Unramified Cohomology Research Articles

Unramified degree three invariants for reductive groups of type A

Let G be a reductive group over an algebraically closed field F of characteristic zero such that the Dynkin diagram of G is the disjoint union of diagrams of type A. We prove that the third unramified cohomology group of the classifying space of G with coefficients in Q/Z is trivial.

Cohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5

We consider geometrically cellular varieties $X$ over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group $H^3_{nr}(X,\mathbb{Q}/\mathbb{Z}(2))$ by its constant part. For $X$ a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient if finite. For $X$ a del Pezzo surface of degree $\geq 5$, we show that this quotient is zero, unless $X$ is a del Pezzo surface of degree 8 of a special type.

Degree three unramified cohomology of adjoint semisimple groups

We compute degree 3 unramified cohomology of adjoint semisimple group. New classes of non-rational adjoint semisimple groups are found.

The derived functors of unramified cohomology

We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.

Torsion 1-cycles and the coniveau spectral sequence

We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree 2{\dim}(X)-3 . We also describe in terms of \mathcal H -cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.

Birational Motives, II: Triangulated Birational Motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".

Modules de cycles et classes non ramifiées sur un espace classifiant

Let G be a finite group of exponent m and let k be a field of characteristic prime to m, containing the m-th roots of unity. For any Rost cycle module M over k, we construct exact sequences detecting the unramified elements in Serre's group of invariants of G with values in M in terms of ''residue morphisms associated to pairs (D,g), where D runs through the subgroups of G and g runs through the homomorphisms \mu_m\to G whose image centralises D. This allows us to recover results of Bogomolov and Peyre on the unramified cohomology of fields of invariants of G, and to generalise them.

Degree three unramified cohomology groups

Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G) by k-automorphisms defined by h⋅xg=xhg for any g,h∈G. Denote by k(G)=k(xg:g∈G)G, the fixed subfield. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. The unramified Brauer group Brnr(C(G)) and the unramified cohomology Hnr3(C(G),Q/Z) are obstructions to the rationality of C(G) (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G such that |G|=p12, Brnr(C(G))={0}, but Hnr3(C(G),Q/Z)≠{0}; thus C(G) is not stably C-rational [12]. Using Peyre's method, we are able to find groups G with |G|=p9 where p is an odd prime number such that Brnr(C(G))={0}, Hnr3(C(G),Q/Z)≠{0}.

Unirational threefolds with no universal codimension $$2$$ 2 cycle

We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of 1-cycles. This implies that its third unramified cohomology group is not universally trivial.

Weak Approximation for Tori Over p-adic Function Fields

We study local–global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus being weak approximation of rational points. We construct a 9-term Poitou–Tate-type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus.

Chow–Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles

Let k k be a field, and let Ω \Omega be a universal domain over k k . Let f : X → S f:X \rightarrow S be a dominant morphism defined over k k from a smooth projective variety X X to a smooth projective variety S S of dimension ≤ 2 \leq 2 such that the general fibre of f Ω f_\Omega has trivial Chow group of zero-cycles. For example, X X could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that X X has dimension ≤ 4 \leq 4 . Then we prove that X X has a self-dual Murre decomposition, i.e., that X X has a self-dual Chow-Künneth decomposition which satisfies Murre’s conjectures (B) and (D). Moreover, we prove that the motivic Lefschetz conjecture holds for X X and hence so does the Lefschetz standard conjecture. We also give new examples of 3 3 -folds of general type which are Kimura finite dimensional, new examples of 4 4 -folds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.

Cohomological invariants of algebraic tori

Given a field F of arbitrary characteristic and an algebraic torus T/F we calculate degree 2 and 3 cohomological invariants of T with values in Q/Z(1) and Q_p/Z_p(2) respectively, the latter for p not equal to 2, char(F) and generalize the former to other algebraic groups. Moreover, we obtain descriptions of the corresponding unramified cohomology groups, and in particular of H 3 nr (F(T), mn \otimes 2 ) for n prime to 2 and char(F) . In the process, we construct a useful short exact sequence for cohomological invariants and make connections with recent results on Chow groups of codimension 2.

Cycles de codimension 2 etH3non ramifié pour les variétés sur les corps finis

AbstractLetXbe a smooth projective variety over a finite field$\mathbb{F}$. We discuss the unramified cohomology groupH3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds,H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefoldXequipped with a fibration onto a curveC, the generic fibre of which is a smooth projective surfaceVover the global field$\mathbb{F}$(C), the vanishing ofH3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors onXimplies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles onV. This sheds new light on work started thirty years ago.

Unramified cohomology, [formula omitted]-connectedness, and the Chevalley–Warning problem in Grothendieck ring

We study the Chevalley–Warning problem in the Grothendieck ring K0(Var/k). We show that the A1-homotopy theory yields well-defined invariants on K0(Var/k)/L, in particular the Brauer group is such an invariant. We use this to give a concrete counter-example to the Chevalley–Warning conjecture over a C1-field (Brown and Schnetz, 2011 [6]). This also gives a negative answer to the question in Bilgin (2011) [5, Ques. 3.8].

Cohomologie non ramifiée et conjecture de Hodge entière

Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of Voisin implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Colliot-Thélène and Ojanguren implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fiber.

Unramified cohomology of alternating groups

Abstract We prove vanishing results for the unramified stable cohomology of alternating groups.

ℤ[1/p]-motivic resolution of singularities

AbstractThe main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

Classes non ramifiées sur un espace classifiant

In this Note, we establish a general formula for the unramified cohomology of fields of linear invariants by finite groups. Such formulas are known in degree 2 and 3.

Weight structures vs.t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)

AbstractIn this paper we introduce a new notion ofweight structure (w)for a triangulated categoryC; this notion is an important natural counterpart of the notion oft-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.Theheartofwis an additive categoryHw⊂C. We prove that a weight structure yields Postnikov towers for anyX∈ObjC(whose 'factors'Xi∈ObjHw). For any (co)homological functorH:C→A(Ais abelian) such a tower yields aweight spectral sequenceT : H(Xi[j]) ⇒H(X[i + j]); Tis canonical and functorial inXstarting fromE2.Tspecializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motivesDMeffgm(if we considerw = wChowwithHw=Choweff) and to Atiyah-Hirzebruch spectral sequences in topology.We prove that there often exists an exact conservative weight complex functorC→K(Hw). This is a generalization of the functort:DMeffgm→Kb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove thatK0(C) ≅K0(Hw) under certain restrictions.We also introduce the concept of adjacent structures: at-structure isadjacenttowif their negative parts coincide. This is the case for the Postnikovt-structure for the stable homotopy categorySH(of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow)t-structuretChowforDMeff_⊃DMeffgmwhich is adjacent to the Chow weight structure. We haveHtChow≅ AddFun(Choweffop,Ab);tChowis related to unramified cohomology. Functors left adjoint to those that aret-exact with respect to somet-structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging toC(X,Y): the one that comes from weight truncations ofXwith the one coming fromt-truncations ofY(forX,Y∈ObjC). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E2of coniveau spectral sequences (by Bloch and Ogus).

Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) G . Specializing to the case where G is a central extension of an F p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) G is not rational although its unramified cohomology group of degree 2 is trivial.