Cohomological dimension
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Maximally stretched laminations on geometrically finite hyperbolic manifolds Open
Let [math] be a discrete group. For a pair [math] of representations of [math] into [math] with [math] geometrically finite, we study the set of [math] –equivariant Lipschitz maps from the real hyperbolic space [math] to itself that have m…
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THE EVENTUAL STABILITY OF DEPTH, ASSOCIATED PRIMES AND COHOMOLOGY OF A GRADED MODULE Open
The asymptotic stability of several homological invariants of the graded\npieces of a graded module has attracted quite a lot of attention over the last\ndecades. We provide in this text several stability results together with\nestimates o…
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Integrality in the Steinberg module and the top-dimensional cohomology of SLn OK Open
We prove a new structural result for the spherical Tits building attached to\nSL_n(K) for many number fields K, and more generally for the fraction fields of\nmany Dedekind domains O: the Steinberg module St_n(K) is generated by integral\n…
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A note on bounded-cohomological dimension of discrete groups Open
Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to…
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Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension Open
We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois com…
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Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces Open
We use the relations between the braid and mapping class groups of a compact, connected, non-orientable surface without boundary and those of its orientable double covering to study embeddings of these groups and their (virtual) cohomo…
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HYPERBOLIC MANIFOLDS THAT FIBRE ALGEBRAICALLY UP TO DIMENSION 8 Open
We construct some cusped finite-volume hyperbolic n -manifolds $M^n$ that fibre algebraically in all the dimensions $5\leq n \leq 8$ . That is, there is a surjective homomorphism $\pi _1(M^n) \to {\mathbb {Z}}$ with finitely generated kern…
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Calculating the virtual cohomological dimension of the automorphism group of a RAAG Open
We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right‐angled Artin group. The algorithm works in the relative setting; in particular, it also applies to untwisted automorphism groups and …
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On the cohomological dimension of the moduli space of Riemann surfaces Open
The moduli space of Riemann surfaces of genus $g\\geq 2$ is (up to a finite\n\\'etale cover) a complex manifold and so it makes sense to speak of its\nDolbeault cohomological dimension. The conjecturally optimal bound is $g-2$.\nThis expec…
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Hyperbolic manifolds that fiber algebraically up to dimension 8 Open
We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $π_1(M_n) \to \mathbb Z$ with finitely generated kernel. The…
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Topological 4-manifolds with right-angled Artin fundamental groups Open
We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by …
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Milnor excision for motivic spectra Open
We prove that the ∞ {\infty} -category of motivic spectra satisfies Milnor excision: if A → B {A\to B} is a morphism of commutative rings sending an ideal I ⊂ A {I\subset A} isomorphically onto an ideal of B , then a motivic spec…
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The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most $1$ Open
We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finall…
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Amenable covers of right‐angled Artin groups Open
Let be the right‐angled Artin group associated with a finite flag complex . We show that the amenable category of equals the virtual cohomological dimension of the right‐angled Coxeter group . In particular, right‐angled Artin groups satis…
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Homogeneous spaces, algebraic $K$-theory and cohomological dimension of\n fields Open
Let $q$ be a non-negative integer. We prove that a perfect field $K$ has\ncohomological dimension at most $q+1$ if, and only if, for any finite extension\n$L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected\nalgebr…
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On the topological complexity of aspherical spaces Open
The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. I…
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Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups Open
This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-gr…
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A bound on the cohomology of quasiregularly elliptic manifolds Open
We show that a closed, connected and orientable Riemannian manifold of dimension $d$ that admits a quasiregular mapping from $\mathbb R^d$ must have bounded cohomological dimension independent of the distortion of the map. The dimension of…
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A spectral sequence for Dehn fillings Open
We study how the cohomology of a type $F_\infty$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients of group pairs). For sufficiently long Dehn fillings where the quotient pair $(\bar{G},\bar{\ma…
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The Second Vanishing Theorem for Local Cohomology Modules Open
We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our van…
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Topological finiteness properties of monoids. Part 1: Foundations Open
We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory whe…
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Framed cohomological Hall algebras and cohomological stable envelopes Open
There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver Q to the Yangian $$Y^{Q}_\textrm{MO}$$ by Maulik–Okounkov, whose construction is b…
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Dynamic characterizations of quasi-isometry, and applications to cohomology Open
We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous re…
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The Rouquier Dimension of Quasi-Affine Schemes Open
We prove that for $X$ a regular quasi-affine scheme of dimension $d$, $\mathcal{O}_X$ is a $d$-step generator of $D^b_{coh}(X)$, establishing Orlov's conjecture in this case. We prove something weaker in the projective case. The main techn…
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On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$ Open
Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of th…
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Bounded cohomology of finitely generated groups: vanishing, non-vanishing, and computability Open
We provide new computations in bounded cohomology:
A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-ame…
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The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most 1 Open
We show that a strong vanishing conjecture for n -fold Massey products holds for fields of virtual cohomological dimension at most 1 using a theorem of Haran. We also prove the same for PpC fields, using results of Haran–Jarden. Finally we…
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On the cohomology of congruence subgroups of GL3 over the Eisenstein integers Open
Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices …
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Weak global dimension and CM-freeness of rings Open
Let $R$ be a ring with finite weak global dimension. We prove that $R$ is (strongly) CM-free. As applications,we obtain some triangulated equivalences between some compactly-generated homotopy categories and relative derived categories.We …
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Cohomological dimensions of specialization-closed subsets and subcategories of modules Open
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 mod…