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Register a new userCohomology of torsion and completion of N-complexes
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Xiaoyan Yang
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We introduce the notions of Koszul $N$-complex, $check{mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $mathbf{D}_N(R)$ of $N$-complexes using the $check{mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the category of cohomologically $mathfrak{a}$-torsion $N$-complexes and the category of cohomologically $mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.
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Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $mathbb{Z}$-subgroups. Let $partial Q$ be the subcomplex of $Q$ with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of $G$. We show that independent of the chain, the normalized torsion limits to the torsion of $partial Q$, shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of $G$ in terms of the torsion of stabilizers and topology of $partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows they satisfy a torsion analogue of the Luck approximation theorem.
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $egeq log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as generalizing a theorem due to C. Miller from finitely generated modules to arbitrary modules. We also show that if $R$ is a complete intersection ring then the vanishing of Tor$_i^R({}^e R, M)$ for single positive values of $i$ and $e$ is sufficient to imply $M$ has finite flat dimension. This extends a result of L. Avramov and C. Miller.
We continue investigating the interaction between flatness and $mathfrak{a}$-adic completion for infinitely generated modules over a commutative ring $A$. We introduce the concept of $mathfrak{a}$-adic flatness, which is weaker than flatness. We prove that $mathfrak{a}$-adic flatness is preserved under completion when the ideal $mathfrak{a}$ is weakly proregular. We also prove that when $A$ is noetherian, $mathfrak{a}$-adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring $A$, with a weakly proregular ideal $mathfrak{a}$, for which the completion $hat{A}$ is not flat. We also study $mathfrak{a}$-adic systems, and prove that if the ideal $mathfrak{a}$ is finitely generated, then the limit of any $mathfrak{a}$-adic system is a complete module.
We give sufficient conditions for F-injectivity to deform. We show these conditions are met in two common geometrically interesting setting, namely when the special fiber has isolated CM-locus or is F-split.
In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.
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