(1− t5
)−144(1− t6
)360
×
(
1− t8
)−2259(1− t9
)3920 · · · , (2.34)
whence
ζ(−5) = −9!
7!
· “163−164−951446−360822599−3920 · · · ”.
Second part
3 Koszulness and infinite resolutions
3.1 The Veronese ring
3.1.1. Let us fix an integer b ≥ 0, and consider the Veronese embedding
ib : P1 −→ Pb+1
defined in coordinates by the formula
ib(u0 : u1) = (x0 : · · · : xb+1) :=
(
ub+1
0 : ub
0u1 : · · · : ub+1
1
)
.
If t = u1/u0 ∈ A1 ⊂ P1, then
ib(t) =
(
t, t2, . . . , tb+1
)
∈ Ab+1 ⊂ Pb+1.
The image Xb := ib(P1) ⊂ Pb+1 of this embedding is called the Veronese curve (or the moment
curve, in view of the last formula).
This curve can be defined in Pb+1 by the equations xixj−xkxl = 0 if i+j = k+ l. A minimal
system of equations consists of b(b + 1)/2 quadratic equations:
fij := xixj − xi+1xj−1 = 0, 0 ≤ i ≤ b− 1, i + 2 ≤ j ≤ b + 1.
Example 3.1. i0 = IdP1 . If b = 1, X1 ⊂ P2 = {(x0 : x1 : x2)} is defined by one equation:
x0x2 − x2
1 = 0.
The curve X3 ⊂ P4 is defined by 3 equations
x0x2 − x2
1 = 0; x0x3 − x1x2 = 0; x1x3 − x2
2 = 0.
3.1.2. We have i∗bOPb+1(1) = OP1(b + 1), all the higher cohomology of this sheaf vanish, and
Xb = Spec Proj Ab, where
Ab =
∞
⊕
n=0
Γ
(
P1,O((b + 1)n
)
= C[x0, . . . , xb+1]/(fij).
The Hilbert series of the above algebra is as follows:
H(Ab; t) =
∞∑
n=0
((b + 1)n + 1)tn =
1 + bt
(1− t)2
. (3.1)
18 V. Gorbounov and V. Schechtman
It is not difficult to figure out the q-analogue of (3.1): if we set [b]q := (qb − q−b)/(q − q−1),
then:
Hq(Ab; t) :=
∞∑
n=0
[(b + 1)n + 1]qtn =
(
q − q−1
)−1 ·
∑
n
{
qq(b+1)ntn − q−1q−(b+1)ntn
}
=
(
q − q−1
)−1 ·
(
q
1− qb+1t
− q−1
1− q−b−1t
)
=
1 + [b]qt
(1− qb+1t)(1− q−b−1t)
. (3.2)
We will use these formulas later in Section 3.4.4.
Example 3.2. For b = 1 we have,
H(A1; t) =
1 + t
(1− t)2
=
1− t2
(1− t)3
, (3.3)
the ring A1 is a cone:
A1 = C[x0, x1, x2]/
(
x0x2 − x2
1
)
= B/(f).
It admits a dga resolution of length 1: the Koszul complex:
K·(B; f) : 0 −→ B · e −→ B −→ A −→ 0, d(e) = f.
Therefore
K·(B; f) = B ⊗ Λ·〈e〉 = C[x0, x1, x2; e]
as a graded algebra; the homological degree of xi is 0 and of e is 1. These correspond to the
exponents −3 and 1 in (3.3).
There is another interpretation of the Koszul complex. Let V be a vector space
2
⊕
i=0
C · xi,
then B = S·V . Consider the dual space V ∗ =
2
⊕
i=0
C · yi. Define a graded Lie algebra L on 3
generators yi, i = 0, 1, 2, of degrees 1, obeying 5 relations:
[y0, y0] = 0, [y0, y1] + [y1, y0] = 0, [y0, y2] + [y1, y1] + [y2, y0] = 0,
[y1, y2] + [y2, y1] = 0, [y2, y2] = 0.
It is concentrated in degrees 1 and 2: L = L1 ⊕ L2, where L1 = V ∗ and L2 = C · [y1, y1].
If L′ = C · y1 ⊕C · [y1, y1] ⊂ L is a Lie subalgebra generated by y1, then L′ is free (sic!); it is
an ideal, and the quotient algebra L̄ = L/L′ is Abelian, on 2 generators ȳ0, ȳ2.
Consider the complex of the Chevalley cochains:
C ·(L) = Λ·(L∗) = S·L∗1 ⊗ Λ·L∗2 = S·V ⊗ Λ·〈e〉, e = [y1, y1]∗
then one can identify C ·(L) with K(B; f).
Example 3.3. Starting from b = 2 the product decomposition of the Hilbert series becomes
infinite: for example, if b = 2
H(A2; t) =
1 + 2t
(1− t)2
=
(1− t2)3(1− t4)3(1− t6)11(1− t8)30(1− t10)105 · · ·
(1− t)4(1− t3)2(1− t5)6(1− t7)18(1− t9)56 · · ·
.
The first two exponents are “koszul”: the number of unknowns and the number of equations.
The exponents grow exponentially.
Homological Algebra and Divergent Series 19
3.2 Gauss cyclotomic identity
3.2.1. Necklaces. The necklace polynomial is defined as
Mn(x) =
1
n
∑
d|n
µ(d)xn/d.
Example 3.4. If p is a prime, then Mp(x) = (xp − x)/p.
Let a necklace c be made of n beads; suppose that each bead can be one of m colors. A necklace
is called primitive if it is not of the form c = dc′ where d|n, d > 1.
Theorem 3.5 (C. Moreau, 1872). The number of primitive necklaces made of n beads in b
colors is equal to Mn(b).
The proof is an exercise. C. Moreau was an artillery captain from Constantine, cf. [21].
Corollary 3.6. The number of all necklaces made of n beads in b colors is equal to Φn(b), where
Φn(x) =
1
n
∑
d|n
φ(d)xn/d,
φ(d) is the Euler function.
Proof. If this number is C(n; b) then
C(n; b) =
∑
l|n
Ml(b) =
∑
l|n
∑
d|l
µ(d)
d
· bl/d
l/d
(we set p = l/d)
=
∑
p|n
bp ·
1
p
∑
d|(n/p)
µ(d)
d
or, ∑
d|(n/p)
µ(d)
d
=
φ(n/p)
n/p
,
which proves the corollary. �
3.2.2. A theorem of Pólya. Following Polyakov (cf. [28]), one can consider the same numbers
from the point of view of Pólya theory [27].
Suppose we are given two finite sets X and Y as well as a weight function w : Y → N. If
n = |X| , without loss of generality we can assume that X = {1, 2, . . . , n}. Consider the set of
all mappings F = {f | f : X → Y }. We can define the weight of a function f ∈ F to be
w(f) =
∑
x∈X
w (f(x)) .
Every subgroup of the symmetric group on n elements, Sn, acts on X through permutations.
If A is one such subgroup, an equivalence relation ∼A on F is defined as f ∼A g ⇐⇒ f = g ◦ a
for some a ∈ A.
Denote by [f ] = {g ∈ F | f ∼A g} the equivalence class of f with respect to this equivalence
relation. [f ] is also called the orbit of f . Since each a ∈ A acts bijectively on X, then
w(g) =
∑
x∈X
w (g(x)) =
∑
x∈X
w (g(a ◦ x))) =
∑
x∈X
w (f(x)) = w(f).
20 V. Gorbounov and V. Schechtman
Therefore we can safely define w([f ]) = w(f). In other words, permuting the summands of
a sum does not change the value of the sum.
Let ck = |{y ∈ Y | w(y) = k}| be the number of elements of Y of weight k. The generating
function by weight of the source objects is c(t) =
∑
k ck · tk. Let Ck = |{[f ] | w([f ]) = k}| be
the number of orbits of weight k. The generating function of the filled slot configurations is
C(t) =
∑
k Ck · tk.
Theorem 3.7. Given all the above definitions, Pólya’s enumeration theorem asserts that
C(t) = Z(A)
(
c(t), c
(
t2
)
, . . . , c(tn)
)
,
where Z(A) is the cycle index (Zyklenzeiger) of A
Z(A)(t1, t2, . . . , tn) =
1
|A|
∑
g∈A
t
j1(g)
1 t
j2(g)
2 · · · tjn(g)
n .
Consider the group of cyclic permutations G
∼= Z/n as a subgroup of the symmetric group Sn.
Define, after Pólya, the cycle index polynomial of G, in n variables, as
PG(x1, . . . , xn) =
1
n
∑
σ∈G
n∏
i=1
x
ci(σ)
i ,
where ci(σ) is the number of cycles of length i in σ.
In other words, one associates to each σ ∈ G a monomial. For example, for n = 6 there are
following permutations:
σ0 = (1)(2)(3)(4)(5)(6), the corresponding monomial is x6
1,
σ1 = (123456) : x6,
σ2 = σ2
1 = (135)(246) : x2
3,
σ3 = σ3
1 = (14)(25)36) : x3
2,
σ4 = σ4
1 = (153)(264) : x2
3,
σ5 = σ5
1 = (654321) : x6.
The Zyklenzeiger is equal to PZ/6(x) = 1
6(x6
1 + x3
2 + 2x2
3 + 2x6).
In general,
PZ/n(x1, . . . , xn) =
1
n
∑
d|n
φ(d)xn/d
d .
After a change of variables xi =
b∑
j=1
yi
j , one obtains a polynomial
WG(y1, . . . , yb) := PG
(∑
yj ,
∑
y2
j , . . . ,
∑
yn
j
)
.
As a special case of the theorem of Pólya we obtain that the number of necklaces made of n
beads in b colors is equal to WZ/n(1, . . . , 1) = PZ/n(b, . . . , b) = Φn(b).
Homological Algebra and Divergent Series 21
3.2.3. Cyclotomic identity.
Theorem 3.8 (Gauss, cf. [11]). One has the following formula:
1− bt =
∞∏
n=1
(1− tn)Mn(b) .
It is proved by the application of the Möbius inversion. There is a useful generalization of
this identity, found by Pieter Moree [22]. We will need it later studying the equivariant Hilbert
series, see Section 3.4.4.
Theorem 3.9. Let f(q) ∈ C[q, q−1] be an arbitrary Laurent polynomial. Introduce the polyno-
mials
Mn(f ; q) =
1
n
∑
d|n
µ(d)f(qd)n/d.
Then
1− f(q)t =
∞∏
n=1
∞∏
i=−∞
(
1− qitn
)ain ,
where for each n, the number ain are defined by the equations∑
i
ainqi = Mn(f ; q).
Proof. Applying − log to the both sides of the above formula we obtain:
− log(1− f(q)t) =
∞∑
m=1
f(q)mtm
m
and
− log
∏
i,n
(
1− qitn
)ain
=
∑
i
∞∑
n,k=1
ain
qiktnk
k
=
∞∑
m=1
tm ·
∑
n|m
∑
i
ain
qim/n
m/n
,
therefore
f(q)m =
∑
n|m
∑
i
nainqim/n
for each m = 1, 2, . . .. Make a change of variables: p = qm,
f(p1/m)m =
∑
n|m
∑
i
nainpi/n.
Using the Möbius inversion we get:∑
i
nainpi/n =
∑
d|n
µ(n/d)f(p1/d)d,
and making a substitution: q = p1/n, we obtain∑
i
nainqi =
∑
d|n
µ(n/d)f(qn/d)d,
which is the formula required. �
22 V. Gorbounov and V. Schechtman
Moreover, O. Ogievetsky noticed that the theorem can be generalized to the case of several
variables:
Theorem 3.10. Let f(q1, . . . , qp) be a Laurent polynomial in p variables. Introduce the Laurent
polynomials
Mn(f ; q1, . . . , qp) =
1
n
∑
d|n
µ(d)f
(
qd
1 , . . . , qd
p
)n/d
.
Then
1− f(q, . . . , qp)t =
∞∏
n=1
∞∏
i=−∞
(
1− qi1
1 · · · q
ip
p tn
)ai1...ip;n ,
where for each n, the numbers ai1...ip;n are the coefficients of Mn:∑
i
ai1...ip;nqi1
1 · · · q
ip
p = Mn(f ; q1, . . . , qp).
The proof is the same as before.
Example 3.11. Take f(q) = −q. Then
1 + qt = (1− qt)−1
(
1− q2t2
)
,
where
1
n
∑
d|n
(−1)dµ(d)
is equal to 0 if n ≥ 3, −1 if n = 1, and 1 if n = 2.
3.2.4. Application: the ζ function of the aff ine line. Let A := Fp[x]; this ring is similar
in many ways to Z.
Nonzero ideals I ⊂ A are in bijection with unitary polynomials f(x), I = (f), and principal
ideals correspond to irreducible ones. Set
N(I) := ](A/I) = pdeg f ,
and define
ζ(A; s) =
∑
I⊂A, I 6=0
N(I)−s =
∑
f unitary
p−s deg f .
There is pn unitary polynomials of degree n, therefore
ζ(A; s) =
∞∑
n=1
pn · p−sn =
1
1− p · p−s
=
1
1− pT
, (3.4)
where T := p−s.
The Euler product formula for ζ(A; s) can be written in the following form
ζ(A; s) =
∏
f unitary, irreducible
1
1− p− deg f ·s
Homological Algebra and Divergent Series 23
=
∞∏
d=1
∏
f un., irr., deg f=d
1
1− p−ds
=
∞∏
d=1
1
(1− T d)Nd(p)
,
where Nd(p) denotes the number of unitary irreducible polynomials of degree d in A.
On the other hand, applying the cyclotomic identity to (3.4), one gets
ζ(A; s) =
1
1− pT
=
1
∞∏
d=1
(1− T d)Md(p)
,
proving therefore
Theorem 3.12 (Gauss [11]). The number Nd(p) of irreducible unitary polynomials of degree
d in Fp[x] is equal to
Md(p) =
1
d
∑
l|d
µ(l)pd/l.
Corollary 3.13. For d ≥ 1, Nd(p) > 0, i.e. for each d ≥ 1 there is an irreducible polynomial of
degree d.
It is interesting to compare this theorem of Gauss with Riemann’s explicit formula:
π(x) =
∞∑
n=1
µ(n)
n
Li(x1/n) +
∑
Im(ρ)>0
(xρ/n + x1−ρ) +
∫ ∞
x1/n
dt
t(t2 − 1) log t
− log 2
.
Here π(x) = the number of primes p ≤ x, the summation is over the non-trivial roots of ζ(s),
and
Lix =
∫ x
1
dt
log t
.
3.3 Witt formula
La Nature est un temple où de vivants piliers
Laissent parfois sortir de confuses paroles . . .
Ch. Baudelaire
Theorem 3.14 (Witt [35]). Let L be a free Lie algebra on b generators. Then
dim Ln = Mn(b) =
1
n
∑
d|n
µ(d)bn/d.
In other words,
H(L; t) :=
∞∑
n=0
dim Ln · tn = −
∞∑
m=1
µ(m)
m
log
(
1− btm
)
.
Proof. If x1, . . . , xb are the generators of L, then the universal enveloping algebra UL =
C〈x1, . . . , xb〉 (a free associative algebra), has the following Hilbert series
H(UL; t) :=
∞∑
i=0
dim ULi · ti =
1
1− bt
.
24 V. Gorbounov and V. Schechtman
On the other hand, due to Poincaré–Birkhoff–Witt, UL = S·L, therefore, if one denotes an :=
dim Ln, then
H(UL; t) =
∞∏
n=1
1
(1− tn)an
,
and applying the cyclotomic identity finishes the proof. �
3.3.1. This theorem admits a nice generalization. Let L be a free Lie superalgebra with b even
generators x1, . . . , xb and c odd generators y1, . . . , yc.
Equip L with a grading, assigning to xi and to yj the degree 1, and setting deg([a, b]) =
deg(a) + deg(b). If Ln ⊂ L is a subspace of the elements of the degree n, then L =
∞
⊕
n=1
Ln.
Explicitly, the Lie monomial α = [a1, [a2, . . . an] . . .], where ai = xj or yk, has the degree n.
On the other hand, its parity is equal to the number of y’s among ai. For example, [xi, yj ] is an
element odd of degree 2 (sic!).
Each homogenous summand Ln therefore is decomposed into two summands: even and odd
elements, Ln = Lp
n ⊕ Li
n. One defines the super-dimension as dim∼ Ln = dim Lp
n − dim Li
n.
Theorem 3.15 (Petrogradsky [25]). One has the formula
dim∼ Ln = Mn(b− c) =
1
n
∑
d|n
µ(d)(b− c)n/d. (3.5)
Example 3.16. Let b = 0, c = 1; then L = C · y ⊕ C · [y, y]. In the right hand side, one sees
that Mn(−1) = −1 if n = 1, 1 if n = 2, and 0 if n ≥ 3, cf. Examples 3.11 and 3.2.
3.3.2. Hochschild homology. Let A be an associative unitary algebra. Recall that the
Hochschild homology HHi(A) of A is defined as the homology of the complex
CH·(A) : · · · −→ A⊗A⊗A
b−→ A⊗A
b−→ A −→ 0,
where CHn(A) = A⊗n+1, and the differential b is defined by
b(a1 ⊗ a2 · · · ⊗ an) = a1a2 ⊗ a3 ⊗ · · · ⊗ an − a1 ⊗ a2a3 ⊗ a4 · · · ⊗ an + · · ·
+ (−1)n−1a1 ⊗ · · · ⊗ an−1an + (−1)nana1 ⊗ a2 ⊗ · · · ⊗ an−1.
In particular, d(a ⊗ b) = ab − ba and HH0(A) = A/[A,A], where [A,A] ⊂ A is the subspace
generated by the elements ab− ba.
In fact, one can replace A by Ā := A/C: the complex CH·(A) is quasi-isomorphic to
C̄H ·(A) : · · · −→ A⊗ Ā⊗ Ā −→ Ā⊗A −→ A −→ 0.
One has
HHi(A) = TorA⊗Ao
i (A,A).
Let us introduce the operators B : A⊗n+1 −→ A⊗n+2 by the formula:
B(a0, . . . , an) =
n−1∑
i=0
{
(−1)ni(1, ai, . . . , an, a0, . . . , ai−1)
− (−1)n(i−1)(ai−1, 1, ai, . . . , an, a0, . . . , ai−2)
}
.
Homological Algebra and Divergent Series 25
For example,
B(a0) = (1, a0) + (a0, 1). (3.6)
Then Bb + bB = 0 (cf. [17, 2.1]); therefore it induces the morphisms
B : HHn(A) −→ HHn+1(A). (3.7)
Recall that the cyclic homology can be defined as the homology of the bi-complex
HCi(A) = Hi
(
CH·(A) B−→ CH·(A)[1] B−→ CH·(A)[2] B−→ · · ·
)
.
3.3.3. Now one can give an algebraic interpretation of the polynomials Φn(x) =
∑
d|n
φ(d)xn/d/n.
Let A = C〈x1, . . . , xb〉, then HH0(A) inherits a grading from A, HH0(A) =
∞
⊕
n=0
HH0(A)n; if
one has Ā = ⊕
n≥1
An, then:
H̄H0(A) = ⊕
n≥1
HH0(A)n = Ā/[Ā, Ā].
One can think of HH0(A)n as a space of cyclic words xi1xi2 . . . xin of length n in letters xi,
where two words are identified if one is a cyclic permutation of another. Therefore cyclic words
are identified with necklaces of n beads in b colors.
Hence HH0(A)n can be viewed as a linear space with a basis indexed by necklaces of n beads
in b colors. It follows that
dim HH0(A)n = Φn(b) =
1
n
∑
d|n
φ(d)bn/d, n ≥ 1,
(cf. Corollary 3.6), or
H(H̄H0(A); t) = −
∞∑
m=1
φ(m)
m
log(1− btm)
(the Polyakov formula).
3.3.4. Let V =
b
⊕
i=1
C · xi, therefore A = TV . For each n ≥ 1 one defines an automorphism
τ : V ⊗n −→ V ⊗n as
τ(v1 ⊗ · · · ⊗ vn) = vn ⊗ v1 ⊗ v2 ⊗ · · · ⊗ vn−1.
One observes that
HH0(TV )n = V ⊗n
τ := Coker(1− τ).
Following [17, 3.1], one defines a complex of length 1:
CHsm
· (TV ) : 0 −→ TV ⊗ V −→ TV −→ 0
equipped with a differential d(a ⊗ v) = av − va. Define also a morphism of complexes φ :
CH·(TV ) −→ CHsm
· (TV ) as: φ0 = IdTV ,
φ1(a⊗ v1 . . . vn) =
n∑
i=1
vi+1 · · · vnav1 · · · vi−1 ⊗ vi. (3.8)
26 V. Gorbounov and V. Schechtman
On the other hand, there is an evident inclusion ι : CHsm
· (TV ) −→ CH·(A) such that φ◦ ι = Id,
and, as one can check, ι ◦ φ is homotopic to the identity.
It follows that
HH1(TV )n =
(
V ⊗n
)τ := Ker(1− τ)
and HHi(TV ) = 0 for i ≥ 2.
Instead of the Hochschild homology one can also consider the (reduced) cyclic homology.
Then: HC0(TV ) = HH0(TV ) and H̄Ci(TV ) = 0 for i > 0, cf. [23].
3.3.5. Partial derivatives. Let m = . . . xixjxk . . . be a cyclic word (i.e. m ∈ TV/[TV, TV ])
such that the letter xi appears once in it; then one can define a usual word (i.e. an element
de TV ) ∂m/∂xi, by “cutting” m and removing the letter xi:
∂m/∂xi = xjxk . . . .
When xi appears in a word several times, the result will be a sum, by the Leibnitz rule, cf. [19].
In this way we define a map
∂
∂xi
: TV/[TV, TV ] −→ TV.
Consider an operator
b∑
i=1
xi∂/∂xi (cf. (3.8)). It respects the polynomial degree, and it is not
hard to verify that its image is contained in TV τ . For example:
e(xyz) = xyz + yzx + zxy.
One observes that the map
b∑
i=1
xi
∂
∂xi
: HH0(TV ) = TVτ −→ TV τ = HH1(TV ) (3.9)
coincides with homomorphism B, cf. (3.6), (3.7). (Compare [23, the line before (14), p. 8].)
It was pointed out to us by V. Ginzburg that there is another interpretation of the above
map. Consider the composition
d : TVτ = TV/[TV, TV ]
∑
xi∂/∂xi−→ TV τ ⊂ TV ⊗ V. (3.10)
It was already Quillen who studied this map in the 80’s, cf. [30]. In the notations of [30, § 3]
TV/[TV, TV ] = TV\ and TV ⊗ V = Ω1
TV,\; one can consider these spaces as the space of cyclic
functions (differentiable 1-forms respectively) over the “non-commutative space Spec TV ”; the
morphism (3.10) is called “the Karoubi–de Rham differential”.
3.4 Koszul duality
3.4.1. Let A be an associative quadratic algebra, which means that, it is a quotient A =
TV/(R) of the free associative algebra over the space V of finite dimension by the two-sided
ideal (R) generated by a subspace of relations R ⊂ V ⊗ V .
Recall that the quadratic dual algebra is defined as A! = TV ∗/(R⊥), where R⊥ ⊂ V ∗⊗V ∗ ∼=
(V ⊗ V )∗ is the annihilator of R.
We are interested in the case when A is commutative; in this case R ⊃ Λ2V , and R⊥ ⊂
Λ2V ⊥ = S2V ∗. In other words, if {xi} form a basis of V , {yi} form a basis of V ∗, then R⊥
Homological Algebra and Divergent Series 27
is contained in S2V ∗ ⊂ V ⊗2; in other words it is generated by odd commutators [yi, yj ] =
yiyj + yiyj . Define a Lie algebra L as a graded Lie algebra generated by yi of degree 1 and
relations g = 0, g ∈ R⊥. Then A! = UL by definition. The Lie algebra L is called the Koszul,
or Quillen dual of A.
For example, if A = H∗(X; C) is the cohomology ring of a simply connected topological
space X, then L is its homotopy Lie algebra, ⊕πi(X)C, under some additional conditions,
cf. [29].
Therefore we have, L =
∞
⊕
n=1
Ln; moreover one observes that L1 = V ∗, L2 = S2V ∗/R⊥. Set
L≤2 = L1 ⊕ L2; it is a quotient Lie algebra of L.
3.4.2. The Chevalley cochain complex of L is by definition the space
C ·(L) = SL∗1 ⊗ ΛL∗2 ⊗ SL∗3 ⊗ · · ·
equipped with the Chevalley differential. This complex is double graded:
C ·(L) =
∞
⊕
a,b=0
C ·(L)ab,
where the first (homological) degree of L∗i is set to be equal i− 1, and the second (polynomial)
degree of L∗i is i, both gradings are compatible with the product. The Chevalley differential
preserves the second grading and decreases the first by 1:
dC ·(L)ab ⊂ C ·(L)a−1,b.
Here are the components of C ·(L) of polynomial degree ≤ 3:
L∗3 −→ L∗2 ⊗ L∗1 −→ S3L∗1,
L∗2 −→ S2L∗1, (3.11)
L∗1.
One has: C ·(L)0· = SL∗1 and the complex starts as:
· · · −→ (L∗3 ⊕ Λ2L∗2)⊗ SL∗1 −→ L∗2 ⊗ SL∗1 −→ SL∗1. (3.12)
Since L∗1 = V and L∗2 = R ⊂ S2V , the first differential in (3.12) is a SV -linear map which sends
f ⊗ 1 ∈ L∗2 to f ∈ R ⊂ S2L∗1.
One observes that:
(a) there is a natural augmentation C ·(L) −→ A = SV/(R), cf. [24].
(A similar picture arises in the construction of the mixed Tate motivic cohomology of a field k,
cf. [4]. There, the analogues of complexes C ·(L)n· are the Beilinson motivic complexes Z(n), the
analogue of A is the Milnor K-theory KMiln
· (k); the analogue of L is (the Lie algebra of) the
“mixted Tate motivic fundamental group”.)
(b) If one chooses a basis {f1, . . . , fd} of R, then one can identify the Chevalley complex of
L≤2 with the Koszul complex
C ·(L≤2)
∼= K(SV ; (f)),
cf. Example 3.2.
(c) There is a natural inclusion C ·(L≤2) ⊂ C ·(L) compatible with the augmentation to A.
Define a generating series
H(C ·(L);u, t) :=
∞∑
a,b=0
dim C ·(L)abu
atb
28 V. Gorbounov and V. Schechtman
and its specialization, the Euler–Hilbert series:
EH(C ·(L); t) := H(C ·(L);−1, t) =
∞∑
n=0
EP (C ·(L)?,n)tn,
where EP stands for the Euler–Poincaré characteristic. One observes immediately that
EH(SL∗n) = (1− tn)− dim Ln and EH(ΛL∗n) = (1− tn)dim Ln ,
where
EH(C ·(L); t) =
∞∏
n=1
(1− tn)(−1)n dim Ln . (3.13)
3.4.3. For example, consider the Veronese ring Ab, cf. Section 3.1.1. Recall that Ab is generated
by x0, . . . , xb+1, subject to relations xixj − xkxl = 0 if i + j = k + l. Then A! is generated by
yi = x∗i , 0 ≤ i ≤ b + 1 obeying the relations∑
i+j=k
[yi, yj ] = 0, k = 0, . . . , 2(b + 1), (3.14)
where [yi, yj ] = yiyj +yjyi (sic!). Therefore if one defines a graded Lie algebra Lb with generators
yi ∈ Lb
1 = A∗
b1 and relations (3.14), then A!
b = ULb.
This Lie algebra has a nice structure, cf. [13] (the case b = 1 was considered in Example 3.2).
It admits an involution i(yj) = yb+1−j . Let L̂b ⊂ Lb be a Lie subalgebra generated by y1, . . . , yb.
Then L̂b = [Lb, Lb] is a Lie ideal (stable under i); as a graded Lie algebra it is free.
The quotient algebra L̄ = Lb/L̂b is a graded Abelian Lie algebra on 2 generators ȳ0, ȳb+1.
One sees that L̂b
≥2 = Lb
≥2.
We have noticed in Example 3.2 that L1 is finite-dimensional; as opposed to that, if b ≥ 2,
than Lb is infinite-dimensional (with an exponential growth). Namely, one has Lb =
∞
⊕
n=1
Lbn,
where the dimensions of the homogenous components can be calculated easily using an odd
analogue of the Witt theorem, see Theorem 3.14. One obtains:
dim L1 = −M1(−b) + 2 = b + 2,
dim∼ Ln = dim∼ L̂n = Mn(−b) =
1
n
∑
d|n
µ(d)(−b)n/d (n ≥ 2).
Here L̂b1 is generated by b odd elements, therefore
dim∼ L̂n = (−1)n dim L̂n.
One observes that the signs od dim∼ Ln alternate. For example, dim L2 = M2(−b) = (b2 + b)/2,
dim L3 = −M3(−b) = (b3 − b)/3, etc.
Now consider the Chevalley complex of Lb: (3.13) implies that its Euler–Hilbert is as follows:
EH
(
C ·(Lb
)
; t
)
= (1− t)− dim Lb1
(
1− t2
)dim Lb2
(
1− t3
)− dim Lb3 · · ·
= (1− t)−2
∞∏
n=1
(1− tn)Mn(−b) =
1 + bt
(1− t)2
. (3.15)
It is clear that it coincides with the Hilbert series of the ring Ab, (3.1). the last equality is due
to the cyclotomic identity. It is not surprising.
In fact, a deep theorem of Bezrukavnikov says:
Homological Algebra and Divergent Series 29
Theorem 3.17 (cf. [6]). The algebra Ab is Koszul.
Formula (3.15) can be viewed as a “numerical evidence” that the theorem holds.
Corollary 3.18. There are natural isomorphisms A!
b
∼= Ext·Ab
(C, C), Ab
∼= Ext·
A!
b
(C, C) =
H ·(L; C).
It follows that A∼
b := C ·(Lb) is a dga resolution of Ab, cf. [24].
3.4.4. Characters. The Lie algebra sl(2) acts on Ab in such a way that Ab1 is an irreducible
sl(2)-module; therefore its character is
Ch(Ab1) = [b + 1]q =
qb+1 − q−b−1
q − q−1
.
The character of Ab is given by the equivariant Hilbert series, cf. (3.2):
Hq(Ab; t) =
1 + [b]qt
(1− qb+1t)(1− q−b−1t)
.
The above action induces an action of sl(2) on Lb and therefore on A∼
b := C ·(Lb). Applying
now the “q-cyclotomic” identity: Theorem 3.9 for f(q) = −[b]q, one obtains the sl(2)-character
Ch(A∼
b ).
On the other hand:the subalgebra Lie L̂b ⊂ Lb is free on b generators, therefore the Lie algebra
gl(b) acts on it. One can use the theorem of Ogievetsky (see Theorem 3.10) for f(q1, . . . , qb) =
−Chgl(b)(L̂b1) for calculation of the character Chgl(b)(C ·(L̂b)).
The gl(b)-character of the free Lie algebra on odd b generators was calculated by Angeline
Brandt in [9].
The free group on b generators is isomorphic to the fundamental group of the Riemann sphere
with b + 1 points removed; its nilpotent completion is a fundamental object of the theory of
Grothendieck–Drinfeld–Ihara, cf. [10].
3.4.5. Returning to the case of an arbitrary commutative quadratic algebra, one can show that
a complete intersection is Koszul, cf. [26]. The other way around, if A is commutative Koszul,
and L is its dual Lie algebra, then A is a complete intersection if and only if L = L≤2.
One can say that commutative (or maybe also noncommutative?) Koszul algebras are natural
generalizations of the quadratic complete intersections; and one would expect that all the results
which hold for quadratic complete intersections will generalize to Koszul algebras.
Acknowledgements
We thank Fedor Malikov who read thoroughly the first part and helped to correct many signs;
some calculations made with him have been the starting point of the second part. We are grate-
ful to Vladimir Hinich for very interesting discussions about Golod rings and Koszul algebras;
to Oleg Ogievetsky for an important remark; to Alexander Polishchuk for very useful consul-
tations; to Hossein Abbaspour and Thomas Tradler who taught us about the string topology,
and especially to Victor Ginzburg for his numerous explanations, questions and bibliographical
comments.
This article was finished during our stay at Max-Planck-Institut für Mathematik and the
Hausdorff Institut für Mathematik in June and July 2008; we are grateful to both institutions
for the excellent working atmosphere.
30 V. Gorbounov and V. Schechtman
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http://arxiv.org/abs/math.QA/0404218
1 Introduction
First part
2 Numerical invariants and regularization
2.1 Semi-free resolutions and Hilbert series
2.2 Möbius inversion
2.3 Dirichlet series
2.4 Values at negative points
Second part
3 Koszulness and infinite resolutions
3.1 The Veronese ring
3.2 Gauss cyclotomic identity
3.3 Witt formula
3.4 Koszul duality
References
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| language | English |
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| spelling | Gorbounov, V. Schechtman, V. 2019-02-19T17:48:11Z 2019-02-19T17:48:11Z 2009 Homological Algebra and Divergent Series / V. Gorbounov, V. Schechtman // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 36 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 13D02; 14N99 https://nasplib.isofts.kiev.ua/handle/123456789/149155 We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits. This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. We thank Fedor Malikov who read thoroughly the first part and helped to correct many signs; some calculations made with him have been the starting point of the second part. We are grateful to Vladimir Hinich for very interesting discussions about Golod rings and Koszul algebras; to Oleg Ogievetsky for an important remark; to Alexander Polishchuk for very useful consultations; to Hossein Abbaspour and Thomas Tradler who taught us about the string topology, and especially to Victor Ginzburg for his numerous explanations, questions and bibliographical comments. This article was finished during our stay at Max-Planck-Institut f¨ur Mathematik and the Hausdorf f Institut f¨ur Mathematik in June and July 2008; we are grateful to both institutions for the excellent working atmosphere. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Homological Algebra and Divergent Series Article published earlier |
| spellingShingle | Homological Algebra and Divergent Series Gorbounov, V. Schechtman, V. |
| title | Homological Algebra and Divergent Series |
| title_full | Homological Algebra and Divergent Series |
| title_fullStr | Homological Algebra and Divergent Series |
| title_full_unstemmed | Homological Algebra and Divergent Series |
| title_short | Homological Algebra and Divergent Series |
| title_sort | homological algebra and divergent series |
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