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We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of a finitely generated parafree augmented algebra of infinite cohomological dimension . Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and give a sufficient condition for an augmented algebra to be residually nilpotent in terms of its relations.

We study the mod- p cohomology of the group O u t ( F n ) of outer automorphisms of the free group F n in the case n = 2 ( p − 1 ) which is the smallest n for which the p -rank of this group is 2 . For p = 3 we give a complete computation, at least above the virtual cohomological dimension of O u t ( F 4 ) (which is 5). More precisely, we calculate the equivariant cohomology of the p -singular part of outer space for p = 3 . For a general prime p > 3 we give a recursive description in terms of the mod- p cohomology of A u t ( F k ) for k ≤ p − 1 . In this case we use the O u t ( F 2 ( p − 1 ) ) -equivariant cohomology of the poset of elementary abelian p -subgroups of O u t ( F n ) .

Using Auslander’s G-dimension, we assign a numerical invariant to any group Γ . It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ . We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.

Let Γ be a group of finite virtual cohomological dimension n. In this paper, we show that Γ has a multiple complex, i.e. Γ has a projective resolution that is the tensor product of γ complexes which are resolutions of some period after n-steps, where γ is the maximum of the p-ranks of the finite elementary abelian p-subgroups of Γ. This is an extension of the results of Benson and Carlson (Math. Z. 195 (1987) 221) and Benson and Greenless (J. Algebra 192 (1997) 678). Finally, we present a result on gaps in Farrell cohomology using a multiple complex. This result is a generalization of the result of Benson et al. (J. Algebra 131 (1990) 40) and Juan-Pineda (J. Pure and Appl. Algebra 111 (1996) 213).

Let p be an odd prime. We compute the p -primary cohomology of Out ( F n ) above its virtual cohomological dimension, for the first three cases for which the p -rank of Out ( F n ) equals one, namely n = p −1, p and p +1.

In this paper we present algorithms that compute certain local cohomology modules associated to ideals in a ring of polynomials containing the rational numbers. In particular, we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of D-modules.

Let Γ be a discrete group of finite virtual cohomological dimension, and let V be a f.g. Z -torsion free Γ module. In this situation, the Farrell cohomology of Γ with coefficients in V is defined and we prove in this paper cohomological non-vanishing results for these groups similar to those existing for finite groups and Tate cohomology, i.e. that we either have that these groups vanish for all dimensions or there are infinitely many nontrivial. The proof is based on a geometric approach to these groups, the study of Euler characteristics, minimal resolutions, and the notion of exponent.

This can be regarded as a generalization of Serre’s Theorem that every torsion-free group of finite virtual cohomological dimension has finite cohomological dimension (see [2]). For suppose that G is torsion-free and that H has cohomological dimension n < co. Then the nth syzygy in any projective resolution of Z over ZG is projective as ZH-module, and Moore’s Conjecture implies at once that it is also projective as ZG-module. One of the special cases of the conjecture which we prove here also implies Serre’s Theorem. It is natural to generalize the conjecture, because, in addition to group rings, it makes perfect sense for crossed products and in fact for strongly group-graded rings. Recall that a G-graded ring is a ring A with a direct sum decomposition

In this paper, we investigate negative degree elements of group cohomology and their products. We give a definition for arbitrary groups, which generalizes the definitions of Tate for finite groups and Farrell for groups of finite virtual cohomological dimension. We find that for finite groups, if we take coefficients in a field of characteristics p, very often all products between elements of negative degree vanish. In particular, this happens when the depth of the positive cohomology ring is greater than one. The latter is the case, for example, when a Sylow p-subgroup has a non-cyclic center.

Let S be a locally spectral topological space. Under a minor technical assumption on S (which is empty if S is spectral) it is proved that the sheaf-theoretic cohomological dimension of S is not larger than the combinatorial (‘Krull’) dimension. To this end a weakening of the notion of augmented simplicial space is studied. A typical example of such a quasi-augmented simplicial space X.→S is the simplicial space of (finite) specialization chains in a locally spectral space S. It is shown that there are natural inverse and direct image functors between sheaves on X. and sheaves on S which form a topos morphism. The notion of cohomological descent allows to study sheaf cohomology on X. instead of S. Criteria are provided for cohomological descent to hold, which are then applied to give the result announced above.

A characterization of dcc categories of cohomological dimension one is given from which a characterization of dcc posets of cohomological dimension one is deduced.

The mapping class group of a closed orientable surface of genus g, denoted Γg, is defined as the group of path components of the group of orientation preserving diffeomorphisms of the surface. We completely calculate the p-torsion of the Farrell-Tate cohomology of Γp − 1. The Farrell-Tate and ordinary cohomologies of Γg coincide above the virtual cohomological dimension 4g − 5.The basic method is to describe for Zp⊂Γp − 1 the quotient group N(Z / p) / Z / p as a finite extension of the pure mapping class group, where N(·) is the normalizer in Γp−1. Then putting a result of Cohen about the cohomology of the pure mapping class group into the Lyndon- Hochschild-Serre spectral sequence, we obtain the ordinary (and Farrell-Tate) cohomology of the normalizer group. The result about the p-torsion of the Farrell-Tate cohomology of Γp−1 then follows.

In this paper free and proper discontinuous actions on Rm × Sn − 1 are constructed of groups with infinite virtual cohomological dimension.

We define and develop the rudiments of a cohomology theory suitable for the treatment of the absolute Galois groups of formally real fields. Although these groups have torsion, a sensible notion of cohomological dimension can be defined in this theory.

The study made in this paper was motivated by the following question: given n 2 0, is the class of fields of cohomological dimension in elementary? It is relatively easy to see that this class is closed under elementary substructures, see Proposition 2.5; for n12 it is however unknown whether it is closed under ultraproducts, or even ultrapowers. Klingen showed in [6] that the class of fields having a class formation is elementary. As fields having a class formation have cohomological dimension 2, this suggests the following type of question: given an elementary class generated by fields of cohomological dimension 5 2, do all its elements have cohomological dimension 12? In this paper we propose to show that the elementary class S generated by all (totally imaginary) global fields and their algebraic extensions contains only fields of cohomological dimension ~2 (Theorem 3.6). This implies in particular that cd(Q*(i)) =2. We also study, for Q* a non-standard extension of Q, some of the properties of G(Q*); in contrast with the standard case, G(Q*) contains non-pro-cyclic abelian subgroups (Proposition 3.7), and cd(Q*(p))> 1 (Corollary 3.8). However, as in the standard case, we have cd(Q*“b)= 1 (Theorem 3.11). Our study allows us, given a prime number p, to find a sentence ulp such that for every field KES

culim : Abe+ Ab . It is not difficult to see (and it will fail out below) that cdCat hd Calways. In [4), Dana Latch showed, using a spectral sequence, that if C has at most Hn morphisms, then (04 cdChdC

In this paper we introduce and study the cohomology of a small category with coefficients in a natural system. This generalizes the known concepts of Watts [23] (resp. of Mitchell [17]) which use modules (resp. bimodules) as coefficients. We were led to consider natural systems since they arise in numerous examples of linear extensions of categories; in Section 3 four examples are discussed explicitly which indicate deep connection with algebraic and topological problems: (1) The category of Z//p2-modules, p prime. (2) The homotopy category of Moore spaces in degree n, n22. (3) The category of group rings of cyclic groups. (4) The homotopy category of Eilenberg-MacLane fibrations. We prove the following results on the cohomology with coefficients in a natural system: (5) An equivalence of small categories induces an isomorphism in cohomology. (6) Linear extensions of categories are classified by the second cohomology group HZ. (7) The group H’ can be described in terms of derivations. (8) Free categories have cohomological dimension 5 1, and category of fractions preserve dimension one. (9) A double cochain complex associated to a cover yields a method of computation for the cohomology; two examples are given. The results (7) and (8) correspond to known properties of the Hochschild-Mitchell cohomology, see [7] and [ 171. In the final section we discuss the various notions of cohomology of small categories, and we show that all these can be described in terms of Ext functors studied in the classical paper [l l] of Grothendieck.

A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G/G′≃Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G′ then G′ should be free in the special case when the commutator G′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FPn-1 and G/N≃Z then N is of homological type FPn and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)⩽cd(G)-1.Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds.

Let R = ⨁ n ∈ N 0 R n be a Noetherian homogeneous ring with local base ring ( R 0 , m 0 ) and irrelevant ideal R + , let M be a finitely generated graded R -module. In this paper we show that H m 0 R 1 ( H R + 1 ( M ) ) is Artinian and H m 0 R i ( H R + 1 ( M ) ) is Artinian for each i in the case where R + is principal. Moreover, for the case where ara ( R + ) = 2 , we prove that, for each i ∈ N 0 , H m 0 R i ( H R + 2 ( M ) ) is Artinian if and only if H m 0 R i + 2 ( H R + 1 ( M ) ) is Artinian. We also prove that H m 0 d ( H R + c ( M ) ) is Artinian, where d = dim ( R 0 ) and c is the cohomological dimension of M with respect to R + . Finally we present some examples which show that H m 0 R 2 ( H R + 1 ( M ) ) and H m 0 R 3 ( H R + 1 ( M ) ) need not be Artinian.

We study, by using the theory of algebraic D -modules, the local cohomology modules supported on a monomial ideal I of the local regular ring R=k[[x 1 ,…,x n ]] , where k is a field of characteristic zero. We compute the characteristic cycle of H I r (R) and H m p (H I r (R)) , where m is the maximal ideal of R and I is a squarefree monomial ideal. As a consequence, we can decide when the local cohomology module H I r (R) vanishes and compute the cohomological dimension cd(R,I) in terms of the minimal primary decomposition of the monomial ideal I . We also give a Cohen–Macaulayness criterion for the local ring R/I and compute the Lyubeznik numbers λ p,i (R/I)=dim k Ext R p (k,H I n−i (R)) .

Let G be a group, spli Z G the supremum of the projective lengths of the injective Z G -modules and silp Z G the supremum of the injective lengths of the projective Z G -modules. The invariants spli Z G and silp Z G were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] in connection with the existence of complete cohomological functors. If spli Z G is finite then silp Z G = spli Z G = findim Z G [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and cd ¯ Z G ≤ spli Z G ≤ cd ¯ Z G + 1 , where cd ¯ Z G is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that cd ¯ Z G = vcd G if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ , Arch. Math. 89 (1) (2007) 24–32] that if spli Z G is finite then G admits a finite dimensional model for E ¯ G , the classifying space for proper actions. We conjecture that spli Z G = cd ¯ Z G + 1 for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type FP ∞ .

This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category of Artin-Tate motives with the derived category of mixed Artin-Tate motives. Update in second version: a functorial and strict weight filtration for mixed Artin-Tate motives is established. Moreover, two minor corrections have been performed: first, the category of Artin-Tate motives is now defined to be the triangulated category generated by direct factors of $f_* \mathbf 1(n)$---as opposed to the thick category generated by these generators. Secondly, the exactness of $f^*$ for a finite map is only stated for etale maps $f$.

It is proved that if E is an idempotent matrix with entries in the group algebra KG of a group G of finite cohomological dimension over a field K of characteristic zero, then the partial augmentations of the trace of E corresponding to the elements of infinite order are all zero, provided G contains a free Abelian subgroup of rank cd K G − 1 in its centre.