Cohen‐macaulay modules over cohen‐macaulay rings

We prove that the cuspidal $p$-adic eigencurve $C_{cusp}$ is etale over the weight space at any classical weight $1$ Eisenstein point $f$. Further we show that $C_{cusp}$ meets transversely at $f$ each of the two Eisenstein components of the eigencurve $C$ passing through that point and give a new proof of Greenberg--Ferrero's theorem on the order of vanishing of the Kubota-Leopoldt $p$-adic $L$-function. Finally we prove that the local ring of $C$ at $f$ is Cohen-Macaulay but not Gorenstein and compute the $q$-expansions of a basis of overconvergent weight $1$ modular forms lying in the same generalised eigenspace as $f$.

On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution ...

Trimming complexes and applications to resolutions of determinantal facet ideals

The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are $R$ and $S$. Besides, the other generalizations of Gorenstein properties are also explored.

Almost Gorenstein rings arising from fiber products

Using SAGBI basis techniques, we find Grobner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

Rees Algebras of Closed Determinantal Facet Ideals

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

Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.

Vanishing of Avramov Obstructions for Products of Sequentially Transverse Ideals

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman. We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can give bounds on the minimal number of generators $\mu(I)$ of $I$ depending only on $s$; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all $s\geq 3$. Finally, we study the Tor-algebra structure of $R/I$. It is shown that these rings have Tor algebra class $G(r)$ for $s \leq r \leq 2s-1$. Furthermore, we produce ideals $I$ for all $s \geq 3$ and all $r$ with $s \leq r \leq 2s-1$ such that $\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)$ and $R/I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov.

Keller VandeBogert

Papers

Applications and homological properties of local rings with decomposable maximal ideals

Trimming complexes and applications to resolutions of determinantal facet ideals

Applications and homological properties of local rings with decomposable maximal ideals

Vanishing of Avramov Obstructions for Products of Sequentially Transverse Ideals

Structure Theory for a Class of Grade 3 Homogeneous Ideals Defining Type 2 Compressed Rings