Dagger closure and solid closure in graded dimension two (Q2849034)

There are many closure operations on ideals of commutative rings such as solid closure, plus closure and tight closure. There is a characteristic-free approach to the tight closure called dagger closure [\textit{M. Hochster} and \textit{C. Huneke}, J. Pure Appl. Algebra 71, No. 2--3, 233--247 (1991; Zbl 0734.13003)]. Suppose that the ring \(R\) is normal, two-dimensional, positively graded and of finite type over a field \(k\). In [Am. J. Math. 128, No. 2, 531--539 (2006; Zbl 1102.13002)], \textit{H. Brenner} proved that tight closure and plus closure are the same if \(k\) is the algebraic closure of a finite field. This condition is needed to deduce that the moduli space of semistable vector bundle over a curve is finite.NEWLINENEWLINEUnder the above assumption, this paper shows that dagger closure and solid closure are the same. The proof is much more interesting in the characteristic zero case. This algebraic result has a geometric argument. To give an idea of the proof, let \(I=(f_1,\dots, f_n)\vartriangleleft R\) and \(f\in R_m\) be homogeneous. There are two reduction steps. First, \(I\) is \(R_+\)-primary. Second the syzygy bundle \(\mathrm{syz}(\underline{f}_i)\) is strongly semistable. This reduction follows by a result of \textit{A. Langer} [Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)] and the Harder-Narasimhan filtration. Look at the cohomology class \(c:=\delta(f)\in H^1(\mathrm{syz}(\underline{f}_i)(m))\), where \(\delta:R_m\to H^1(\mathrm{syz}(\underline{f}_i)(m))\) is the connecting homomorphism. This cohomology class defines a torsor call it \(T(c)\). In terms of torsor, one has the following translation to geometry: 1) \(f\) is in the solid closure of \(I\) if and only if \(T(c)\) is not affine. 2) \(f\) is in the dagger closure of \(I\) if and only if \(\delta(f)\) is almost zero.NEWLINENEWLINEAlmost zero means that for all \( \epsilon> 0\) there exists a finite morphism \(\varphi : Y' \to Y :=Proj(R)\) between smooth projective curves and a line bundle \(L\) on \(Y'\) with a non-zero global section \(s\) such that \(\deg L/\deg \varphi < \epsilon\) and \(0=s\varphi^*(c)\in H^1(Y',\varphi^*(\mathrm{syz}(\underline{f}_i)(m))\otimes L)\).NEWLINENEWLINEThus things reduce to show \(1)\Leftrightarrow 2)\). Assume \(c\neq 0\). Affineness of \(T(c)\) restate by showing that the slope of \(\mathrm{syz}(\underline{f}_i)(m)\) is not positive. Recall that the strongly semistable condition says the maximal slope and the minimal slope are the same after pull-back over finite morphisms. Let \(G\) denote a locally free sheaf on \(Y\). The bridge between \(1)\) and \( 2)\) may be read as follows: \(\overline{\mu}_{\min}(G) > 0\) if and only if there is \(\epsilon > 0\) such that for every finite morphism \(\varphi : Y'\to Y\) and every invertible quotient sheaf \(\varphi^\ast (G)\to L \to 0\), one has \(\deg (L)/ \deg (\varphi) \geq\epsilon > 0\), see [\textit{H. Brenner}, Trans. Am. Math. Soc. 356, No. 1, 371--392 (2004; Zbl 1041.13002)].NEWLINENEWLINEConcerning the assumption on the dimension, it may be worth to recall from [\textit{H. Brenner} and \textit{A. Stäbler}, J. Algebra 370, 176--185 (2012; Zbl 1279.13013)] that dagger closure is trivial in regular domains containing a field. But solid closure in equal characteristic zero is not trivial in three-dimensional regular rings by [\textit{P. C. Roberts}, Contemp. Math. 159, 351--356 (1994; Zbl 0818.13010)].