Seminormal or \(t\)-closed schemes and Rees rings
A commutative ring \(A\) with identity is called decent if its total quotient ring is absolutely flat. For example, integral domains, reduced Noetherian rings, and reduced nontrivial positively graded rings with \(\text{Min} (A)\) compact are all decent rings. The author defines a reduced scheme \(X\) to be decent if there is an open affine base \((U_\lambda)\) of \(X\) such that every ring of sections \(\Gamma(U_\lambda,{\mathcal O}_X)\) is a decent ring. For a positively graded ring \(A\), the projective scheme \(X=\text{Proj}(A)\) is said to be canonically decent if \(\Gamma(D_+(f),{\mathcal O}_X)=A_{(f)}\) is decent for each homogeneous \(f\in A\) of positive degree. For such schemes, the author defines total quotient schemes and uses them to define the normalization, seminormalization, and \(t\)-closure of a decent scheme. The author also investigates seminormal and \(t\)-closed schemes and Rees rings. (Recall that \(A\) is seminormal if whenever \(a^3=b^2\) for \(a,b\in A\), there is a \(t\in A\) such that \(a=t^2\) and \(b=t^3\); and that \(A\) is \(t\)-closed if the annihilator of each element of \(A\) is generated by an idempotent and whenever \(a^3+rab-b^2=0\) for \(r,a,b\in A\), then \(a=t^2-rt\) and \(b=t^3-rt^2\) for some \(t\in A.)\) For example, if \(A\) is a decent ring and \(F=(I_k)\) is a regular filtration on \(A\), then the associated Rees ring \(R=\bigoplus_{k\in\mathbb{N}}I_kX^k\) is decent and \(\text{Proj}(R)\) is canonically decent.
scientific article; zbMATH DE number 1268664