Equivariant algebraic enhanced de Rham functor

The principal objective in this chapter is a proof of the de Rham theorem, one version of which we have stated in 4.17. In its most complete form it asserts that the homomorphism from the de Rham cohomology ring to the differentiable singular cohomology ring given by integration of closed forms over differentiable singular cycles is a ring isomorphism. The approach will be to exhibit both the de Rham cohomology and the differentiable singular cohomology as special cases of sheaf cohomology and to use a basic unique­ness theorem for homomorphisms of sheaf cohomology theories to prove that the natural homomorphism between the de Rham and differentiable singular theories is an isomorphism. As an added dividend of this approach we shall also obtain the existence of canonical isomorphisms of the de Rham and differentiable singular cohomology theories with the continuous singular theory, the Alexander-Spanier theory, and the Čech cohomology theory for differentiable manifolds. From these isomorphisms we shall conclude that the de Rham cohomology theory is a topological invariant of a differentiable manifold.

We prove that ifWandW′are non-zeroB-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a characterμsuch thatW(μ−1) andW′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that ifWis aB-pair and ifFis a Schur functor (for example Symnor Λn) such thatF(W) is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank ofWis sufficiently large, then there is a characterμsuch thatW(μ−1) is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply top-adic representations.

The title refers to the nilcommutative or NC -schemes introduced by M. Kapranov in ‘Noncommutative Geometry Based on Commutator Expansions’, J. Reine Angew. Math 505 (1998) 73–118. The latter are noncommutative nilpotent thickenings of commutative schemes. We also consider the parallel theory of nil-Poisson or NP -schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for NC - and NP -schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck in Crystals and the de Rham Cohomology of Schemes, Dix exposés sur la cohomologie des schémas , Masson, Paris (1968), pp. 306–358. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme Y as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme X (Theorems 6.1, 6.4, 6.7). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.2) and of Feigin and Tsygan (Corollary 6.8) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.

We study ‐gauges for de Rham cohomology of smooth algebraic varieties in characteristic . Applying to good reductions of Shimura varieties of Hodge type, we recover the Ekedahl–Oort stratifications by constructing universal de Rham ‐gauges with ‐structure. We also study the cohomology of de Rham ‐gauges on these varieties. In particular, in the PEL‐type case and when the weights of the flat automorphic vector bundles are ‐small, we determine the ‐gauge structure on their de Rham cohomology by the associated dual Bernstein‐Gelfand‐Gelfand (BGG) complexes.

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech–de Rham spectral sequence by making use of the factor map which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus T θ T_\theta is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over T θ T_\theta .

In this chapter, we study the topology of C ∞-manifolds. We define the de Rham cohomology of a manifold, which is the vector space of closed differential forms modulo exact forms. After sheafifying the construction, we see that the de Rham complex forms a so-called acyclic resolution of the constant sheaf ℝ. We prove a general result that sheaf cohomology can be computed using such resolutions, and deduce a version of de Rham’s theorem that de Rham cohomology is sheaf cohomology with coefficients in ℝ. It follows that de Rham cohomology depends only on the underlying topology. Using a different acyclic resolution that is dual to the de Rham complex, we prove Poincaré duality. This duality makes cohomology, which is normally contravariant, into a covariant theory. We devote a section to explaining these somewhat mysterious covariant maps, called Gysin maps. We end this chapter with the remarkable Lefschetz trace formula, which in principle, calculates the number of fixed points for a map of a manifold to itself.

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincar\'e Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result.

The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry , in Algebraic topology: the Abel symposium 2007 , Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k -algebra and k has characteristic 0, we show that two objects, S 1 ⊗ A and ϵ( A ), determine one another, functorially in A . The object S 1 ⊗ A is the S 1 -equivariant simplicial k -algebra obtained by tensoring A by the simplicial group S 1 := B ℤ, while the object ϵ( A ) is the de Rham algebra of A , endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S 1 -equivariant k -algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ( S 1 ⊗ A )∼ϵ( A ) . We deduce from this the comparison mentioned above, identifying the S 1 -equivariant functions on the derived loop space LX of a smooth k -scheme X with the algebraic de Rham cohomology of X / k . As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k -schemes. By construction, these decompositions are moreover compatible with the S 1 -action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

The concept of acyclic models, developed by Eilenberg and MacLane [2], appears to be one of the most convenient tools of algebraic topology. It is the purpose of this paper to describe a generalization of the theory of Eilenberg and MacLane, and to apply the theory to proving de Rham's theorem. The generalized theory has other applications; in particular the proof of the spectral sequence theorem due to Gugenheim and Moore [4] could be slightly simplified by the results of this paper. The reader is assumed to know the theory of functors and categories; the general reference for this topic will be Eilenberg and Steenrod [3, Chapter IV, pp. 108-113]. The first two chapters are devoted to establishing the existence or uniqueness of chain and cochain maps; the third chapter is devoted to cup products. In the fourth chapter the singular Cr cubical cohomology theory is defined, and it is shown that for paracompact manifolds the result is independent of r. In the last chapter the de Rham cohomology is defined and shown to be isomorphic to the singular cohomology (de Rham's theorem) by means of the Stokes' map f* (integration over chains); it is also shown that the Stokes' map preserves products, i.e., that f*(ao3) = (f*a)U(f*). The relation of acyclic models to products will be treated at length in another paper.

We show the vanishing of adjoint Bloch–Kato Selmer groups of automorphic Galois representations over CM fields. This proves their rigidity in the sense that they have no deformations that are de Rham. In order for this to make sense, we also prove that automorphic Galois representations over CM fields are de Rham themselves. Our methods draw heavily from the 10 author paper, where these Galois representations were studied extensively. Another crucial piece of inspiration comes from the work of P. Allen who used the smoothness of certain local deformation rings in characteristic $0$ to obtain rigidity in the polarized case.

We give an account of the construction of the Bhatt--Morrow--Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic $p$ and the connections to crystalline and de Rham--Witt theory.

Lorentz invariant derivative interactions for a single spin-2 field are investigated, up to the cubic order. We start from the most general Lorentz invariant terms involving two spacetime derivatives, which are polynomials in the spin-2 field as well as its first derivatives. Using a perturbative Arnowitt-Deser-Misner analysis, we determined the parameters such that the corresponding Hamiltonian possesses a Lagrange multiplier, which would signify there are at most 5 degrees of freedom that are propagating. The resulting derivative terms are linear combinations of terms coming from the expansion of the Einstein-Hilbert Lagrangian around a Minkowski background, as well as the cubic ``pseudolinear derivative term'' identified in Hinterbichler [J. High Energy Phys. 10 (2013) 102]. We also derived the compatible potential terms, which are linear combinations of the expansions of the first two de Rham--Gabadadze--Tolley mass terms in unitary gauge.

For a smooth foliated manifold $(M,\mathcal F)$, the basic and the foliated cohomologies are defined by using the de Rham complex of $M$. These cohomologies are related with the cohomology of the manifold by the de Rham spectral sequence of $\mathcal F$. A foliated manifold is an example of a space with two topologies, one coarser than the other. For these spaces one can define a continuous cohomology that, for a foliated manifold, corresponds to the continuous foliated (or leafwise) cohomology. In this paper we introduce a construction for spaces with two topologies based upon the Alexander-Spanier continuous cochains. It allows us to define a spectral sequence, similar to the de Rham spectral sequence for a foliation. In particular, continuous basic and foliated cohomologies are defined and related with the cohomology of the space. For a smooth foliated manifold, we also consider Alexander-Spanier differentiable cochains. We compare the continuous and differentiable cohomologies, and the latter with the de Rham cohomology. We prove that all three spectral sequences are isomorphic from $E_2$ onwards if $\mathcal F\/$ is a Riemannian foliation. As a consequence, we conclude that this spectral sequence is a topological invariant of the Riemannian foliation. We also compute some examples. In particular, we give an isomorphism between the $E_2$ term for a $G$-Lie foliation and the reduced cohomology of $G$ (in the sense of S.-T. Hu) with coefficients in the reduced foliated cohomology of $\mathcal F$.

Let M and N be smooth manifolds and let B ¯ ( A ) \bar B (A) be the reduced bar construction on the de Rham complex Λ ( M ) \Lambda (M) or a suitable subcomplex A of M. For every smooth map f : N → M × M f:N \to M \times M , the tensor product Λ ( N ) ⊗ B ¯ ( A ) \Lambda (N) \otimes \bar B(A) , equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration P ( M ) → M × M P(M) \to M \times M via the smooth map F. Moreover, Λ ( N ) ⊗ B ¯ ( A ) \Lambda (N) \otimes \bar B(A) can be taken as a de Rham subcomplex of the pullback space.