arXiv subjects

Richard Wicklein

Publications and source records attributed to Richard Wicklein.

6 recordsLinked to original sources

arXiv2016

Adic semidualizing complexes

We introduce and study a class of objects that encompasses Christensen and Foxby's semidualizing modules and complexes and Kubik's quasi-dualizing modules: the class of $\mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.

math.AC↗

arXiv2016

Adically Finite Chain Complexes

We investigate the similarities between adic finiteness and homological finiteness for chain complexes over a commutative noetherian ring. In particular, we extend the isomorphism properties of certain natural morphisms from homologically finite complexes to adically finite complexes. We do the same for characterizations of certain homological dimensions. In addition, we study transfer of adic finiteness along ring homomorphisms, all with a view toward subsequent applications.

math.AC↗

arXiv2016

Adic Finiteness: Bounding Homology and Applications

We prove a versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring $R$ of prime characteristic is regular if and only if for some proper ideal $\mathfrak b$ the derived local cohomology complex $\mathbf{R}Γ_{\mathfrak{b}}(R)$ has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.

math.AC↗

arXiv2016

Extended Local Cohomology and Local Homology

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion $\widehat R^{\mathfrak a}$ of a commutative noetherian ring $R$ with respect to a proper ideal $\mathfrak a$. In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over $\widehat R^{\mathfrak a}$, not just over $R$. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.

math.AC↗

arXiv2016

Adic Foxby Classes

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special converses of these results. We also investigate change of rings and local-global properties of these classes.

math.AC↗

arXiv2015

Support and adic finiteness for complexes

Let $X$ be a chain complex over a commutative noetherian ring $R$, that is, an object in the derived category $\mathcal{D}(R)$. We investigate the small support and co-support of $X$, introduced by Foxby and Benson, Iyengar, and Krause. We show that the derived functors $M \otimes_R^{\mathbf{L}} -$ and $\mathbf{R}\operatorname{Hom}_R(M,-)$ can detect isomorphisms in $\mathcal{D}(R)$ between complexes with restrictions on their supports or co-supports. In particular, the derived local (co)homology functors $\mathbf{R}Γ_{\mathfrak{a}}(-)$ and $\mathbf{L}Λ_{\mathfrak{a}}(-)$ with respect to an ideal $\mathfrak{a}\subsetneq R$ have the same ability. Furthermore, we give reprove some results of Benson, Iyengar, and Krause in our setting, with more direct proofs. Also, we include some computations of co-supports, since this construction is still quite mysterious. Lastly, we investigate "$\mathfrak{a}$-adically finite" $R$-complexes, that is, the $X\in\mathcal{D}(R)$ that are $\mathfrak{a}$-cofinite \textit{à la} Hartshorne. For instance, we characterize these complexes in terms of a finiteness condition on $\mathbf{L}Λ_{\mathfrak{a}}(X)$.

math.AC↗