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Cohomological Invariants in Positive Characteristic
Abstract
We determine the mod p cohomological invariants for several affine group schemes G in characteristic p. These are invariants of G-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod p Milnor K-theory of fields.
... Proposition 4.12 (1)]. The idea of considering tame subgroups goes back to the works due to Kato [18], Izhboldin [14], Garibaldi et al. [9], Auel et al. [2] and Totaro [25]. In Sect. ...
... Hence, the collection of Zgraded abelian groups i≥0 H i+1,i (K ) K does not form a cycle module in the sense of Rost [22]. To remedy the situation, we will consider tame subgroups of H i+1,i (K ) (cf. [2,9,14,18,25]). ...
... where K t is a maximal tamely ramified extension of K (cf. [25,4] ...
Auel–Bigazzi–Böhning–Graf von Bothmer proved that if a proper smooth variety X over a field k of characteristic p>0 has universally trivial Chow group of 0-cycles, the cohomological Brauer group of X is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod p étale motivic cohomology groups. We also see that the properness assumption on the variety X can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
... When the discretely valued field is henselian of characteristic > 0, Kato [Kat89, Page 110] and Izhboldin [Izh96] analyzed the wild quotient of +1, ( ) = +1 ( , Z/ ( )) = 1 ( , Ω ,log ) (Section 2.3), which is the quotient of this group by its tamely ramified part (defined below). Totaro [Tot22] generalizes the result to arbitrary discrete valuation fields. Kato defined an increasing filtration of +1, ( ) as follows: For ≥ 0, let be the subgroup of +1, ( ) generated by elements of the form 1 1 ∧ · · · ∧ with ∈ , 1 , . . . ...
... It says that, for of characteristic > 0, the subgroups of closed forms in Ω is generated by the exact forms together with the forms of the form ( 1 / 1 ) ∧ · · · ∧ ( / ) [Izh96, Lemma 1.5.1]. To describe the subgroup 0 , we need to describe tame extensions of [Tot22]. We fix a discrete valuation as above. ...
We use Kato's Swan conductor to study the Brauer p-dimension of fields of characteristic . We mainly investigate two types of fields: henselian discretely valued fields and semi-global fields. While investigating the Brauer p-dimension of semi-global fields, we use a Gersten-type sequence to analyse the ramification behavior of a Brauer class in a 2-dimensional regular local ring. Using this result, we give a partial result on the Brauer p-dimension of function fields of algebraic curves over with good reduction.
... Finally, to show that 1, ur is P 1 -invariant, we apply Lemma 6.6. Condition (1) is due to Izhboldin [15] (see also [29,Theorem 4.4]), and (2) is a part of Theorem 6.3. Lemma 6.6. ...
... and the same for the right vertical map. The statement for the upper horizontal map in the case when = 1 is a consequence of [29,Theorem 4.3]. Indeed, given any discrete valuation ring over k, the cited theorem shows that there is an exhaustive filtration 1, ( ) = −1 ⊂ 0 ⊂ 1 ⊂ · · · ⊂ 1, (Frac ) whose graded quotients are described solely in terms of the residue field. ...
Let X be a smooth proper variety over a field k and suppose that the degree map is isomorphic for any field extension K/k . We show that is an isomorphism for any -invariant Nisnevich sheaf with transfers G . This generalises a result of Binda, Rülling and Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge–Witt cohomology is a -invariant Nisnevich sheaf with transfers.
... x , W n Ω i X,log ) = 0, and the same for the right vertical map. The statement for the upper horizontal map is a consequence of [28,Theorem 2.3]. Indeed, given any discrete valuation ring over k, the cited theorem shows that there is an exhaustive filtration F i (R) = U −1 ⊂ U 0 ⊂ U 1 ⊂ · · · ⊂ F i (Frac R) whose graded quotients are described solely in terms of the residue field. ...
... Finally, to show F i is P 1 -invariant we apply Lemma 6.5 below. The condition (1) is due to Izhboldin [14] (see also [28,Theorem 2.4], [22,Proposition 4.9]), and (2) is a part of Theorem 6.3. Lemma 6.5. ...
Let X be a smooth proper variety over a field k and suppose that the degree map is isomorphic for any field extension K/k. We show that is an isomorphism for any -invariant Nisnevich sheaf with transfers G. This generalize a result of Binda-R\"ulling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a -invariant Nisnevich sheaf with transfers.
... It says that, for a field of characteristic > 0, ∈ N, the subgroups of closed forms in Ω is generated by the exact forms together with the forms of the form ( 1 / 1 ) ∧ · · · ∧ ( / ) [Izh96, Lemma 1.5.1]. To describe the subgroup 0 , we need to describe tame extensions of [Tot22]. We fix a discrete valuation as above. ...
We use Kato's Swan conductor to systematically investigate the Brauer p-dimension of henselian discretely valued fields of residual characteristic . We transform the period-index problem of these fields into the symbol length problem for certain abelian groups relating to Kahler differentials of residue fields.
We determine the group of cohomological invariants of algebraic tori of degree and weight 2.
Let G be a finite group and F a field. We give an elementary proof that the group of normalized cohomological invariants of G over F with values in the Brauer group is isomorphic to .
Let G be a commutative affine algebraic group over a field F, and let H:FieldsF→AbGrps be a functor. A (homomorphic) H-invariant of G is a natural transformation Tors(−,G)→H, where Tors(−,G) is the functor FieldsF→AbGrps taking a field extension L/F to the group of isomorphism classes of GL-torsors over Spec(L). The goal of this paper is to compute the group Invhom1(G,H) of H-invariants of G when G is a group of multiplicative type, and H is the functor taking a field extension L/F to L×⊗ZQ/Z.
We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture.
The first version of this paper was written in the spring of 1996.
This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. The book is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory.
Let G be a commutative affine algebraic group over a field F, and let H:FieldsF→AbGrps be a functor. A (homomorphic) H-invariant of G is a natural transformation Tors(−,G)→H, where Tors(−,G) is the functor FieldsF→AbGrps taking a field extension L/F to the group of isomorphism classes of GL-torsors over Spec(L). The goal of this paper is to compute the group Invhom1(G,H) of H-invariants of G when G is a group of multiplicative type, and H is the functor taking a field extension L/F to L×⊗ZQ/Z.
Notre thèse s'intéresse aux invariants cohomologiques des groupes algébriques linéaires, lisses et connexes sur un corps quelconque. Plus spécifiquement on étudie les invariants de degré 2 à coefficients dans le complexe de faisceaux galoisiens Q/Z(1), c'est-à-dire des invariants à valeurs dans le groupe de Brauer. Pour se faire on utilise la cohomologie étale des faisceaux sur les schéma simpliciaux. On obtient une description de ces invariants pour tous les groupes linéaires, lisses et connexes, notamment les groupes non réductifs sur un corps imparfait (par exemple les groupes pseudo-réductifs ou unipotents).On se sert de la description établie pour étudier le comportement du groupe des invariants à valeurs dans le groupe de Brauer par des opérations sur les groupes algébriques. On explicite aussi ce groupe d'invariants pour certains groupes algébriques non réductifs sur un corps imparfait
Let F be a field of characteristic and G be a smooth finite algebraic group over F. We compute the essential dimension of G at p. That is, we show that
\[
{\mathrm{ed}}_{F}(G;p)=\{\begin{array}{cc}1,\hfill & \text{if}\phantom{\rule{0.25em}{0ex}}p\phantom{\rule{0.25em}{0ex}}\text{divides}\phantom{\rule{0.25em}{0ex}}|G|,\phantom{\rule{0.25em}{0ex}}\text{and}\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}
\]
This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of noncrossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.
We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.
We study the degree 3 cohomological invariants with coefficients in ℚ/ℤ(2) of a split reductive group over an arbitrary field. As an application, we compute the group of reductive indecomposable degree 3 invariants of all split simple algebraic groups.
We give a lower bound for the essential dimension of a split simple algebraic group of "adjoint" type over a field of characteristic 2. We also compute the essential dimension of orthogonal and special orthogonal groups in characteristic 2.
We study the degree 3 cohomological invariants with coefficients in Q=Z.2/ of a semisimple group over an arbitrary field. A list of all invariants of adjoint groups of inner type is given.
Let G be an algebraic group over a field F. As defined by Serre, a cohomological invariant of G of degree n with values in Q/Z(j) is a functorial-in-K collection of maps of sets TorsG(K) → Hn(K,Q/Z(j)) for all field extensions K/F, where TorsG (K) is the set of isomorphism classes of G-torsors over Spec K. We study the group of degree 3 invariants of an algebraic torus with values in Q/Z(2). In particular, we compute the group Hnr3(F (S), Q/Z(2)) of unramified cohomology of an algebraic torus S.
Group cohomology reveals a deep relationship between algebra and topology, and its recent applications have provided important insights into the Hodge conjecture and algebraic geometry more broadly. This book presents a coherent suite of computational tools for the study of group cohomology and algebraic cycles. Early chapters synthesize background material from topology, algebraic geometry, and commutative algebra so readers do not have to form connections between the literatures on their own. Later chapters demonstrate Peter Symonds's influential proof of David Benson's regularity conjecture, offering several new variants and improvements. Complete with concrete examples and computations throughout, and a list of open problems for further study, this book will be valuable to graduate students and researchers in algebraic geometry and related fields.
The paper considers generalities for localization complexes for varieties. Examples of these complexes are given by the Gersten resolutions in various contexts, in particular in K-theory and in étale cohomology. The paper gives a general notion of coefficient systems for such complexes, the so-called cycle modules. There are the corresponding “complexes of cycles with coefficients” and their homology groups, the “Chow groups with coefficients”. For these some general constructions are developed: proper push-forward, flat pull-back, spectral sequences for fibrations, homotopy invariance and intersection theory. If one specializes the material to the case of Milnor’s K-theory as coefficient system, one obtains in particular an elementary development of intersections for the classical Chow groups. This treatment is somewhat different to former approaches. The main tool is still the deformation to the normal cone. The major different is that homotopy invariance is not established alone for the Chow groups, but for the “cycle complex with coefficients in Milnor’s K-theory”. This enables one to keep control in fibered situations. The proof of associativity of intersections is based on a doubled version of the deformation to the normal cone.
This volume concerns invariants of G-torsors with values in mod p Galois cohomology - in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology - for various simple algebraic groups G and primes p. The author determines the invariants for the exceptional groups F4 mod 3, simply connected E6 mod 3, E7 mod 3, and E8 mod 5. He also determines the invariants of Spinn mod 2 for n </= 12 and constructs some invariants of Spin14. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of Spinn is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.
Let k k be a field, X X over k k a smooth variety with function field K K and E E a quadratic vector bundle over X X . Assuming that the generic fibre q q of E E is in I 3 K ⊂ W ( K ) I^3K\subset W(K) , we compute the image of its Arason invariant in C H 2 ( X ) / 2 CH^2(X)/2 by the d 2 d_2 differential of the Bloch-Ogus spectral sequence. This gives an obstruction to e 3 ( q ) e^3(q) being a global cohomology class.
We show that for a field k of characteristic p, H
i
(k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K
n
M
(k)?K
n
(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K
n
M
(k) and K
n
(k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K
n
(X,ℤ/p
r
)=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings
with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture.
We show that operations in Milnor K-theory mod p of a field are spanned by
divided power operations. After giving an explicit formula for divided power
operations and extending them to some new cases, we determine for all fields
k and all prime numbers p, all the operations
commuting with field extensions over the base field k. Moreover, the integral
case is discussed and we determine the operations for
smooth schemes over a field.
We explain how to exploit Rost’s theory of Chow groups with coefficients to carry some computations of cohomological invariants. In particular, we use the idea of the “stratification method ” introduced by Vezzosi. We recover a number of known results, with very different proofs. We obtain some new information on spin groups. Status: This paper has been accepted for publication in Documenta Mathematica. In what follows, k is a base field, and all other fields considered in this paper will be assumed to contain k. We fix a prime number p which is different from char(k), and for any field K/k, we write H i (K) for H i (Gal(Ks/K),Z/p(i)). Here Ks is a separable closure of K, and Z/p(i) is the i-th Tate twist of Z/p
Schémas en Groupes (SGA 3), 3 vols
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