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Sean Sather-Wagstaff

Publications and source records attributed to Sean Sather-Wagstaff.

At least 19 recordsLinked to original sources
arXiv2018

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen-Macaulay fiber products of finite Cohen-Macaulay type.

math.AC↗
arXiv2017

Geometric aspects of representation theory for {DG} algebras: answering a question of Vasconcelos

We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules $M$ over a finite dimensional, positively graded, commutative DG algebra $U$. In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group $\operatorname{YExt}^1_U(M,M)$ and a quotient of tangent spaces coming from an algebraic group action on an algebraic variety. As an application, we answer a question of Vasconcelos from 1974 by showing that a local ring has only finitely many semidualizing complexes up to shift-isomorphism in the derived category $\mathcal{D}(R)$.

math.AC↗
arXiv2016

Vanishing of Ext and Tor over fiber products

Consider a non-trivial fiber product $R=S\times_kT$ of local rings $S$, $T$ with common residue field $k$. Given two finitely generate $R$-modules $M$ and $N$, we show that if $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$ for some $i\geq 5$, then $\operatorname{pd}_R(M)\leq 1$ or $\operatorname{pd}_R(N)\leq 1$. From this, we deduce several consequence, for instance, that $R$ satisfies the Auslander-Reiten Conjecture.

math.AC↗
arXiv2016

Extension Groups for DG Modules

Let $M$ and $N$ be differential graded (DG) modules over a positively graded commutative DG algebra $A$. We show that the Ext-groups $\operatorname{Ext}^i_A(M,N)$ defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups $\operatorname{YExt}^i_A(M,N)$ given in terms of equivalence classes of extensions. On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective.

math.AC↗
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

Testing for the Gorenstein property

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $(R,\mathfrak m)$ is Gorenstein if and only if it admits an integrally closed $\mathfrak m$-primary ideal of finite Gorenstein dimension. This is accomplished through a detailed study of certain test complexes. Along the way we construct such a test complex that detect finiteness of Gorenstein dimension, but not that of projective dimension.

math.AC↗
arXiv2015

Homology over trivial extensions of commutative DG algebras

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.

math.AC↗
arXiv2015

Coherence conditions in flat regular pullbacks

We investigate the behavior of four coherent-like conditions in regular conductor squares. In particular, we find necessary and sufficient conditions in order that a pullback ring be a finite conductor ring, a coherent ring, a generalized GCD ring, or quasi-coherent ring. As an application of these results, we are able to determine exactly when the ring of integer-valued polynomials determined by a finite subset possesses one of the four coherent-like properties.

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↗
arXiv2015

Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective $R$-modules need not be closed under tensor products, even when $R$ is local and artinian; (2) the class of Gorenstein injective $R$-modules need not be closed under torsion products, even when $R$ is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when $R$ is a local, complete hypersurface.

math.AC↗
arXiv2015

Using semidualizing complexes to detect Gorenstein rings

A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $\mathcal{F}_C$-projective dimension, and finite $\mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.

math.AC↗
arXiv2014

Path Ideals of Weighted Graphs

We introduce and study the weighted $r$-path ideal of a weighted graph $G_ω$, which is a common generalization of Conca and De Negri's $r$-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted graph. Over a field, we explicitly describe primary decompositions of these ideals, and we characterize Cohen-Macaulayness of these ideals for trees (with arbitrary $r$) and complete graphs (for $r=2$).

math.AC↗
arXiv2014

On the structure of $S_2$-ifications of complete local rings

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $\operatorname{Spec}(R)-V(J)$, and the (finite simple) graph $Γ_R$ with vertex set $\operatorname{Min}(R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of maximal ideals of $T$ is equal to the number of connected components of $Γ_R$. We further investigate this graph by exhibiting a technique for showing that a given graph $G$ can be realized as one of the form $Γ_R$.

math.AC↗