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.


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 R 0 = k[x 0 , . . . , x N ] be a polynomial algebra over a field k, f 1 , . . . , f p ∈ R 0 homogeneous elements of degrees d i = deg f i > 0 which generate the ideal I = (f 1 , . . . , f p ). Consider the quotient algebra A = R 0 /I; it is graded Geometrically the projection R 0 −→ A corresponds to a closed embedding i : X := Proj A ֒→ P N := Proj R 0 ; the projective variety X is defined in P N by the equations f 1 = 0, . . . , f p = 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 R j i (so R j i · R m l ⊂ R j+m i+l ); the indexes i and j will be called the homological degree and the polynomial degree respectively. We set R i := ⊕ j R j i , R j := ⊕ i R 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 ∈ R i , y ∈ R l we have xy = (−1) il yx. R should be equipped with a differential d : R −→ R, d 2 = 0, such that d(R j i ) ⊂ R j i−1 and for x ∈ R i , y ∈ R, d(xy) = dx · y + (−1) i x · dy.
(ii) R 0 should be equipped with a morphism of algebras ǫ : R 0 −→ A such that ǫ(R j 0 ) ⊂ A j . (iii) For each j ≥ 0, R j should be a resolution of A j , i.e. the complexes 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 where Λ denotes the exterior algebra; the differential is defined by the formula dξ j = f j . The requirements (i), (ii), (iv) are fulfilled (the homological degree of x i is 1). The requirement (iii) holds iff the sequence f = {f 1 , . . . , f M } is regular.
If this is the case, the Krull dimension of A is given by a simple formula 1) where N + 1 is the number of x i and M is the number of ξ j ; one can say that N + 1 − M is the "superdimension" of the polynomial ring k[x 0 , . . . , x N ] ⊗ Λ ξ 1 · · · ξ M (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 = P 1 ֒→ P N . This part 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 R 0 = C[λ 1 , . . . , λ 16 ] by the ideal generated by 10 quadratic elements The corresponding projective variety i : X = Proj A ֒→ P 15 is the hermitian symmetric space The Hilbert series of A is equal to L i 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 (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 One can define a differential on this algebra that will make a dga whose cohomology will be The dga C 1 (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 L j . 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 where F i = S if i is even and Λ otherwise. One expects that it is possible to introduce a differential on C i (L) which provides a dga whose cohomology is H * (L ≥i ). For example, an algebra quasi-isomorphic to C 3 (L) has been studied in a very interesting paper [1].
The algebra of syzygies Q is a remarkable object. It is finite-dimensional over C, Q = 3 ⊕ i=0 Q i and admits, after [13] a scalar product Q i ⊗ Q 3−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 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 C 1 (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 string 2 .
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 e 2 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?
be a semi-free resolution of A. For R 1 one can take R 1 = M ⊕ l=1 R 0 ξ l , where the variables ξ l are odd and dξ l = f l , so ξ l ∈ R d i 1 .
Example 2.1. Suppose that f 1 , . . . , f M is a regular sequence, that is, X is a complete intersection. Then one can take for R the Koszul complex, The exact sequence (2.1) immediately gives an expression of the Hilbert series: Let us return to the general case. Recall that R as an algebra is a polynomial superalgebra in homogeneous variables. Let us define integer numbers a n = 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 This formula shows in particular that the numbers a n 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 = "a 1 + a 2 + a 3 + · · · ", (2.6) log(deg X) = −"(log 1) · a 1 + (log 2) · a 2 + (log 3) · a 3 + · · · ". (2.7) It is natural also to consider the "higher moments": n l a n ", 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 series 3 . There are several ways of doing the summation. For example, one can write a Lambert series This series diverges if |t| is small; but one can show that if |t| is sufficiently big then the series absolutely converges and t −1 f (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 " a n " : 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.

Möbius inversion
2.2.1. Take the logarithm of (2.5): log and then the derivative: In other words, if one denotes Recall that the Möbius function µ : 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 The Möbius inversion formula says that if f : N + −→ C is a function and if a function g : N + −→ C is defined by (2.14) Applying this formula to the function f (m) = ma m , we get

Dirichlet series
2.3.1. Let P (t) be a polynomial with complex coefficients such that P (0) = 1, so one can write Consider the product (2.5) for P (t): On the other hand One puts It follows that if ρ(P ) > 1, then |a m | grows as fast as m −1 ρ(P ) m .

Let us consider the Dirichlet series
∞ n=1 a n n −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]: Now if we rewrite, after Riemann, and replace s by s + 1:

It follows:
∞ n=1 a n n −s = 1

Consider the last integral
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.
2.3.4. Following Riemann and using the notations of [34], consider the integral denotes the integral along the contour where ǫ > 0 is sufficiently small. Here (−t) s = e s log(−t) , and we use the branch of the logarithm π ≥ arg t ≥ −π on the small circle. Then cf. [34, 12.22]. But J P (s) is an entire function. Therefore, if one sets 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 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 the quotation marks mean that the sum is regularized. We define "the Dirichlet summation": A n " := z(X, L; 0), and " ∞ n=1 log n · A n " := −z ′ (X, L; 0), 2.3.6. We have ζ(0) = −1/2 and ζ ′ (0) = − log √ 2π = 0 (cf. [33, Ch. VII, § 9]). The integral I P (0) is well defined (below we shall compute its value), so the formula (2.23) gives z P (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 contribution into z ′ (X, L; 0), is The first factor gives 1, whereas Example 2.2 (Veronese curve and the Weil zeta function). Let X = P 1 , L = O(q + 1), q ≥ 1. and It is well known that if q is a prime power then p m (q) is equal to the number of monic irreducible polynomials of degree m in F q [T ]. In fact, we have -this is the Euler product of the zeta function of the affine line over the finite field F q .
(The idea that the Hilbert series of a variety is equal to the Weil zeta function of another one seems very strange. Cf. [12] So if Q(t) = q 0 + q 1 t/1! + q 2 t 2 /2! + · · · is the Taylor series at 0 then Here are some first values:

One can say this in a different way: if
and we imagine the numbers c i as "Chern classes" then q m will be the coefficients of the logarithm of the Todd genus . . .

Now applying (2.22)
: for m = 1, the function ζ(s) has a simple pole at s = 1 with the residue 1, whence which happens rather often. In this case it is easy to see that

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, where the Bernoulli numbers are defined by the generating series Here are some first values: It follows: So if the numbers a n are defined by (2.24) then " ∞ n=1 n 2l a n " = z(X, L; −2l -a formula found empirically in [5, (4.21)].
(c) If s = −1, the series (2.27) in our example is still "Eisenstein summable", and on gets a rational value: This is an easy corollary of the decomposition of cot z into simple fractions 5 . But this value is different from −q 0 = −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 A n = Γ(X, L ⊗n ), and we can switch on the Riemann-Roch-Hirzebruch [15]: where T X denotes the tangent bundle, Td the Todd genus, given for line bundle E by the formula, so Td (T X ) = 1 + Td 1 (T X ) + Td 2 (T X ) + · · · , where One writes: e nc 1 (L) = ∞ i=0 c 1 (L) i n i /i!, so h n = R(n), where 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 where e 0 , e 1 , . . . , e d ∈ Z. For example, On the other hand it is known that It follows: so one finds the degree of X. Next One can put these calculations into a formal "generating function" for the moments n l a n ".
Note that from the formulas above and using the expansion one concludes the formal identity: The following elegant formula is due to F. Hirzebruch [16]: the polynomial P (t) is a characteristic number defined as where c 1 , p 1 , p 2 , . . . 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 (e t ) = log 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 is a series in t equal to log P (e t ) − log P (1). This gives the desired generating function.
Example 2.5. Let X g be the moduli space of semi-stable vector bundles of rank 2 with trivial determinant over a Riemann surface of genus g; it carries the canonical determinant line bundle L g , cf. [2]. Consider a graded algebra H 0 X g , L ⊗i g (we thank Peter Zograf who proposed to consider this example). The coefficients of the Hilbert series H g (t) = H(A g ; t) can be calculated using the Verlinde formula [2]. Here are some first examples, cf. [36]:
The above discussion gives a strange expression of this number as a regularized infinite product (starting from g = 4), of the form ∞ n=1 n an . For example, the beginning of the product for P 4 (t) will be: If t = u 1 /u 0 ∈ A 1 ⊂ P 1 , then The image X b := i b (P 1 ) ⊂ P b+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 P b+1 by the equations x i x j − x k x l = 0 if i + j = k + l. A minimal system of equations consists of b(b + 1)/2 quadratic equations: x 2 )} is defined by one equation: The curve X 3 ⊂ P 4 is defined by 3 equations , all the higher cohomology of this sheaf vanish, and The Hilbert series of the above algebra is as follows: It is not difficult to figure out the q-analogue of (3.1): if we set [b] q := (q b − q −b )/(q − q −1 ), then: We will use these formulas later in Section 3.4.4.
Example 3.2. For b = 1 we have, the ring A 1 is a cone: It admits a dga resolution of length 1: the Koszul complex: as a graded algebra; the homological degree of x i 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 If L ′ = C · y 1 ⊕ C · [y 1 , y 1 ] ⊂ L is a Lie subalgebra generated by y 1 , then L ′ is free (sic!); it is an ideal, and the quotient algebraL = L/L ′ is Abelian, on 2 generatorsȳ 0 ,ȳ 2 .
Consider the complex of the Chevalley cochains: The first two exponents are "koszul": the number of unknowns and the number of equations. The exponents grow exponentially.

Gauss cyclotomic identity
3.2.1. Necklaces. The necklace polynomial is defined as 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. 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
Proof . If this number is C(n; b) then 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 Every subgroup of the symmetric group on n elements, S n , 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 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 c k = |{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 c k · t k . Let C k = |{[f ] | w([f ]) = k}| be the number of orbits of weight k. The generating function of the filled slot configurations is Theorem 3.7. Given all the above definitions, Pólya's enumeration theorem asserts that Consider the group of cyclic permutations G ∼ = Z/n as a subgroup of the symmetric group S n . Define, after Pólya, the cycle index polynomial of G, in n variables, as In other words, one associates to each σ ∈ G a monomial. For example, for n = 6 there are following permutations: The Zyklenzeiger is equal to P Z/6 (x) = 1 6 (x 6 1 + x 3 2 + 2x 2 3 + 2x 6 ). In general, After a change of variables x i = b j=1 y i j , one obtains a polynomial W G (y 1 , . . . , y b ) := P G y j , y 2 j , . . . , y n 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 W Z/n (1, . . . , 1) = P Z/n (b, . . . , b) = Φ n (b).

Cyclotomic identity.
Theorem 3.8 (Gauss, cf. [11]). One has the following formula: 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 polynomials Then where for each n, the number a in are defined by the equations i a in q i = M n (f ; q).
Proof . Applying − log to the both sides of the above formula we obtain: Using the Möbius inversion we get: and making a substitution: q = p 1/n , we obtain which is the formula required.
Moreover, O. Ogievetsky noticed that the theorem can be generalized to the case of several variables: Then where for each n, the numbers a i 1 ...ip;n are the coefficients of M n : The proof is the same as before.
Example 3.11. Take f (q) = −q. Then where N d (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 , proving therefore Theorem 3.12 (Gauss [11]). The number It is interesting to compare this theorem of Gauss with Riemann's explicit formula: Here π(x) = the number of primes p ≤ x, the summation is over the non-trivial roots of ζ(s), and

Ch. Baudelaire
Theorem 3.14 (Witt [35]). Let L be a free Lie algebra on b generators. Then

In other words,
Proof . If x 1 , . . . , x b are the generators of L, then the universal enveloping algebra U L = C x 1 , . . . , x b (a free associative algebra), has the following Hilbert series

V. Gorbounov and V. Schechtman
On the other hand, due to Poincaré-Birkhoff-Witt, U L = S · L, therefore, if one denotes a n := dim L n , then and applying the cyclotomic identity finishes the proof. Each homogenous summand L n therefore is decomposed into two summands: even and odd elements, L n = L p n ⊕ L i n . One defines the super-dimension as dim ∼ L n = dim L p n − dim L i n .

Let
C · x i , therefore A = T V . For each n ≥ 1 one defines an automorphism One observes that Following [17, 3.1], one defines a complex of length 1: Define also a morphism of complexes φ : On the other hand, there is an evident inclusion ι : CH sm · (T V ) −→ CH · (A) such that φ • ι = Id, and, as one can check, ι • φ is homotopic to the identity.
It follows that and HH i (T V ) = 0 for i ≥ 2.
3.3.5. Partial derivatives. Let m = . . . x i x j x k . . . be a cyclic word (i.e. m ∈ T V /[T V, T V ]) such that the letter x i appears once in it; then one can define a usual word (i.e. an element de T V ) ∂m/∂x i , by "cutting" m and removing the letter x i : When x i 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

Consider an operator
. It respects the polynomial degree, and it is not hard to verify that its image is contained in T V τ . For example: e(xyz) = xyz + yzx + zxy.

One observes that the map
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 It was already Quillen who studied this map in the 80's, cf. [30]. In the notations of [30, one can consider these spaces as the space of cyclic functions (differentiable 1-forms respectively) over the "non-commutative space Spec T V "; the morphism (3.10) is called "the Karoubi-de Rham differential".

Koszul duality
3.4.1. Let A be an associative quadratic algebra, which means that, it is a quotient A = T V /(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 We are interested in the case when A is commutative; in this case R ⊃ Λ 2 V , and R ⊥ ⊂ ; in other words it is generated by odd commutators [y i , y j ] = y i y j + y i y j . Define a Lie algebra L as a graded Lie algebra generated by y i of degree 1 and relations g = 0, g ∈ R ⊥ . Then A ! = U L 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, ⊕ π i (X) C , under some additional conditions, cf. [29].

The Chevalley cochain complex of L is by definition the space
equipped with the Chevalley differential. This complex is double graded: 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: Here are the components of C · (L) of polynomial degree ≤ 3: One has: C · (L) 0· = SL * 1 and the complex starts as: Since L * 1 = V and L * 2 = R ⊂ S 2 V , the first differential in (3.12) is a SV -linear map which sends f ⊗ 1 ∈ L * 2 to f ∈ R ⊂ S 2 L * 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 K Miln · (k); the analogue of L is (the Lie algebra of) the "mixted Tate motivic fundamental group".) where [y i , y j ] = y i y j +y j y i (sic!). Therefore if one defines a graded Lie algebra L b with generators y i ∈ L b 1 = A * b1 and relations (3.14), then A ! b = U L b . This Lie algebra has a nice structure, cf. [13] (the case b = 1 was considered in Example 3.2). It admits an involution i(y j ) = y b+1−j . LetL b ⊂ L b be a Lie subalgebra generated by y 1 , . . . , y b . ThenL b = [L b , L b ] is a Lie ideal (stable under i); as a graded Lie algebra it is free.
The quotient algebraL = L b /L b is a graded Abelian Lie algebra on 2 generatorsȳ 0 ,ȳ b+1 . One sees thatL b ≥2 = L b ≥2 . We have noticed in Example 3.2 that L 1 is finite-dimensional; as opposed to that, if b ≥ 2, than L b is infinite-dimensional (with an exponential growth). Namely, one has L b = HereL b1 is generated by b odd elements, therefore dim ∼L n = (−1) n dimL n .
One observes that the signs od dim ∼ L n alternate. For example, dim L 2 = M 2 (−b) = (b 2 + b)/2, dim L 3 = −M 3 (−b) = (b 3 − b)/3, etc. Now consider the Chevalley complex of L b : (3.13) implies that its Euler-Hilbert is as follows: It is clear that it coincides with the Hilbert series of the ring A b , (3.1). the last equality is due to the cyclotomic identity. It is not surprising. In fact, a deep theorem of Bezrukavnikov says: Theorem 3.17 (cf. [6]). The algebra A b is Koszul.
Formula (3.15) can be viewed as a "numerical evidence" that the theorem holds.
It follows that A ∼ b := C · (L b ) is a dga resolution of A b , cf. [24].
3.4.4. Characters. The Lie algebra sl(2) acts on A b in such a way that A b1 is an irreducible sl(2)-module; therefore its character is The character of A b is given by the equivariant Hilbert series, cf. (3.2): The above action induces an action of sl(2) on L b and therefore on A ∼ b := C · (L b ). 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 LieL b ⊂ L b 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 (q 1 , . . . , q b ) = −Ch gl(b) (L b1 ) for calculation of the character Ch gl(b) (C · (L b )).
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.