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In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.

We establish a “second vanishing theorem” for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

In this paper, we provide number-theoretic formulas for Farrell–Tate cohomology for SL 2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander–Milnor conjecture, (III) cohomology of SL 2 over number fields.

L.N. Vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a Witt group structure. R.A. Rao and W. van der Kallen showed that the Vaserstein symbol need not be injective over three dimensional affine algebras over the real field, but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic ≠ 2 , 3 . R.G. Swan, R.A. Rao and J. Fasel gave another example of a real affine algebra of dimension three for which the Vaserstein symbol is not injective. We demonstrate an uncountably infinite family of non-isomorphic affine algebras of dimension three over the real field for which the Vaserstein symbol is not injective.

If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology functors finitary for all sufficiently large $n$. We establish sufficient conditions for a group $G$ possessing a finite dimensional model for $e.g.$ to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if $G$ is a locally (polycyclic-by-finite) group, then $G$ has cohomology almost everywhere finitary if and only if $G$ has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of $G$ is finitely generated., 26 pages

We show how Mislin's theorem on group homomorphisms that induce an isomorphism in cohomology can be proved based on ideas of Alperin and the use of Lannes's T -functor. This enables us to extend the class of groups to include groups of finite virtual cohomological dimension and profinite groups.

Let G be a profinite group, p a prime number and A a profinite Z ˆ 〚 G 〛 -module of finite projective p-dimension pd G , p ( A ) . Let N be a closed normal subgroup of G such that N is of finite cohomological p-dimension cd p ( N ) , H k ( N , F p ) is nonzero and finite for k = min { cd p ( N ) , pd G , p ( A ) } and N acts trivially on A. Then the virtual projective p-dimension vpd G / N , p ( A ) of A as a Z ˆ 〚 G / N 〛 -module is finite and vpd G / N , p ( A ) = pd G , p ( A ) − k . In the case when A = Z ˆ this generalizes the main result from [Th. Weigel, P.A. Zalesskii, Profinite groups of finite cohomological dimension, C. R. Acad. Sci. Paris Ser. I 338 (2004) 353–358].

Ž . 2 2 Ž . Let F s, t s X s q 2 X st q X t 1 F i F 3 be a collection of bii i0 i1 i2 nary quadratic forms with indeterminate coefficients over a field k and let w x R s k X . The aim of this paper is to compute the cohomoi, j 1F iF 3, 0 F jF 2 logical dimension of the ideal I ; R generated by the following Sylvester Ž . Ž . Ž . Ž . resultants: Res F , F , Res F , F , Res F , F , and Res F q F , F . 1 2 1 3 2 3 1 2 3 The closed points of the affine variety defined by I correspond to coefficients of the three forms having a common zero. Thus, in the w x Ž terminology of L1 , I is a resultant system for F , F , and F compare 1 2 3 w x. with Theorem 1.b in L1 . In general, if F , . . . , F are binary forms of 1 s degrees d G d G ??? G d , one can show that a resultant system is 1 2 s s Ž . Ž w x. generated by at least 1 q Ý 1 q d elements Theorem 1 in L1 and ts3 t one would like to know if this lower bound is sharp. In our case, where s s 3 and d s d s d s 2, the question is whether 1 2 3 4Ž . I can be generated by three elements. This would imply that H R s 0 I and this is the motivation behind the main result of this paper. 4Ž . It turns out that it is possible to show that H R s 0 working in the I Ž . subring of joint invariants of G s GL 2, k acting on R via change of Ž . variables in s, t . This calculation is based on some recent results by Lyubeznik on F-modules, which include an algorithm for deciding whether local cohomology modules over regular rings of prime characteristic p vanish. To the best of the author’s knowledge, Theorem 9 in this paper is the first application of this algorithm. Yan has recently produced a 4Ž . w x topological argument showing that H R vanishes in characteristic 0 Y . I

Let G be a connected algebraic Q-group, S, the Steinberg representation of G= G(Q). Recall that SG may be realized on the reduced integral homology of the Tits building of parabolic Q-subgroups of G (see Section 1). In this article, we combine facts about SG with the Borel-Serre Duality Theorem (see (3.1)) and basic Lie theory to derive new results on the cohomology of arithmetic subgroups r of G. These are summarized below. Our sharpest results apply to H’(T, -) (v is the virtual cohomological dimension of r) where G is split over E. Some of our work generalizes that of Ash, Ash and Rudolph, and Lee and Szczarba on &5,(Z). See [Al, A-R, L-S, L-S1 1. In earlier papers on this topic, an important ingredient has been a simplicial complex Y of dimension equal to vcd S&(E) on which &5,(Z) acts cocompactly with finite cell stabilizers. The existence of such a Y for a general Chevalley group has not been verified. (There has been recent success with SP,(Z) in [M-M].) Roughly speaking, we avoid this issue by using SG as a substitute for the top chain group of Y. Assume for now the G is semisimple, split over H, and has no factor of We 4.

A pro-aflne algebraic group G, over the algebraically closed field k, is an inverse limit of affine algebraic groups over k [5, p. 11271. This paper studies the cohomology, in the category of rational modules, of prounipotent groups when k has characteristic zero (a finite-dimensional rational module V for a pro-affine group G is an abstract G-module such that the corresponding homomorphism G -P GL( v) is a morphism of pro-atfine groups and in general a rational G-module is a direct limit of finitedimensional ones). This theory closely parallels that of the cohomology of pro-p groups [ 12, Chap. I]: the free prounipotent groups turn out to be precisely those of cohomological dimension ones, and the dimension of the first and second cohomology groups give numbers of generators and relations. In addition, one-relator groups turn out to have cohomological dimension two, parallel to the situation for discrete groups [9]. Finally, we apply our theory to the universal pro-affine hull A of a group r to conclude that if r has a free subgroup of finite index, then the prounipotent radical of A is a free prounipotent group. The paper is divided into five sections. The first contains preliminary material on: the pro-variety structure of prounipotent groups, the existence and description of injective rational modules and cohomology, and various technical results used in the rest of the paper. In the second section we define free prounipotent groups and characterize them as groups of cohomological dimension one. This allows us to show that pro-affine subgroups of free prounipotent groups are free prounipotent. In Section 3 we interpret the lowdimensional cohomology of a free prounipotent group with coefficients in the

Our purpose here is to obtain generalizations of several results on cohomological dimension of algebras due to M. Auslander, Eilenberg, Rosenberg, and Zelinsky in which the notion of separable algebra plays a significant role. The generalization consists of substituting the more general notion of separable K -rings for that of separable algebra. We prove also that the cohomological dimension of a semi-primary purely inseparable algebra over a field is infinite. In addition, we obtain a relative version of one inequality for projective and cohomological dimensions by Hochschild.

Let G be a group and ZG its integral group ring. If H is a subgroup of the group 6, the (right) augmentation ideal d,(H) is the right ideal of ZG generated by the elements h 1 with h E H. If HI and Hz are two subgroups of G and H, n H2 = H, then clearly d,(H) C A,(H,) n A&H,). Theorem 1 below characterizes the situation in which A,(H) = A,(H,) n A&H,). As a consequence we obtain a necessary condition for the generalized free product of two finitely generated free groups to be again free and an upper bound for the cohomological dimension of a generalized free product.

We prove that the sheaf of arithmetic differential operators with overconvergent coefficients, introduced by P. Berthelot, has finite cohomological dimension. A similar geometrical proof shows that the weak p -adic completion of the Weyl algebra has also finite cohomological dimension. Moreover, this algebra can be naturally endowed with a filtration which is compatible with the Frobenius.

The celebrated theorem that a group of cohomological dimension one is free was first proved by Stallings [8] under the assumption that the group is finitely generated. Later Swan [lo] was able to obtain this theorem without such an assumption. Thus we have that the cohomological dimension (cd) of a group G is the same as its category and geometric dimension with the only possible exception when cd G is two (2). Groups of cd 2 are by no means rare [3, Section 8.31. This paper stems from an attempt to classify such groups. We find that the simple infinite groups constructed by Camm [l] are of cd 2 (Section 4), and a locally nilpotent group of cd 2 turns out to be Z @ Z or a noncyclic subgroup of Q (Section 3, Theorem C). Recent work of Gruenberg [3] and Stammbach [9] on locally nilpotent groups and solvable groups, relate the cd of such groups to their Hirsch numbers. We obtain that a torsion-free solvable group is of finite cd if and only if it has finite Hirsch number (Section 2, Theorem I). With this grouptheoretical interpretation of cd we shall prove that a locally nilpotent group is of finite cd if and only if its abelian subgroups are of finite cd. (Theorem A). Moreover, in this case, it is merely a subgroup of some unitriangular group T&J of various degrees d (Theorem B). The Mal’cev completion G* of a torsion-free locally nilpotent group G is used as the basic tool throughout this paper.

In [lo] and [7], Serre and Oort have shown that the category of commutative group schemes of finite type over an algebraically closed field has homological dimension one if the field has characteristic zero and two otherwise. We extend their result by relating the homological dimension of this category of group schemes over any perfect field to the cohomological dimension of the Galois group of the field. In particular, if ;Z and B are abelian varieties over a finite field then our result implies that Exti(-4, B) = 0 for i > 2, and this completes the computation of these groups (see [4]). Also, Oort and Oda [8] have shown that Exts(A, B) = 0 if ;1 and B are abelian varieties over an algebraically closed field. We give a short proof of

Let G be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by Colliot-Thelene, Gille and Parimala. Basing on their results, we compute the group of classes of R-equivalence G(k)/R, the defect of weak approximation AΣ (G), the first Galois cohomology H 1 (k, G), and the Tate–Shafarevich kernel x 1 (k, G) (for suitable k) in terms of the algebraic fundamental group π1(G). We prove that the groups G(k)/R and AΣ (G) and the set x 1 (k, G) are stably k-birational invariants of G.