Almost Gorenstein rings arising from fiber products

Naoki Endo, Shiro Goto, Ryotaro Isobe

doi:10.4153/s000843952000051x

<jats:title>Abstract</jats:title><jats:p>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<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline1.png" /><jats:tex-math>$R \times _T S$</jats:tex-math></jats:alternatives></jats:inline-formula>of Cohen–Macaulay local rings<jats:italic>R</jats:italic>,<jats:italic>S</jats:italic>of the same dimension<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline2.png" /><jats:tex-math>$d>0$</jats:tex-math></jats:alternatives></jats:inline-formula>over a regular local ring<jats:italic>T</jats:italic>with<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline3.png" /><jats:tex-math>$\dim T=d-1$</jats:tex-math></jats:alternatives></jats:inline-formula>is an almost Gorenstein ring if and only if so are<jats:italic>R</jats:italic>and<jats:italic>S</jats:italic>. In addition, the other generalizations of Gorenstein properties are also explored.</jats:p>

Almost Gorenstein rings arising from fiber products

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