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The notion of a Buchsbaum ring (and module) is a generalization of that of a [[Cohen–Macaulay ring|Cohen–Macaulay ring]] (respectively, module). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302901.png" /> denote a Noetherian [[Local ring|local ring]] (cf. also [[Noetherian ring|Noetherian ring]]) with [[Maximal ideal|maximal ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302903.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302904.png" /> be a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302905.png" />-module with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302906.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302907.png" /> is called a Buchsbaum module if the difference
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is independent of the choice of a parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b1302909.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029011.png" /> is a system of parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029013.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029014.png" />) denotes the length of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029015.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029016.png" /> (respectively, the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029017.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029018.png" />). When this is the case, the difference
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The notion of a Buchsbaum ring (and module) is a generalization of that of a [[Cohen–Macaulay ring|Cohen–Macaulay ring]] (respectively, module). Let $A$ denote a Noetherian [[Local ring|local ring]] (cf. also [[Noetherian ring|Noetherian ring]]) with [[Maximal ideal|maximal ideal]] $\mathfrak{m}$ and $d = \operatorname { dim } A$. Let $M$ be a finitely-generated $A$-module with $\dim_AM = s$. Then $M$ is called a Buchsbaum module if the difference
  
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\begin{equation*} \operatorname{l} _ { A } ( M / \mathfrak{q}M ) - e _ { \mathfrak{q} } ^ { 0 } ( M ) \end{equation*}
  
is called the Buchsbaum invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029020.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029021.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029022.png" /> is a Cohen–Macaulay module if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029023.png" /> for some (and hence for any) parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029025.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029026.png" /> is a Cohen–Macaulay <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029027.png" />-module if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029028.png" /> is a Buchsbaum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029029.png" />-module with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029030.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029031.png" /> is said to be a Buchsbaum ring if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029032.png" /> is a Buchsbaum module over itself. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029033.png" /> is a Buchsbaum ring, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029034.png" /> is a Cohen–Macaulay ring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029035.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029036.png" />.
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is independent of the choice of a parameter ideal $\mathfrak { q } = ( a _ { 1 } , \ldots , a _ { s } )$ of $M$, where $a _ { 1 } , \dots , a _ { s }$ is a system of parameters of $M$ and ${\bf l} _ { A } ( M / \text{q}M )$ (respectively, $e _ { \mathfrak{q} } ^ { 0 } ( M )$) denotes the length of the $A$-module $M / \mathfrak { q } M$ (respectively, the multiplicity of $M$ with respect to $\text{q}$). When this is the case, the difference
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\begin{equation*} I ( M ) = {\bf l } _ { A } ( M / \mathfrak { q } M ) - e _ { \mathfrak { q } } ^ { 0 } ( M ) \end{equation*}
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is called the Buchsbaum invariant of $M$. The $A$-module $M$ is a Cohen–Macaulay module if and only if $\text{l} _ { A } ( M / \text{q}M ) = e _ { \text{q} } ^ { 0 } ( M )$ for some (and hence for any) parameter ideal $\text{q}$ of $M$, so that $M$ is a Cohen–Macaulay $A$-module if and only if $M$ is a Buchsbaum $A$-module with $I ( M ) = 0$. The ring $A$ is said to be a Buchsbaum ring if $A$ is a Buchsbaum module over itself. If $A$ is a Buchsbaum ring, then $A _ { \mathfrak{p} }$ is a Cohen–Macaulay ring with $\operatorname { dim } A _ { \mathfrak { p } } = \operatorname { dim } A - \operatorname { dim } A / \mathfrak { p }$ for every $\mathfrak{p} \in \operatorname { Spec } A \backslash \{ \mathfrak{m} \}$.
  
 
A typical example of Buchsbaum rings is as follows. Let
 
A typical example of Buchsbaum rings is as follows. Let
  
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\begin{equation*} A = B / ( X _ { 1 } , \dots , X _ { d } ) \bigcap ( Y _ { 1 } , \dots , Y _ { d } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029038.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029039.png" />, denotes the [[Formal power series|formal power series]] ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029040.png" /> variables over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029042.png" /> is a Buchsbaum ring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029044.png" />.
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where $B = k [ [ X _ { 1 } , \dots , X _ { d } , Y _ { 1 } , \dots , Y _ { d } ]]$, with $d \geq 1$, denotes the [[Formal power series|formal power series]] ring in $2 d$ variables over a [[Field|field]] $k$. Then $A$ is a Buchsbaum ring with $\operatorname { dim } A = d$ and $I ( A ) = d - 1$.
  
A, not necessarily local, [[Noetherian ring|Noetherian ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029045.png" /> is said to be a Buchsbaum ring if the local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029046.png" /> are Buchsbaum for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029047.png" />.
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A, not necessarily local, [[Noetherian ring|Noetherian ring]] $R$ is said to be a Buchsbaum ring if the local rings $R _ { \mathfrak{p} }$ are Buchsbaum for all $\mathfrak { p } \in \operatorname { Spec } R$.
  
The theory of Buchsbaum rings and modules dates back to a question raised in 1965 by D.A. Buchsbaum [[#References|[a3]]]. He asked whether the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029048.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029049.png" /> a parameter ideal, is an invariant for any Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029050.png" />. This is, however, not the case and a counterexample was given in [[#References|[a28]]]. Thereafter, in 1973 J. Stückrad and W. Vogel published the classic paper [[#References|[a29]]], from which the history of Buchsbaum rings and modules started. In [[#References|[a29]]] they gave a characterization of Buchsbaum rings in terms of the following property of systems of parameters: A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029051.png" />-dimensional Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029052.png" /> with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029053.png" /> is Buchsbaum if and only if every system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029054.png" /> of parameters for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029055.png" /> forms a weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029057.png" />-sequence, that is, the equality
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The theory of Buchsbaum rings and modules dates back to a question raised in 1965 by D.A. Buchsbaum [[#References|[a3]]]. He asked whether the difference $ \operatorname{l} _ { A } ( A / \mathfrak { q } ) - e _ { \mathfrak { q } } ^ { 0 } ( A )$, with $\text{q}$ a parameter ideal, is an invariant for any Noetherian local ring $A$. This is, however, not the case and a counterexample was given in [[#References|[a28]]]. Thereafter, in 1973 J. Stückrad and W. Vogel published the classic paper [[#References|[a29]]], from which the history of Buchsbaum rings and modules started. In [[#References|[a29]]] they gave a characterization of Buchsbaum rings in terms of the following property of systems of parameters: A $d$-dimensional Noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ is Buchsbaum if and only if every system $a _ { 1 } , \dots , a _ { d }$ of parameters for $A$ forms a weak $A$-sequence, that is, the equality
  
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\begin{equation*} ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } = ( a _ { 1 } , \dots , a _ { i - 1 } ) : \mathfrak{m} \end{equation*}
  
holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029059.png" />. Therefore, systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029060.png" /> of parameters in a Buchsbaum local ring need not be regular sequences, but the differences
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holds for all $1 \leq i \leq d$. Therefore, systems $a _ { 1 } , \dots , a _ { d }$ of parameters in a Buchsbaum local ring need not be regular sequences, but the differences
  
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\begin{equation*} [ ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } ] / ( a _ { 1 } , \dots , a _ { i - 1 } ) ,\; 1 \leq i \leq d, \end{equation*}
  
are very small and only finite-dimensional vector spaces over the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029063.png" />. Weak sequences are closely related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029064.png" />-sequences introduced by C. Huneke [[#References|[a21]]]. Actually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029065.png" /> is a Buchsbaum ring if and only if every system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029066.png" /> of parameters for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029067.png" /> forms a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029069.png" />-sequence, that is, the equality
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are very small and only finite-dimensional vector spaces over the residue class field $A /_{ \mathfrak{m}}$ of $A$. Weak sequences are closely related to $d$-sequences introduced by C. Huneke [[#References|[a21]]]. Actually, $A$ is a Buchsbaum ring if and only if every system $a _ { 1 } , \dots , a _ { d }$ of parameters for $A$ forms a $d$-sequence, that is, the equality
  
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\begin{equation*} ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } a _ { j } = ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { j } \end{equation*}
  
holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029071.png" />.
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holds for all $1 \leq i \leq j \leq d$.
  
 
One of the fundamental results on Buchsbaum rings and modules is the surjectivity criterion. Let
 
One of the fundamental results on Buchsbaum rings and modules is the surjectivity criterion. Let
  
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\begin{equation*} H _ { \mathfrak{m} } ^ { i } ( M ) = \operatorname { lim } _ { n \rightarrow \infty } \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } ^ { n } , M ) \quad ( i \in \mathbf{Z} ) \end{equation*}
  
denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029073.png" />th [[Local cohomology|local cohomology]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029074.png" /> with respect to the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029075.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029076.png" /> is a Buchsbaum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029077.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029078.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029079.png" /> and the equality
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denote the $i$th [[Local cohomology|local cohomology]] of $M$ with respect to the maximal ideal $\mathfrak{m}$. If $M$ is a Buchsbaum $A$-module, then $\mathfrak { m } \cdot H _ { \mathfrak { m } } ^ { i } ( M ) = ( 0 )$ for all $i \neq s$ and the equality
  
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\begin{equation*} \operatorname {I} ( M ) = \sum _ { i = 0 } ^ { s - 1 } \left( \begin{array} { c } { s - 1 } \\ { i } \end{array} \right) .\operatorname { l}_ { A } ( H _ {\frak m } ^ { i } ( M ) ) \end{equation*}
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029081.png" />.
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holds, where $s = \operatorname { dim } _ { A } M$.
  
Unfortunately, the vanishing does not characterize Buchsbaum modules. Modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029082.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029083.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029084.png" /> are called quasi-Buchsbaum and constitute a class which is strictly larger than that of Buchsbaum modules. However, if the canonical homomorphism
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Unfortunately, the vanishing does not characterize Buchsbaum modules. Modules $M$ with $\mathfrak { m } \cdot H _ { \mathfrak { m } } ^ { i } ( M ) = ( 0 )$ for all $i \neq \operatorname { dim } _ { A } M$ are called quasi-Buchsbaum and constitute a class which is strictly larger than that of Buchsbaum modules. However, if the canonical homomorphism
  
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\begin{equation*} \varphi _ { M } ^ { i } : \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } , M ) \rightarrow H _ {\frak m } ^ { i } ( M ) = \operatorname { lim } _ { n \rightarrow \infty } \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } ^ { n } , M ) \end{equation*}
  
is surjective for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029086.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029087.png" /> is a Buchsbaum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029088.png" />-module. The converse is also true if the base ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029089.png" /> is regular (cf. also [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]).
+
is surjective for all $i \neq \operatorname { dim } _ { A } M$, then $M$ is a Buchsbaum $A$-module. The converse is also true if the base ring $A$ is regular (cf. also [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]).
  
After the establishment of the surjectivity criterion, by Stückrad and Vogel [[#References|[a30]]] in 1978, the development of the theory became rather rapid. The ubiquity of Buchsbaum normal local rings was established by S. Goto [[#References|[a6]]] as an application of the Evans–Griffith construction [[#References|[a5]]]. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029091.png" /> be integers. Then there exists a Buchsbaum local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029092.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029094.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029096.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029098.png" />), one may choose the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b13029099.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290100.png" /> is an [[Integral domain|integral domain]] (respectively, a [[Normal ring|normal ring]]). See [[#References|[a1]]] for progress in the research about the ubiquity of Buchsbaum homogeneous integral domains. Besides, Buchsbaum local rings of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290101.png" /> have been classified [[#References|[a8]]]. Also, certain famous isolated singularities are Buchsbaum (cf. [[#References|[a23]]]).
+
After the establishment of the surjectivity criterion, by Stückrad and Vogel [[#References|[a30]]] in 1978, the development of the theory became rather rapid. The ubiquity of Buchsbaum normal local rings was established by S. Goto [[#References|[a6]]] as an application of the Evans–Griffith construction [[#References|[a5]]]. Namely, let $d \geq 1$ and $\{ h _ { i } \} _ { 0 \leq i \leq d - 1 }$ be integers. Then there exists a Buchsbaum local ring $A$ with $\operatorname { dim } A = d$ and $\operatorname{l} _ { A } ( H _ { \text{m} } ^ { i } ( A ) ) = h _ { i }$ for $0 \leq i \leq d - 1$. If $h _ { 0 } = 0$ (respectively, $d \geq 2$ and $h _ { 0 } = h _ { 1 } = 0$), one may choose the ring $A$ so that $A$ is an [[Integral domain|integral domain]] (respectively, a [[Normal ring|normal ring]]). See [[#References|[a1]]] for progress in the research about the ubiquity of Buchsbaum homogeneous integral domains. Besides, Buchsbaum local rings of multiplicity $2$ have been classified [[#References|[a8]]]. Also, certain famous isolated singularities are Buchsbaum (cf. [[#References|[a23]]]).
  
The theory of Buchsbaum rings and modules is closely related to that of Cohen–Macaulayness in blowing-ups. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290102.png" /> be an ideal of positive height in a Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290103.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290104.png" /> and call it the Rees algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290105.png" />. Then the canonical morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290106.png" /> is the blowing-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290107.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290108.png" /> (cf. also [[Blow-up algebra|Blow-up algebra]]). If the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290109.png" /> is Cohen–Macaulay, then the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290110.png" /> naturally is locally Cohen–Macaulay. The problem when the Rees algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290111.png" /> is Cohen–Macaulay has been intensively studied from the 1980s onwards ([[#References|[a18]]], [[#References|[a38]]], [[#References|[a16]]], [[#References|[a39]]], [[#References|[a17]]]).
+
The theory of Buchsbaum rings and modules is closely related to that of Cohen–Macaulayness in blowing-ups. Let $I$ be an ideal of positive height in a Noetherian local ring $A$. Let $R ( I ) = \oplus _ { n \geq 0 }  I ^ { n }$ and call it the Rees algebra of $I$. Then the canonical morphism $\operatorname{Proj} R ( I ) \rightarrow \operatorname{Spec} A$ is the blowing-up of $A$ with centre $I$ (cf. also [[Blow-up algebra|Blow-up algebra]]). If the ring $R ( I )$ is Cohen–Macaulay, then the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290110.png"/> naturally is locally Cohen–Macaulay. The problem when the Rees algebra $R ( I )$ is Cohen–Macaulay has been intensively studied from the 1980s onwards ([[#References|[a18]]], [[#References|[a38]]], [[#References|[a16]]], [[#References|[a39]]], [[#References|[a17]]]).
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290112.png" /> is Cohen–Macaulay if the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290113.png" /> is generated by a regular sequence and if the base ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290114.png" /> is Cohen–Macaulay [[#References|[a2]]]. However, the converse is not true even for parameter ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290115.png" />. Actually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290116.png" /> is a Buchsbaum ring if and only if the Rees algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290117.png" /> is a Cohen–Macaulay ring for every parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290118.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290119.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290120.png" /> is an integral domain with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290121.png" />. This insightful result of Y. Shimoda [[#References|[a35]]] in 1979 opened the door towards a further development of the theory. Firstly, Goto and Shimoda [[#References|[a19]]] showed that a Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290122.png" /> is a Buchsbaum ring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290123.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290124.png" />) if and only if the Rees algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290125.png" /> is a Cohen–Macaulay ring for every parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290126.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290127.png" />. When this is the case, the Rees algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290128.png" /> are also Cohen–Macaulay for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290129.png" />. In 1981, Buchsbaum rings were characterized in terms of the blowing-ups of parameter ideals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290130.png" /> be a Noetherian local ring with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290132.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290133.png" /> is a Buchsbaum ring if and only if the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290134.png" /> is locally Cohen–Macaulay for every parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290135.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290136.png" /> [[#References|[a7]]]. Subsequently, Goto [[#References|[a10]]] proved that the associated graded rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290137.png" /> of parameter ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290138.png" /> in a Buchsbaum local ring are always Buchsbaum. In addition, Stückrad showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290139.png" /> is a Buchsbaum ring for every parameter ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290140.png" /> in a Buchsbaum local ring [[#References|[a36]]]. The systems of parameters in Buchsbaum local rings behave very well and enjoy the monomial property [[#References|[a10]]].
+
The ring $R ( I )$ is Cohen–Macaulay if the ideal $I$ is generated by a regular sequence and if the base ring $A$ is Cohen–Macaulay [[#References|[a2]]]. However, the converse is not true even for parameter ideals $I$. Actually, $A$ is a Buchsbaum ring if and only if the Rees algebra $R ( \mathfrak { q } )$ is a Cohen–Macaulay ring for every parameter ideal $\text{q}$ in $A$, provided that $A$ is an integral domain with $\operatorname { dim } A = 2$. This insightful result of Y. Shimoda [[#References|[a35]]] in 1979 opened the door towards a further development of the theory. Firstly, Goto and Shimoda [[#References|[a19]]] showed that a Noetherian local ring $A$ is a Buchsbaum ring with $H _ { \mathfrak{m} } ^ { i } ( A ) = ( 0 )$ ($i \neq 1 , \operatorname { dim } A$) if and only if the Rees algebra $R ( \mathfrak { q } )$ is a Cohen–Macaulay ring for every parameter ideal $\text{q}$ in $A$. When this is the case, the Rees algebras $R ( \mathfrak{q} ^ { n } )$ are also Cohen–Macaulay for all $n \geq 1$. In 1981, Buchsbaum rings were characterized in terms of the blowing-ups of parameter ideals. Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $d = \operatorname { dim } A \geq 1$. Then $A / H _ { \mathfrak{m} } ^ { 0 } ( A )$ is a Buchsbaum ring if and only if the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290134.png"/> is locally Cohen–Macaulay for every parameter ideal $\text{q}$ in $A$ [[#References|[a7]]]. Subsequently, Goto [[#References|[a10]]] proved that the associated graded rings $G ( \mathfrak { q } ) = \oplus _ { n  \geq 0} \mathfrak { q } ^ { n } / \mathfrak { q } ^ { n + 1 }$ of parameter ideals $\text{q}$ in a Buchsbaum local ring are always Buchsbaum. In addition, Stückrad showed that $R ( \mathfrak { q } )$ is a Buchsbaum ring for every parameter ideal $\text{q}$ in a Buchsbaum local ring [[#References|[a36]]]. The systems of parameters in Buchsbaum local rings behave very well and enjoy the monomial property [[#References|[a10]]].
  
 
Buchsbaum rings are yet (2000) the only non-trivial case for which the monomial conjecture, raised by M. Hochster, has been solved affirmatively (except for the equi-characteristic case). See [[#References|[a31]]] for these results, together with geometric applications and concrete examples. See [[#References|[a31]]] for researches on the Buchsbaum property in affine semi-group rings and Stanley–Reisner rings of simplicial complexes.
 
Buchsbaum rings are yet (2000) the only non-trivial case for which the monomial conjecture, raised by M. Hochster, has been solved affirmatively (except for the equi-characteristic case). See [[#References|[a31]]] for these results, together with geometric applications and concrete examples. See [[#References|[a31]]] for researches on the Buchsbaum property in affine semi-group rings and Stanley–Reisner rings of simplicial complexes.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290141.png" /> be a Buchsbaum module over a Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290142.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290143.png" /> is said to be maximal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290144.png" />. Noetherian local rings possessing only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules are said to have finite Buchsbaum-representation type. Buchsbaum representation theory was studied by Goto and K. Nishida [[#References|[a15]]], [[#References|[a11]]], [[#References|[a13]]], and the Cohen–Macaulay local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290145.png" /> of finite Buchsbaum-representation type have been classified under certain mild conditions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290146.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290147.png" /> must be regular [[#References|[a15]]]. The situation is a little more complicated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290148.png" /> [[#References|[a13]]]. In [[#References|[a11]]] (not necessarily Cohen–Macaulay) surface singularities of finite Buchsbaum-representation type are classified.
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Let $M$ be a Buchsbaum module over a Noetherian local ring $A$. Then $M$ is said to be maximal if $\operatorname{dim}_{A} M = \operatorname{dim} A $. Noetherian local rings possessing only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules are said to have finite Buchsbaum-representation type. Buchsbaum representation theory was studied by Goto and K. Nishida [[#References|[a15]]], [[#References|[a11]]], [[#References|[a13]]], and the Cohen–Macaulay local rings $A$ of finite Buchsbaum-representation type have been classified under certain mild conditions. If $\operatorname { dim } A \geq 2$, then $A$ must be regular [[#References|[a15]]]. The situation is a little more complicated if $\operatorname { dim } A = 1$ [[#References|[a13]]]. In [[#References|[a11]]] (not necessarily Cohen–Macaulay) surface singularities of finite Buchsbaum-representation type are classified.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290149.png" /> is a regular local ring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290150.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290151.png" /> be a maximal Buchsbaum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290152.png" />-module. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290153.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290154.png" />-module for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290155.png" />, so that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290156.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290157.png" /> defines a [[Vector bundle|vector bundle]] on the punctured spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290158.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290159.png" />. Thanks to the surjectivity criterion, one can prove the structure theorem of maximal Buchsbaum modules over regular local rings: Every maximal Buchsbaum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290160.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290161.png" /> has the form
+
Suppose that $A$ is a regular local ring with $\operatorname { dim } A = d$ and let $M$ be a maximal Buchsbaum $A$-module. Then $M _ { \mathfrak{p} }$ is a free $A _ { \mathfrak{p} }$-module for all $\mathfrak{p} \in \operatorname { Spec } A \backslash \{ \mathfrak{m} \}$, so that the $A$-module $M$ defines a [[Vector bundle|vector bundle]] on the punctured spectrum $\operatorname {Spec} A \backslash \{ \mathfrak{m} \}$ of $A$. Thanks to the surjectivity criterion, one can prove the structure theorem of maximal Buchsbaum modules over regular local rings: Every maximal Buchsbaum $A$-module $M$ has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290162.png" /></td> </tr></table>
+
\begin{equation*} M \cong \bigoplus _ { i = 0 } ^ { d } E _ { i } ^ { h _ { i } }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290163.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290164.png" />th syzygy module of the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290165.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290167.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290168.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290169.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290170.png" /> is a regular local ring ([[#References|[a4]]], [[#References|[a12]]]). This result has been generalized by Y. Yoshino [[#References|[a40]]] and T. Kawasaki [[#References|[a24]]]. They showed a similar decomposition theorem of a special kind of maximal Buchsbaum modules over Gorenstein local rings; see [[#References|[a32]]] for the characterization of Buchsbaum rings and modules in terms of dualizing complexes. (It should be noted here that the main result in [[#References|[a32]]] contains a serious mistake, which has been repaired in [[#References|[a40]]].)
+
where $E_i$ denotes the $i$th syzygy module of the residue class field $A /_{ \mathfrak{m}}$ of $A$, $h _ { i } = \operatorname { l } _ { A } ( H _ { \mathfrak{m} } ^ { i } ( M ) )$ ($0 \leq i \leq d - 1$), and $h _ { d } = \operatorname { rank } _ { A } M - \sum _ { i = 1 } ^ { d - 1 } \left( \begin{array} { c } { d - 1 } \\ { i - 1 } \end{array} \right) h _ { i }$, if $A$ is a regular local ring ([[#References|[a4]]], [[#References|[a12]]]). This result has been generalized by Y. Yoshino [[#References|[a40]]] and T. Kawasaki [[#References|[a24]]]. They showed a similar decomposition theorem of a special kind of maximal Buchsbaum modules over Gorenstein local rings; see [[#References|[a32]]] for the characterization of Buchsbaum rings and modules in terms of dualizing complexes. (It should be noted here that the main result in [[#References|[a32]]] contains a serious mistake, which has been repaired in [[#References|[a40]]].)
  
A local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290171.png" /> satisfying the condition that all the local cohomology modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290172.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290173.png" />) are finitely generated is said to be an FLC ring (or a generalized Cohen–Macaulay ring). The class of FLC rings includes Buchsbaum rings as typical examples. In fact, a Noetherian local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290174.png" /> is FLC if and only if it contains at least one system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290175.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290176.png" />) of parameters such that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290177.png" /> forms a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290178.png" />-sequence in any order for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290179.png" />. Such a sequence is called an unconditioned strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290181.png" />-sequence (for short, USD-sequence or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290183.png" />-sequence); they have been intensively studied [[#References|[a27]]], [[#References|[a37]]], [[#References|[a20]]]. Recently (1999), Kawasaki [[#References|[a25]]] used the results in [[#References|[a20]]] to establish the arithmetic Cohen–Macaulayfications of Noetherian local rings. Namely, every unmixed local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290184.png" /> contains an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290185.png" /> of positive height with the Cohen–Macaulay Rees algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290186.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290187.png" /> and all the formal fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290188.png" /> are Cohen–Macaulay. Hence, the Sharp conjecture [[#References|[a34]]] concerning the existence of dualizing complexes is solved affirmatively.
+
A local ring $A$ satisfying the condition that all the local cohomology modules $H _ { \mathfrak{m} } ^ { i } ( A )$ ($i \neq \operatorname { dim } A$) are finitely generated is said to be an FLC ring (or a generalized Cohen–Macaulay ring). The class of FLC rings includes Buchsbaum rings as typical examples. In fact, a Noetherian local ring $A$ is FLC if and only if it contains at least one system $a _ { 1 } , \dots , a _ { d }$ ($d = \operatorname { dim } A$) of parameters such that the sequence $\alpha _ { 1 } ^ { n _ { 1 } } , \dots , \alpha _ { d } ^ { n _ { d } }$ forms a $d$-sequence in any order for all integers $n _ { i } \geq 1$. Such a sequence is called an unconditioned strong $d$-sequence (for short, USD-sequence or $d ^ { + }$-sequence); they have been intensively studied [[#References|[a27]]], [[#References|[a37]]], [[#References|[a20]]]. Recently (1999), Kawasaki [[#References|[a25]]] used the results in [[#References|[a20]]] to establish the arithmetic Cohen–Macaulayfications of Noetherian local rings. Namely, every unmixed local ring $A$ contains an ideal $I$ of positive height with the Cohen–Macaulay Rees algebra $R ( I )$, provided $\operatorname { dim } A \geq 1$ and all the formal fibres of $A$ are Cohen–Macaulay. Hence, the Sharp conjecture [[#References|[a34]]] concerning the existence of dualizing complexes is solved affirmatively.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290189.png" /> be a Noetherian graded ring with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290190.png" /> a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290191.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290192.png" /> is a Buchsbaum ring if and only if the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290193.png" /> is Buchsbaum. When this is the case, the local cohomology modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290194.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290195.png" />) are finite-dimensional vector spaces over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290196.png" />. The vanishing of certain homogeneous components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290197.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290198.png" /> may affect the Buchsbaumness in graded algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290199.png" />. For example, if there exist integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290200.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290201.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290202.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290203.png" /> and if
+
Let $R = \oplus _ { n  \geq 0} R _ { n }$ be a Noetherian graded ring with $k = R _ { 0 }$ a field and let $\mathfrak { M } = R _ { + }$. Then $R$ is a Buchsbaum ring if and only if the local ring $R _ {\frak M }$ is Buchsbaum. When this is the case, the local cohomology modules $H _ { \mathfrak{M} } ^ { i } ( R )$ ($i \neq \operatorname { dim } R$) are finite-dimensional vector spaces over the field $k$. The vanishing of certain homogeneous components $[ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { n }$ of $H _ { \mathfrak{M} } ^ { i } ( R )$ may affect the Buchsbaumness in graded algebras $R$. For example, if there exist integers $\{ t _ { i } \} _ { 0 \leq i \leq d - 1}$ ($d = \operatorname { dim } R$) such that $t _ { i } \leq t_{i + 1} + 1$ for all $0 \leq i \leq d - 1$ and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290204.png" /></td> </tr></table>
+
\begin{equation*} [ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { n } = ( 0 ) \end{equation*}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290206.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290207.png" /> is a Buchsbaum ring [[#References|[a9]]]. Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290208.png" /> is a Buchsbaum ring if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290209.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290210.png" /> [[#References|[a33]]]. Hence the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290211.png" /> is arithmetically Buchsbaum if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290212.png" /> is locally Cohen–Macaulay, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290213.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290214.png" /> is equi-dimensional. See [[#References|[a22]]] for the bounds of Castelnuovo–Mumford regularities of Buchsbaum schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290215.png" />.
+
for all $n \neq t_i$ and $0 \leq i \leq d - 1$, then $R$ is a Buchsbaum ring [[#References|[a9]]]. Therefore $R$ is a Buchsbaum ring if $H _ { \mathfrak{M} } ^ { i } ( R ) = [ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { 0 }$ for all $i \neq d$ [[#References|[a33]]]. Hence the scheme $X = \operatorname { Proj } R$ is arithmetically Buchsbaum if $X$ is locally Cohen–Macaulay, provided that $R = k [ R _ { 1 }]$ and $R$ is equi-dimensional. See [[#References|[a22]]] for the bounds of Castelnuovo–Mumford regularities of Buchsbaum schemes $X = \operatorname { Proj } R$.
  
Researches of the Buchsbaumness in Rees algebras recently (1999) started again, although the progress remains tardy (possibly because of the lack of characterizations of Trung–Ikeda type [[#References|[a38]]] for Buchsbaumness). In [[#References|[a14]]] the Buchsbaumness in Rees algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290216.png" /> of certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290217.png" />-primary ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290218.png" /> in Cohen–Macaulay local rings is closely studied in connection with the Buchsbaumness in the associated graded rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290219.png" /> and that of the extended Rees algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290220.png" />. In [[#References|[a26]]], [[#References|[a41]]], [[#References|[a42]]], Buchsbaumness in graded rings associated to certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290221.png" />-primary ideals in Buchsbaum local rings is explored. Especially, the Rees algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290222.png" /> of the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290223.png" /> in a Buchsbaum local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290224.png" /> of maximal embedding dimension (that is, a Buchsbaum local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290225.png" /> for which the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290226.png" /> holds) is again a Buchsbaum ring [[#References|[a42]]].
+
Researches of the Buchsbaumness in Rees algebras recently (1999) started again, although the progress remains tardy (possibly because of the lack of characterizations of Trung–Ikeda type [[#References|[a38]]] for Buchsbaumness). In [[#References|[a14]]] the Buchsbaumness in Rees algebras $R ( I )$ of certain $\mathfrak{m}$-primary ideals $I$ in Cohen–Macaulay local rings is closely studied in connection with the Buchsbaumness in the associated graded rings $G ( I ) = \oplus _ { n  \geq 0} I ^ { n } / I ^ { n + 1 }$ and that of the extended Rees algebras $R ^ { \prime } ( I ) = \oplus _ { n \in \bf Z} I^ { n }$. In [[#References|[a26]]], [[#References|[a41]]], [[#References|[a42]]], Buchsbaumness in graded rings associated to certain $\mathfrak{m}$-primary ideals in Buchsbaum local rings is explored. Especially, the Rees algebra $R ( \mathfrak{m} )$ of the maximal ideal $\mathfrak{m}$ in a Buchsbaum local ring $A$ of maximal embedding dimension (that is, a Buchsbaum local ring $A$ for which the equality $v ( A ) = e _ { \mathfrak{m} } ^ { 0 } ( A ) + \operatorname { dim } A + I ( A ) - 1$ holds) is again a Buchsbaum ring [[#References|[a42]]].
  
 
====References====
 
====References====
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Northcott und Anwendungen" ''Monatsh. Math.'' , '''76''' (1972) pp. 264–271 {{MR|}} {{ZBL|0248.13026}} </TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Eine Verallgemeinerung der Cohen–Macaulay-Ringe und Anwendungen auf ein Problem der Multiplitätstheorie" ''J. Math. Kyoto Univ.'' , '''13''' (1973) pp. 513–528</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Toward a theory of Buchsbaum singularities" ''Amer. J. Math.'' , '''100''' (1978) pp. 727–746 {{MR|0509072}} {{ZBL|0429.14001}} </TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Buchsbaum rings and applications" , Springer (1986) {{MR|0881220}} {{MR|0873945}} {{ZBL|0606.13018}} {{ZBL|0606.13017}} </TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> P. Schenzel, "Dualisierende Komplexe in der lokalen Algebra und Buchsbaum–Ringe" , ''Lecture Notes in Mathematics'' , '''907''' , Springer (1982) {{MR|0654151}} {{ZBL|0484.13016}} </TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> P. Schenzel, "On Veronesean embeddings and projections of Veronesean varieties" ''Archiv Math.'' , '''30''' (1978) pp. 391–397 {{MR|0485849}} {{ZBL|0417.14040}} </TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> R.Y. Sharp, "Necessary conditions for the existence of dualizing complexes in commutative algebra" , ''Lecture Notes in Mathematics'' , '''740''' , Springer (1979) pp. 213–229 {{MR|0563505}} {{ZBL|0421.13003}} </TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> Y. Shimoda, "A note on Rees algebras of two-dimensional local domains" ''J. Math. Kyoto Univ.'' , '''19''' (1979) pp. 327–333 {{MR|0545713}} {{ZBL|0447.13010}} </TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> J. Stückrad, "On the Buchsbaum property of Rees and form modules" ''Beitr. Algebra Geom.'' , '''19''' (1985) pp. 83–103 {{MR|0785248}} {{ZBL|0567.13008}} </TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> N.V. Trung, "Toward a theory of generalized Cohen–Macaulay modules" ''Nagoya Math. J.'' , '''102''' (1986) pp. 1–49 {{MR|}} {{ZBL|0649.13008}} {{ZBL|0637.13013}} </TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> N.V. Trung, S. Ikeda, "When is the Rees algebra Cohen–Macaulay?" ''Commun. Algebra'' , '''17''' (1989) pp. 2893–2922 {{MR|}} {{ZBL|0696.13015}} </TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> W. Vasconcelos, "Arithmetic of blowup algebras" , ''London Math. Soc. Lecture Notes'' , '''195''' , Cambridge Univ. Press (1994) {{MR|1275840}} {{ZBL|0813.13008}} </TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> Y. Yoshino, "Maximal Buchsbaum modules of finite projective dimension" ''J. Algebra'' , '''159''' (1993) pp. 240–264 {{MR|1231212}} {{ZBL|0791.13009}} </TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> K. Yamagishi, "The associated graded modules of Buchsbaum modules with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290228.png" />-primary ideals in equi-I-invariant case" ''J. Algebra'' , '''225''' (2000) pp. 1–27 {{MR|1743648}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> K. Yamagishi, "Buchsbaumness in Rees algebras associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290229.png" />-primary ideals of minimal multiplicity in Buchsbaum local rings" , ''Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999'' (1999) pp. 39–45</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> M. Amasaki, "Existence of homogeneous prime ideals fitting into long Bourbaki sequences" , ''Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999'' (1999) pp. 104–111 {{MR|1610925}} {{ZBL|}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> J. Barshay, "Graded algebras of powers of ideals generated by A-sequences" ''J. Algebra'' , '''25''' (1973) pp. 90–99 {{MR|0332748}} {{ZBL|0256.13017}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> D.A. Buchsbaum, "Complexes in local ring theory" , ''Some Aspects of Ring Theory'' , C.I.M.E. Roma (1965) pp. 223–228 {{MR|}} {{ZBL|0178.37201}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> G. Eisenbud, S. Goto, "Linear free resolutions and minimal multiplicity" ''J. Algebra'' , '''88''' (1984) pp. 89–133 {{MR|0741934}} {{ZBL|0531.13015}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> E.G. Evans Jr., P.A. Griffith, "Local cohomology modules for normal domains" ''J. London Math. Soc.'' , '''19''' (1979) pp. 277–284 {{MR|0533326}} {{ZBL|0407.13019}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> S. Goto, "On Buchsbaum rings" ''J. Algebra'' , '''67''' (1980) pp. 272–279 {{MR|0602063}} {{ZBL|0473.13010}} {{ZBL|0473.13009}} {{ZBL|0413.13012}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> S. Goto, "Blowing-up of Buchsbaum rings" , ''Commutative Algebra'' , ''Lecture Notes'' , '''72''' , London Math. Soc. (1981) pp. 140–162 {{MR|0693633}} {{ZBL|0519.13021}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> S. Goto, "Buchsbaum rings with multiplicity 2" ''J. Algebra'' , '''74''' (1982) pp. 494–508 {{MR|0647250}} {{ZBL|0479.13007}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> S. Goto, "Buchsbaum rings of maximal embedding dimension" ''J. Algebra'' , '''76''' (1982) pp. 383–399 {{MR|0661862}} {{ZBL|0482.13012}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> S. Goto, "On the associated graded rings of parameter ideals in Buchsbaum rings" ''J. Algebra'' , '''85''' (1983) pp. 490–534 {{MR|0725097}} {{ZBL|0529.13010}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> S. Goto, "Surface singularities of finite Buchsbaum-representation type" , ''Commutative Algebra: Proc. Microprogram June 15–July 2'' , Springer (1987) pp. 247–263 {{MR|1015521}} {{ZBL|0741.13015}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> S. Goto, "Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized Cohen–Macaulay modules" M. Nagata (ed.) H. Matsumura (ed.) , ''Commutative Algebra and Combinatorics'' , ''Adv. Stud. Pure Math.'' , '''11''' , Kinokuniya (1987) pp. 39–46 {{MR|0951196}} {{ZBL|0649.13009}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> S. Goto, "Curve singularities of finite Buchsbaum-representation type" ''J. Algebra'' , '''163''' (1994) pp. 447–480 {{MR|1262714}} {{ZBL|0807.13007}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> S. Goto, "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity" ''J. Algebra'' , '''213''' (1999) pp. 604–661 {{MR|1673472}} {{ZBL|0942.13003}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> S. Goto, K. Nishida, "Rings with only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules" ''J. Math. Soc. Japan'' , '''40''' (1988) pp. 501–518 {{MR|0945349}} {{ZBL|0657.13022}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> S. Goto, K. Nishida, "The Cohen–Macaulay and Gorenstein Rees algebras associated to filtrations" , ''Memoirs'' , '''526''' , Amer. Math. Soc. (1994) {{MR|1287443}} {{ZBL|0812.13016}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> S. Goto, Y. Nakamura, K. Nishida, "Cohen–Macaulay graded rings associated ideals" ''Amer. J. Math.'' , '''118''' (1996) pp. 1197–1213 {{MR|}} {{ZBL|0878.13002}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> S. Goto, Y. Shimoda, "On the Rees algebras of Cohen–Macaulay local rings" R.N. Draper (ed.) , ''Commutative Algebra, Analytic Methods'' , ''Lecture Notes in Pure Applied Math.'' , '''68''' , M. Dekker (1982) pp. 201–231 {{MR|0655805}} {{ZBL|0482.13011}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> S. Goto, Y. Shimoda, "On Rees algebras over Buchsbaum rings" ''J. Math. Kyoto Univ.'' , '''20''' (1980) pp. 691–708 {{MR|0592354}} {{ZBL|0473.13010}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> S. Goto, K. Yamagishi, "The theory of unconditioned strong $d$-sequences and modules of finite local cohomology" ''Preprint'' (1978)</td></tr><tr><td valign="top">[a21]</td> <td valign="top"> C. Huneke, "The theory of d-sequences and powers of ideals" ''Adv. Math.'' , '''46''' (1982) pp. 249–279 {{MR|0683201}} {{ZBL|0505.13004}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> L.T. Hoa, C. Miyazaki, "Bounds on Castelnuovo–Mumford regularity for generalized Cohen–Macaulay graded rings" ''Math. Ann.'' , '''301''' (1995) pp. 587–598 {{MR|1324528}} {{ZBL|0834.13016}} </td></tr><tr><td valign="top">[a23]</td> <td valign="top"> M.-N. Ishida, "Tsuchihashi's cusp singularities are Buchsbaum singularities" ''Tôhoku Math. J.'' , '''36''' (1984) pp. 191–201 {{MR|742594}} {{ZBL|}} </td></tr><tr><td valign="top">[a24]</td> <td valign="top"> T. Kawasaki, "Local cohomology modules of indecomposable surjective–Buchsbaum modules over Gorenstein local rings" ''J. Math. Soc. Japan'' , '''48''' (1996) pp. 551–566 {{MR|1389995}} {{ZBL|0866.13007}} </td></tr><tr><td valign="top">[a25]</td> <td valign="top"> T. Kawasaki, "Arithmetic Cohen–Macaulayfications of local rings" , ''Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999'' (1999) pp. 88–92</td></tr><tr><td valign="top">[a26]</td> <td valign="top"> Y. Nakamura, "On the Buchsbaum property of associated graded rings" ''J. Algebra'' , '''209''' (1998) pp. 345–366 {{MR|1652142}} {{ZBL|0942.13002}} </td></tr><tr><td valign="top">[a27]</td> <td valign="top"> P. Schenzel, N.V. Trung, N.T. Cuong, "Verallgemeinerte Cohen–Macaulay-Moduln" ''Math. Nachr.'' , '''85''' (1978) pp. 57–73 {{MR|0517641}} {{ZBL|0398.13014}} </td></tr><tr><td valign="top">[a28]</td> <td valign="top"> J. Stückrad, W. Vogel, "Ein Korrekturglied in der Multiplizitätstheorie von D.G. Northcott und Anwendungen" ''Monatsh. Math.'' , '''76''' (1972) pp. 264–271 {{MR|}} {{ZBL|0248.13026}} </td></tr><tr><td valign="top">[a29]</td> <td valign="top"> J. Stückrad, W. Vogel, "Eine Verallgemeinerung der Cohen–Macaulay-Ringe und Anwendungen auf ein Problem der Multiplitätstheorie" ''J. Math. Kyoto Univ.'' , '''13''' (1973) pp. 513–528</td></tr><tr><td valign="top">[a30]</td> <td valign="top"> J. Stückrad, W. Vogel, "Toward a theory of Buchsbaum singularities" ''Amer. J. Math.'' , '''100''' (1978) pp. 727–746 {{MR|0509072}} {{ZBL|0429.14001}} </td></tr><tr><td valign="top">[a31]</td> <td valign="top"> J. Stückrad, W. Vogel, "Buchsbaum rings and applications" , Springer (1986) {{MR|0881220}} {{MR|0873945}} {{ZBL|0606.13018}} {{ZBL|0606.13017}} </td></tr><tr><td valign="top">[a32]</td> <td valign="top"> P. Schenzel, "Dualisierende Komplexe in der lokalen Algebra und Buchsbaum–Ringe" , ''Lecture Notes in Mathematics'' , '''907''' , Springer (1982) {{MR|0654151}} {{ZBL|0484.13016}} </td></tr><tr><td valign="top">[a33]</td> <td valign="top"> P. Schenzel, "On Veronesean embeddings and projections of Veronesean varieties" ''Archiv Math.'' , '''30''' (1978) pp. 391–397 {{MR|0485849}} {{ZBL|0417.14040}} </td></tr><tr><td valign="top">[a34]</td> <td valign="top"> R.Y. Sharp, "Necessary conditions for the existence of dualizing complexes in commutative algebra" , ''Lecture Notes in Mathematics'' , '''740''' , Springer (1979) pp. 213–229 {{MR|0563505}} {{ZBL|0421.13003}} </td></tr><tr><td valign="top">[a35]</td> <td valign="top"> Y. Shimoda, "A note on Rees algebras of two-dimensional local domains" ''J. Math. Kyoto Univ.'' , '''19''' (1979) pp. 327–333 {{MR|0545713}} {{ZBL|0447.13010}} </td></tr><tr><td valign="top">[a36]</td> <td valign="top"> J. Stückrad, "On the Buchsbaum property of Rees and form modules" ''Beitr. Algebra Geom.'' , '''19''' (1985) pp. 83–103 {{MR|0785248}} {{ZBL|0567.13008}} </td></tr><tr><td valign="top">[a37]</td> <td valign="top"> N.V. Trung, "Toward a theory of generalized Cohen–Macaulay modules" ''Nagoya Math. J.'' , '''102''' (1986) pp. 1–49 {{MR|}} {{ZBL|0649.13008}} {{ZBL|0637.13013}} </td></tr><tr><td valign="top">[a38]</td> <td valign="top"> N.V. Trung, S. Ikeda, "When is the Rees algebra Cohen–Macaulay?" ''Commun. Algebra'' , '''17''' (1989) pp. 2893–2922 {{MR|}} {{ZBL|0696.13015}} </td></tr><tr><td valign="top">[a39]</td> <td valign="top"> W. Vasconcelos, "Arithmetic of blowup algebras" , ''London Math. Soc. Lecture Notes'' , '''195''' , Cambridge Univ. Press (1994) {{MR|1275840}} {{ZBL|0813.13008}} </td></tr><tr><td valign="top">[a40]</td> <td valign="top"> Y. Yoshino, "Maximal Buchsbaum modules of finite projective dimension" ''J. Algebra'' , '''159''' (1993) pp. 240–264 {{MR|1231212}} {{ZBL|0791.13009}} </td></tr><tr><td valign="top">[a41]</td> <td valign="top"> K. Yamagishi, "The associated graded modules of Buchsbaum modules with respect to $\mathfrak{m}$-primary ideals in equi-I-invariant case" ''J. Algebra'' , '''225''' (2000) pp. 1–27 {{MR|1743648}} {{ZBL|}} </td></tr><tr><td valign="top">[a42]</td> <td valign="top"> K. Yamagishi, "Buchsbaumness in Rees algebras associated to $\mathfrak{m}$-primary ideals of minimal multiplicity in Buchsbaum local rings" , ''Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999'' (1999) pp. 39–45</td></tr></table>

Revision as of 16:55, 1 July 2020

The notion of a Buchsbaum ring (and module) is a generalization of that of a Cohen–Macaulay ring (respectively, module). Let $A$ denote a Noetherian local ring (cf. also Noetherian ring) with maximal ideal $\mathfrak{m}$ and $d = \operatorname { dim } A$. Let $M$ be a finitely-generated $A$-module with $\dim_AM = s$. Then $M$ is called a Buchsbaum module if the difference

\begin{equation*} \operatorname{l} _ { A } ( M / \mathfrak{q}M ) - e _ { \mathfrak{q} } ^ { 0 } ( M ) \end{equation*}

is independent of the choice of a parameter ideal $\mathfrak { q } = ( a _ { 1 } , \ldots , a _ { s } )$ of $M$, where $a _ { 1 } , \dots , a _ { s }$ is a system of parameters of $M$ and ${\bf l} _ { A } ( M / \text{q}M )$ (respectively, $e _ { \mathfrak{q} } ^ { 0 } ( M )$) denotes the length of the $A$-module $M / \mathfrak { q } M$ (respectively, the multiplicity of $M$ with respect to $\text{q}$). When this is the case, the difference

\begin{equation*} I ( M ) = {\bf l } _ { A } ( M / \mathfrak { q } M ) - e _ { \mathfrak { q } } ^ { 0 } ( M ) \end{equation*}

is called the Buchsbaum invariant of $M$. The $A$-module $M$ is a Cohen–Macaulay module if and only if $\text{l} _ { A } ( M / \text{q}M ) = e _ { \text{q} } ^ { 0 } ( M )$ for some (and hence for any) parameter ideal $\text{q}$ of $M$, so that $M$ is a Cohen–Macaulay $A$-module if and only if $M$ is a Buchsbaum $A$-module with $I ( M ) = 0$. The ring $A$ is said to be a Buchsbaum ring if $A$ is a Buchsbaum module over itself. If $A$ is a Buchsbaum ring, then $A _ { \mathfrak{p} }$ is a Cohen–Macaulay ring with $\operatorname { dim } A _ { \mathfrak { p } } = \operatorname { dim } A - \operatorname { dim } A / \mathfrak { p }$ for every $\mathfrak{p} \in \operatorname { Spec } A \backslash \{ \mathfrak{m} \}$.

A typical example of Buchsbaum rings is as follows. Let

\begin{equation*} A = B / ( X _ { 1 } , \dots , X _ { d } ) \bigcap ( Y _ { 1 } , \dots , Y _ { d } ), \end{equation*}

where $B = k [ [ X _ { 1 } , \dots , X _ { d } , Y _ { 1 } , \dots , Y _ { d } ]]$, with $d \geq 1$, denotes the formal power series ring in $2 d$ variables over a field $k$. Then $A$ is a Buchsbaum ring with $\operatorname { dim } A = d$ and $I ( A ) = d - 1$.

A, not necessarily local, Noetherian ring $R$ is said to be a Buchsbaum ring if the local rings $R _ { \mathfrak{p} }$ are Buchsbaum for all $\mathfrak { p } \in \operatorname { Spec } R$.

The theory of Buchsbaum rings and modules dates back to a question raised in 1965 by D.A. Buchsbaum [a3]. He asked whether the difference $ \operatorname{l} _ { A } ( A / \mathfrak { q } ) - e _ { \mathfrak { q } } ^ { 0 } ( A )$, with $\text{q}$ a parameter ideal, is an invariant for any Noetherian local ring $A$. This is, however, not the case and a counterexample was given in [a28]. Thereafter, in 1973 J. Stückrad and W. Vogel published the classic paper [a29], from which the history of Buchsbaum rings and modules started. In [a29] they gave a characterization of Buchsbaum rings in terms of the following property of systems of parameters: A $d$-dimensional Noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ is Buchsbaum if and only if every system $a _ { 1 } , \dots , a _ { d }$ of parameters for $A$ forms a weak $A$-sequence, that is, the equality

\begin{equation*} ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } = ( a _ { 1 } , \dots , a _ { i - 1 } ) : \mathfrak{m} \end{equation*}

holds for all $1 \leq i \leq d$. Therefore, systems $a _ { 1 } , \dots , a _ { d }$ of parameters in a Buchsbaum local ring need not be regular sequences, but the differences

\begin{equation*} [ ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } ] / ( a _ { 1 } , \dots , a _ { i - 1 } ) ,\; 1 \leq i \leq d, \end{equation*}

are very small and only finite-dimensional vector spaces over the residue class field $A /_{ \mathfrak{m}}$ of $A$. Weak sequences are closely related to $d$-sequences introduced by C. Huneke [a21]. Actually, $A$ is a Buchsbaum ring if and only if every system $a _ { 1 } , \dots , a _ { d }$ of parameters for $A$ forms a $d$-sequence, that is, the equality

\begin{equation*} ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { i } a _ { j } = ( a _ { 1 } , \dots , a _ { i - 1 } ) : a _ { j } \end{equation*}

holds for all $1 \leq i \leq j \leq d$.

One of the fundamental results on Buchsbaum rings and modules is the surjectivity criterion. Let

\begin{equation*} H _ { \mathfrak{m} } ^ { i } ( M ) = \operatorname { lim } _ { n \rightarrow \infty } \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } ^ { n } , M ) \quad ( i \in \mathbf{Z} ) \end{equation*}

denote the $i$th local cohomology of $M$ with respect to the maximal ideal $\mathfrak{m}$. If $M$ is a Buchsbaum $A$-module, then $\mathfrak { m } \cdot H _ { \mathfrak { m } } ^ { i } ( M ) = ( 0 )$ for all $i \neq s$ and the equality

\begin{equation*} \operatorname {I} ( M ) = \sum _ { i = 0 } ^ { s - 1 } \left( \begin{array} { c } { s - 1 } \\ { i } \end{array} \right) .\operatorname { l}_ { A } ( H _ {\frak m } ^ { i } ( M ) ) \end{equation*}

holds, where $s = \operatorname { dim } _ { A } M$.

Unfortunately, the vanishing does not characterize Buchsbaum modules. Modules $M$ with $\mathfrak { m } \cdot H _ { \mathfrak { m } } ^ { i } ( M ) = ( 0 )$ for all $i \neq \operatorname { dim } _ { A } M$ are called quasi-Buchsbaum and constitute a class which is strictly larger than that of Buchsbaum modules. However, if the canonical homomorphism

\begin{equation*} \varphi _ { M } ^ { i } : \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } , M ) \rightarrow H _ {\frak m } ^ { i } ( M ) = \operatorname { lim } _ { n \rightarrow \infty } \operatorname { Ext } _ { A } ^ { i } ( A / \mathfrak { m } ^ { n } , M ) \end{equation*}

is surjective for all $i \neq \operatorname { dim } _ { A } M$, then $M$ is a Buchsbaum $A$-module. The converse is also true if the base ring $A$ is regular (cf. also Regular ring (in commutative algebra)).

After the establishment of the surjectivity criterion, by Stückrad and Vogel [a30] in 1978, the development of the theory became rather rapid. The ubiquity of Buchsbaum normal local rings was established by S. Goto [a6] as an application of the Evans–Griffith construction [a5]. Namely, let $d \geq 1$ and $\{ h _ { i } \} _ { 0 \leq i \leq d - 1 }$ be integers. Then there exists a Buchsbaum local ring $A$ with $\operatorname { dim } A = d$ and $\operatorname{l} _ { A } ( H _ { \text{m} } ^ { i } ( A ) ) = h _ { i }$ for $0 \leq i \leq d - 1$. If $h _ { 0 } = 0$ (respectively, $d \geq 2$ and $h _ { 0 } = h _ { 1 } = 0$), one may choose the ring $A$ so that $A$ is an integral domain (respectively, a normal ring). See [a1] for progress in the research about the ubiquity of Buchsbaum homogeneous integral domains. Besides, Buchsbaum local rings of multiplicity $2$ have been classified [a8]. Also, certain famous isolated singularities are Buchsbaum (cf. [a23]).

The theory of Buchsbaum rings and modules is closely related to that of Cohen–Macaulayness in blowing-ups. Let $I$ be an ideal of positive height in a Noetherian local ring $A$. Let $R ( I ) = \oplus _ { n \geq 0 } I ^ { n }$ and call it the Rees algebra of $I$. Then the canonical morphism $\operatorname{Proj} R ( I ) \rightarrow \operatorname{Spec} A$ is the blowing-up of $A$ with centre $I$ (cf. also Blow-up algebra). If the ring $R ( I )$ is Cohen–Macaulay, then the scheme naturally is locally Cohen–Macaulay. The problem when the Rees algebra $R ( I )$ is Cohen–Macaulay has been intensively studied from the 1980s onwards ([a18], [a38], [a16], [a39], [a17]).

The ring $R ( I )$ is Cohen–Macaulay if the ideal $I$ is generated by a regular sequence and if the base ring $A$ is Cohen–Macaulay [a2]. However, the converse is not true even for parameter ideals $I$. Actually, $A$ is a Buchsbaum ring if and only if the Rees algebra $R ( \mathfrak { q } )$ is a Cohen–Macaulay ring for every parameter ideal $\text{q}$ in $A$, provided that $A$ is an integral domain with $\operatorname { dim } A = 2$. This insightful result of Y. Shimoda [a35] in 1979 opened the door towards a further development of the theory. Firstly, Goto and Shimoda [a19] showed that a Noetherian local ring $A$ is a Buchsbaum ring with $H _ { \mathfrak{m} } ^ { i } ( A ) = ( 0 )$ ($i \neq 1 , \operatorname { dim } A$) if and only if the Rees algebra $R ( \mathfrak { q } )$ is a Cohen–Macaulay ring for every parameter ideal $\text{q}$ in $A$. When this is the case, the Rees algebras $R ( \mathfrak{q} ^ { n } )$ are also Cohen–Macaulay for all $n \geq 1$. In 1981, Buchsbaum rings were characterized in terms of the blowing-ups of parameter ideals. Let $A$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $d = \operatorname { dim } A \geq 1$. Then $A / H _ { \mathfrak{m} } ^ { 0 } ( A )$ is a Buchsbaum ring if and only if the scheme is locally Cohen–Macaulay for every parameter ideal $\text{q}$ in $A$ [a7]. Subsequently, Goto [a10] proved that the associated graded rings $G ( \mathfrak { q } ) = \oplus _ { n \geq 0} \mathfrak { q } ^ { n } / \mathfrak { q } ^ { n + 1 }$ of parameter ideals $\text{q}$ in a Buchsbaum local ring are always Buchsbaum. In addition, Stückrad showed that $R ( \mathfrak { q } )$ is a Buchsbaum ring for every parameter ideal $\text{q}$ in a Buchsbaum local ring [a36]. The systems of parameters in Buchsbaum local rings behave very well and enjoy the monomial property [a10].

Buchsbaum rings are yet (2000) the only non-trivial case for which the monomial conjecture, raised by M. Hochster, has been solved affirmatively (except for the equi-characteristic case). See [a31] for these results, together with geometric applications and concrete examples. See [a31] for researches on the Buchsbaum property in affine semi-group rings and Stanley–Reisner rings of simplicial complexes.

Let $M$ be a Buchsbaum module over a Noetherian local ring $A$. Then $M$ is said to be maximal if $\operatorname{dim}_{A} M = \operatorname{dim} A $. Noetherian local rings possessing only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules are said to have finite Buchsbaum-representation type. Buchsbaum representation theory was studied by Goto and K. Nishida [a15], [a11], [a13], and the Cohen–Macaulay local rings $A$ of finite Buchsbaum-representation type have been classified under certain mild conditions. If $\operatorname { dim } A \geq 2$, then $A$ must be regular [a15]. The situation is a little more complicated if $\operatorname { dim } A = 1$ [a13]. In [a11] (not necessarily Cohen–Macaulay) surface singularities of finite Buchsbaum-representation type are classified.

Suppose that $A$ is a regular local ring with $\operatorname { dim } A = d$ and let $M$ be a maximal Buchsbaum $A$-module. Then $M _ { \mathfrak{p} }$ is a free $A _ { \mathfrak{p} }$-module for all $\mathfrak{p} \in \operatorname { Spec } A \backslash \{ \mathfrak{m} \}$, so that the $A$-module $M$ defines a vector bundle on the punctured spectrum $\operatorname {Spec} A \backslash \{ \mathfrak{m} \}$ of $A$. Thanks to the surjectivity criterion, one can prove the structure theorem of maximal Buchsbaum modules over regular local rings: Every maximal Buchsbaum $A$-module $M$ has the form

\begin{equation*} M \cong \bigoplus _ { i = 0 } ^ { d } E _ { i } ^ { h _ { i } }, \end{equation*}

where $E_i$ denotes the $i$th syzygy module of the residue class field $A /_{ \mathfrak{m}}$ of $A$, $h _ { i } = \operatorname { l } _ { A } ( H _ { \mathfrak{m} } ^ { i } ( M ) )$ ($0 \leq i \leq d - 1$), and $h _ { d } = \operatorname { rank } _ { A } M - \sum _ { i = 1 } ^ { d - 1 } \left( \begin{array} { c } { d - 1 } \\ { i - 1 } \end{array} \right) h _ { i }$, if $A$ is a regular local ring ([a4], [a12]). This result has been generalized by Y. Yoshino [a40] and T. Kawasaki [a24]. They showed a similar decomposition theorem of a special kind of maximal Buchsbaum modules over Gorenstein local rings; see [a32] for the characterization of Buchsbaum rings and modules in terms of dualizing complexes. (It should be noted here that the main result in [a32] contains a serious mistake, which has been repaired in [a40].)

A local ring $A$ satisfying the condition that all the local cohomology modules $H _ { \mathfrak{m} } ^ { i } ( A )$ ($i \neq \operatorname { dim } A$) are finitely generated is said to be an FLC ring (or a generalized Cohen–Macaulay ring). The class of FLC rings includes Buchsbaum rings as typical examples. In fact, a Noetherian local ring $A$ is FLC if and only if it contains at least one system $a _ { 1 } , \dots , a _ { d }$ ($d = \operatorname { dim } A$) of parameters such that the sequence $\alpha _ { 1 } ^ { n _ { 1 } } , \dots , \alpha _ { d } ^ { n _ { d } }$ forms a $d$-sequence in any order for all integers $n _ { i } \geq 1$. Such a sequence is called an unconditioned strong $d$-sequence (for short, USD-sequence or $d ^ { + }$-sequence); they have been intensively studied [a27], [a37], [a20]. Recently (1999), Kawasaki [a25] used the results in [a20] to establish the arithmetic Cohen–Macaulayfications of Noetherian local rings. Namely, every unmixed local ring $A$ contains an ideal $I$ of positive height with the Cohen–Macaulay Rees algebra $R ( I )$, provided $\operatorname { dim } A \geq 1$ and all the formal fibres of $A$ are Cohen–Macaulay. Hence, the Sharp conjecture [a34] concerning the existence of dualizing complexes is solved affirmatively.

Let $R = \oplus _ { n \geq 0} R _ { n }$ be a Noetherian graded ring with $k = R _ { 0 }$ a field and let $\mathfrak { M } = R _ { + }$. Then $R$ is a Buchsbaum ring if and only if the local ring $R _ {\frak M }$ is Buchsbaum. When this is the case, the local cohomology modules $H _ { \mathfrak{M} } ^ { i } ( R )$ ($i \neq \operatorname { dim } R$) are finite-dimensional vector spaces over the field $k$. The vanishing of certain homogeneous components $[ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { n }$ of $H _ { \mathfrak{M} } ^ { i } ( R )$ may affect the Buchsbaumness in graded algebras $R$. For example, if there exist integers $\{ t _ { i } \} _ { 0 \leq i \leq d - 1}$ ($d = \operatorname { dim } R$) such that $t _ { i } \leq t_{i + 1} + 1$ for all $0 \leq i \leq d - 1$ and if

\begin{equation*} [ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { n } = ( 0 ) \end{equation*}

for all $n \neq t_i$ and $0 \leq i \leq d - 1$, then $R$ is a Buchsbaum ring [a9]. Therefore $R$ is a Buchsbaum ring if $H _ { \mathfrak{M} } ^ { i } ( R ) = [ H _ { \mathfrak{M} } ^ { i } ( R ) ] _ { 0 }$ for all $i \neq d$ [a33]. Hence the scheme $X = \operatorname { Proj } R$ is arithmetically Buchsbaum if $X$ is locally Cohen–Macaulay, provided that $R = k [ R _ { 1 }]$ and $R$ is equi-dimensional. See [a22] for the bounds of Castelnuovo–Mumford regularities of Buchsbaum schemes $X = \operatorname { Proj } R$.

Researches of the Buchsbaumness in Rees algebras recently (1999) started again, although the progress remains tardy (possibly because of the lack of characterizations of Trung–Ikeda type [a38] for Buchsbaumness). In [a14] the Buchsbaumness in Rees algebras $R ( I )$ of certain $\mathfrak{m}$-primary ideals $I$ in Cohen–Macaulay local rings is closely studied in connection with the Buchsbaumness in the associated graded rings $G ( I ) = \oplus _ { n \geq 0} I ^ { n } / I ^ { n + 1 }$ and that of the extended Rees algebras $R ^ { \prime } ( I ) = \oplus _ { n \in \bf Z} I^ { n }$. In [a26], [a41], [a42], Buchsbaumness in graded rings associated to certain $\mathfrak{m}$-primary ideals in Buchsbaum local rings is explored. Especially, the Rees algebra $R ( \mathfrak{m} )$ of the maximal ideal $\mathfrak{m}$ in a Buchsbaum local ring $A$ of maximal embedding dimension (that is, a Buchsbaum local ring $A$ for which the equality $v ( A ) = e _ { \mathfrak{m} } ^ { 0 } ( A ) + \operatorname { dim } A + I ( A ) - 1$ holds) is again a Buchsbaum ring [a42].

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How to Cite This Entry:
Buchsbaum ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buchsbaum_ring&oldid=24390
This article was adapted from an original article by Shiro Goto (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article