A Buchsbaum theory for tight closure

A Noetherian local ring ( R , m mathvariant="fraktur" stretchy="false">) (R,\frak {m}) is called Buchsbaum if the difference alttext="script l left-parenthesis slash q right-parenthesis minus e ? / mathvariant="fraktur">q ? e encoding="application/x-tex">\ell (R/\mathfrak {q})-e(\mathfrak {q}, R) , where alttext="German q"> encoding="application/x-tex">\mathfrak {q} an ideal generated by a system of parameters, constant independent . In this article, we study tight closure analog condition. We prove that in unmixed excellent prime characteristic alttext="p greater-than 0"> p &gt; 0 encoding="application/x-tex">p&gt;0 and dimension at least one, alttext="e script Superscript asterisk Baseline ?<!-- ? </mml:msup> encoding="application/x-tex">e(\mathfrak R)-\ell {q}^*) only parameter test alttext="tau Subscript normal p r ?<!-- ? mathvariant="normal">p mathvariant="normal">a mathvariant="normal">r encoding="application/x-tex">\tau _{\mathrm {par}}(R) contains m"> encoding="application/x-tex">\frak {m} also provide characterization condition via derived category which analogous to Schenzel’s criterion for rings.