maps. So we can consider the deRham complex K(∂; N ). Notice that the deRhamcohomology modules H ∗ (∂; N ) are in general only graded K-vector spaces. They are finite dimensional if N is holonomic; [1, Chapter 1, Theorem 6.1]. In particular H ∗ (∂; H I ∗ (R)) are finite dimensional graded K-vector spaces. By [4, Theorem 1] the deRhamcohomology modules H ∗ (∂; H ∗
For a commutative algebra R , its deRhamcohomology is an important invariant of R . In the paper, an infinite chain of deRham-like complexes is introduced where the first member of the chain is the deRham complex. The complexes are called approximations of the deRham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.
Since the existence of Clark-Ocone-type formulae has such significance for the L 2 deRhamcohomology, we have explored a few different techniques of formulating them in more general settings, especially where there is no natural concept of time, nor any intrinsically defined filtration. Such filtrations play a principal role in the standard Itˆ o integration theory, since they give rise to the fundamental notions of measurability and adaptedness. Noteworthy examples where the standard theory on the classical Wiener space does not directly apply include abstract Wiener spaces (where there is no intrinsic temporal structure), and the loop spaces (where there is ambiguity in the definition of time and filtration, since the end point coincides with the start point; usually an enlargement of filtration is required).
ated by the Fourier coefficients of the cusp form; for the ratios of the odd periods the same holds. The cocycles are present in the background, for instance in the period relations. So apart from the Fourier coefficients there are two, possibly tran- scendental, numbers involved in the coefficients of the period polynomials. The arithmetic of the period polynomials, associated with values of L-functions at in- tegral points in the critical strip, are an important area of study in connection with the cocycles attached to automorphic forms. It goes further than the central idea in these notes, which is establishing the relation between automorphic forms and cohomology. Therefore we have not tried to include all papers in this area in the list of references. We mention the concept “modular symbol”; see [86, 113]. We mention also Haberland’s paper , and [62, 48, 49, 118]. In  Zagier de- scribes rather explicitly how to reconstruct a cuspidal Hecke eigenform from its period polynomial.
In this paper, we search for the Dolbeault, Bott-Chern and Aeppli cohomology hodge numbers for a complex S 6 . In 1997, Gray showed that for the Dolbeault hodge numbers, we have h 3,0 = h 0,3 = 0 and h 0,1 ≥ 1. In 2000, Ugarte essentially gave the following for the Dolbeault cohomology on S 6 which we shall summarise shortly in a table. Let a = h 2,0 2 where h 2,0 2 = dim C E 2,0 2 from the Frohlicher spectral sequence. Ugarte shows that h 2,0 2 = h 2,0 − h 1,0 . Now, let c = h 0,2 , and d = h 2,1 . We have Ugarte’s results in the following:
cohomology of the universal cover of the Salvetti complex associated to an arbitrary Artin group (as well as a formula for the cohomology of the Sal- vetti complex with generic, 1-dimensional local coefficients). This can be interpreted as a computation of the ℓ 2 -cohomology of universal covers of hy-
Since ℓ does not divide the order of W the complex C is split and its cohomology is concentrated in degree zero isomorphic to the sign representation of W (cf. [4, 66.28] or [9, §8]). As induction is exact it follows that Y is split with cohomology concentrated in degree zero isomorphic to τ O(Q ⋊ W ), where τ is the automorphism of O(Q ⋊ W )
Our next definition is that of the inertia stack. It is necessary for the construction of the Grothendieck-Riemann-Roch theorem for stacks. It naturally arises in the index theory of orbifolds and its cohomology (with coefficients in C ) is recovers precisely the Chen-Ruan Cohomology  and its additive structure. Intuitively, it is the space of “loops” in your stack, with each object of the inertia stack corresponding to an automorphism of an object of X. In this section we explicitly construct the groupoid of the inertia stack for any stack, and demonstrate a canonical automorphism which acts on any sheaf over the inertia stack, which will be vital in what follows.
(b) when the method is applied to the calculation of supports of local cohomology modules, if the input is given by polynomials with in- teger coefficients, then the calculation of supports modulo different primes p involves polynomials whose degrees can be bounded from above by a constant times p, that constant being independent of p. In [Lyu97]) Gennady Lyubeznik described an algorithm for computing the support of F -finite F -modules. That algorithm requires the calculation for roots of these modules, and this relies on the calculation of Grobner bases; these are often too complex to be computed in practice. Crucially, our algorithm does not involve Gr¨ obner bases, and consists essentially of matrix multiplications together with the listing of terms of polynomials of degrees of order p. It is this that makes our algorithm a practical tool for computing F -finite F -modules. 1
The geometric objects of interest in our physical theory are a space (or orbifold, or stack) denoted X (or X when an orbifold or stack) along with a vector bundle (“gauge bundle”) E over it. The particles manifest themselves as cohomology classes of the sheaf of sections of bundles associated to this bundle E. In the case of a non- stacky space X, we need to impose the following conditions coming from physics:
−p + q + d, then the curvature R is of exotic degree zero. Applying the Chern-Weil theory for superconnections, we obtain characteristic forms with values in Bott- Chern cohomology, which is a refinement of deRham cohomology. Though it is well known in literature, we prove the deRham cohomology classes of a cohesive mod- ule only depends on the Z 2 -graded topological bundle structure by transgressing
Taking the foregoing as motivation we now concentrate on the case G «G L n(k). For feDJ let S = S(G) be the Schur algebra associated with n and f (cf. [Gl]), so that modS is the category o f homogeneous polynomial representations o f degree f o f G. The cohomology theory of S has received attention in recent years. It was proved independently by Akin and Buchsbaum in [AB2], and by Donkin in [Do3], that S has finite global dimension. In [AB2] this is accomplished by giving an inductive procedure for the construction o f finite projective resolutions of Weyl modules. In [Do3] it is proved for a more general class o f algebras (analogues o f the Schur algebras for arbitrary reductive groups), using the machinery o f good filtrations. These generalized Schur algebras (more precisely algebras Morita equivalent to them) have also featured in the work o f Cline, Parshall and Scott, as examples of quasi- hereditary algebras (see [CPS1]). We remark that although the Kempf Vanishing Theorem is used in proving that these algebras are quasi-hereditary, this is not necessary for S: a short direct proof is given in [P]. 2
2. Steenrod operations on cyclic cohomology. Let k be a commutative ring with unit, A a commutative k-Hopf algebra, and Ꮿ a cyclic category (see [10, page 202]). We will denote the k-algebra over Ꮿ by k[ Ꮿ ] and the cyclic category over A by A Ꮿ (see ). We deﬁne an A Ꮿ -structure of cocommutative coalgebra by the formula
Let S be any smooth toric surface. We establish a ring isomorphism between the equivariant extended Chen-Ruan cohomology of the n-fold symmetric product stack [Sym n (S)] of S and the equivariant extremal quantum cohomology of the Hilbert scheme Hilb n (S) of n points in S. This proves a generalization of Ruan’s Cohomological Crepant Resolution Conjecture for the case of Sym n (S).
H ∗ (X). So assume that n ≥ 2. By the naturality of the cup product, we get g ∗ (a k b) = g ∗ (a k )g ∗ (b) and g ∗ (a k ) = g ∗ (a) k , where a ∈ H 1 (X) and b ∈ H 2 (X) are generators of the cohomology algebra H ∗ (X). Clearly, g ∗ (a) = a. If G acts nontrivially on H ∗ (X), then we get g ∗ (b) = a 2 or g ∗ (b) = a 2 + b. If g ∗ (b) = a 2 , then g ∗ (a n b) = a n+2 = 0. This gives a n b = 0, a contradiction. So
One can describe the Hochschild and ordinary cohomology of ﬁnite inverse categories more precisely in terms of products of the Hochschild cohomology and ordinary cohomology of group algebras; this will be an easy consequence of the explicit description in 4.1 below of an iso- morphism between kC and a direct product of matrix algebras over group algebras. For the same reason, standard results on Schur multipliers for ﬁnite groups carry over to ﬁnite inverse categories, such as:
We have already seen that Galois cohomology of rigid fields is always uni- versally Koszul, even if it is not strongly Koszul. Here we would like to show that these algebras have Koszul filtrations, and moreover, show that Koszul filtrations are preserved by direct sums and twisted extensions. Of course, this could easily follow from the fact that Universal Koszulity implies the existence of a Koszul filtration, However, these results can stand in their own terms, as outside of the context of Galois cohomology, there are very few examples of Universally Koszul algebras.