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Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of $\text{GL}_n(\mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $\text{GL}_n(\mathcal{O})$ in its virtual cohomological dimension., 34 pages, minor revision. To appear in Math. Z

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro when $A$ is a field of positive characteristic; our result applies, for example, when $A$ is the ring of integers., Comment: 7 pages

This work is about some features of local cohomology theory. We develop a new tool called Mayer-Vietoris spectral sequence that allows us to study several local cohomology modules supported in different ideals, which led us to generalize or retrieve previous results of several authors and also produce new ones, especially in what concerns multigraded polynomial rings. We also deal with generalized Cohen-Macaulay modules and deficiency modules, providing relations between their Bass and Betti numbers in order to both generalize classical results and produce new ones as a case of the conjecture of Auslander and Reiten in a particular case. Finally, local cohomology is viewed as an important tool for the studying of the interplay between finiteness of homological dimensions and the vanishing of Ext modules. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES Este trabalho trata de algumas características da teoria da cohomologia local. Desenvolvemos uma nova ferramenta chamada sequência espectral de Mayer-Vietoris que nos permite estudar vários módulos de cohomologia local suportados em diferentes ideais, o que nos levou a generalizar ou recuperar resultados anteriores de vários autores e também a produzir novos, especialmente no que diz respeito a anéis polinomiais multigraduados. Também lidamos com módulos Cohen-Macaulay generalizados e módulos de deficiência, fornecendo relações entre números de Bass e Betti destes de modo a tanto generalizar resultados clássicos quanto a provar novos como um caso da a conjectura de Auslander e Reiten. Finalmente, cohomologia local é vista como uma importante ferramenta para o estudo da interação entre a finitude de dimensões homológicas e de anulamento de módulos Ext.

We establish a Grothendieck–Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.

In this paper we study compacta Y that are resolvable by a free p-adic action on a compactum of a lower dimension and focus on compacta Y whose cohomological dimension with respect to the group Z[1/p] is 1.

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n., Comment: 10 pages, to appear in PAMS

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 $H_2$.

For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-Betti numbers of Aut($F_n$) and Out($F_n$) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$., ''Camille No\^us'' is a scientific consortium created to affirm the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community. This scientific collective, like Bourbaki, H. P. de Saint Gervais or A. Besse in mathematics takes on the identity of a scientific personality who embodies the collective contribution of the academic community

Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))

We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I(R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.

Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont., Comment: 11 pages

Let R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the categor...

We give a proof without use of perfectoid geometry of the following vanishing theorem of Scholze: for $X\subset \mathbb{P}^n$ a projective scheme of dimension $d$ over an algebraically closed characteristic $0$ field, and $X_r$ the inverse image of $X$ via the map that assigns $(x_0^{p^r}: \dots : x_n^{p^r})$ to the homogeneous coordinates $(x_0:\ldots :x_n)$, the induced map $H^i(X, {{\mathbb{F}}}_p)\to H^i(X_r, {{\mathbb{F}}}_p)$ on étale cohomology dies for $i>d$ and $r$ large. Our proof holds in characteristic $\ell \neq p$ as well.

We construct free abelian subgroups of the group $U(A_\Gamma)$ of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group $U(A_\Gamma)$ was previously studied by Charney, Stambaugh and the second author, who constructed a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of the principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension., Comment: Improvements in exposition, Lemma 4.27 added to clarify proof of Theorem 4.28, remark 3.8 about the definition of "compatibility" added

We investigate the topological nilpotence degree, in the sense of Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$ is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian $p$-subgroups. By replacing centralizers of elementary abelian $p$-subgroups with components of Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn's result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn's result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our results apply much more generally, for example, we establish results for $p$-local compact groups in the sense of Broto-Levi-Oliver, for connected $H$-spaces with Noetherian mod $p$ cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson's depth conjecture in the case of a Noetherian unstable algebra of minimal depth., 42 pages; comments welcome v2; Significantly revised version, accepted for publication in Selecta Mathematica

An author added. "Camille Noûs" is a scientific consortium created to affirm the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community. This scientific collective, like Bourbaki, Henri Paul de Saint Gervais or Arthur Besse in mathematics, or Isadore Nabi in biology, takes on the identity of a scientific personality who embodies the collective contribution of the academic community. More precisely, Camille Noûs is a collective individual who symbolizes our deep attachment to the values of ethics and probation that are carried by the contradictory debate, she is insensitive to the indicators elaborated by the institutional management of research, she knows what our results owe to collective construction. This is the meaning of the "Noûs", bearing a collegial We but referring above all to the concept of "reason" (or "rational" or "intellect") inherited from Greek philosophy.Some comments added.; International audience; The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional ℓ 2-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the ℓ 2-Betti numbers of Aut(Fn) and Out(Fn) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing ℓ 2-Betti numbers in degree 3 and 2 and we prove for instance that F_2^d × Z has ergodic dimension d + 1.; Le but de cette note est d'introduire une astuce qui relie l'annulation (ou la non-annulation) du nombre de Betti ℓ 2 en dimension maximale des actions d'un groupe avec l'annulation pour ses sous-actions. On fournit trois différents types d'applications : on montre que les nombres de Betti ℓ 2 de Aut(Fn) et Out(Fn) (et de leurs sous-groupe de Torelli) ne s'annulent pas en degréégalà leur dimension cohomologique virtuelle ; on prouve qu'un sous-groupe quelconque du groupe fondamental d'une variété compacte de dimension 3 a ses nombres de Betti ℓ 2 nuls en degré 3 et 2 et enfin, on parvientà déterminer la dimension ergodique de certains produits directs de la forme H × A où A est moyennable infini.

We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this., Comment: manuscript submitted to Canadian Journal of Mathematics in May 2021

For G a group of finite virtual cohomological dimension and a prime p, the p-period of G is defined to be the least positive integer d such that Farrell cohomology groups Hi(G; M) and Hi+d(G; M) have naturally isomorphic ZG modules M. We generalize a result of Swan on the p-period of a finite p-periodic group to a p-periodic infinite group, i.e., we prove that the p-period of a p-periodic group G of finite vcd is 2LCM(|N(axn) / C(axn)|) if the G has a finite quotient whose a p-Sylow subgroup is elementary abelian or cyclic, and the kernel is torsion free, where N(-) and C(-) denote normalizer and centralizer, axn ranges over all conjugacy classes of Z/p subgroups. We apply this result to the computation of the p-period of a p-periodic mapping class group. Also, we give an example to illustrate this formula is false without our assumption.