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Let $(R,frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $frak m$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $frak m$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$.
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We construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{mathfrak{p}}$ does not satisfy Auslanders condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ 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.
We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.
In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.
In this paper we study Cohen-Macaulay local rings of dimension $d$, multiplicity $e$ and second Hilbert coefficient $e_2$ in the case $e_2 = e_1 - e + 1$. Let $h = mu(mathfrak{m}) - d$. If $e_2 eq 0$ then in our case we can prove that type $A geq e - h -1$. If type $A = e - h -1$ then we show that the associated graded ring $G(A)$ is Cohen-Macaulay. In the next case when type $A = e - h$ we determine all possible Hilbert series of $A$. In this case we show that the Hilbert Series of $A$ completely determines depth $G(A)$.
A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $Rto Tgets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the emph{connected sum} $R#_TS$ is defined to be the local ring obtained by factoring out the diagonal image of $V$ in the fiber product $Rtimes_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R#_TS$. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra $Ext^*_{R#_kS}(k,k)$ is an amalgam of the algebras $Ext^*_{R}(k,k)$ and $Ext^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring $R#_TS$ almost never is complete intersection.
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