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Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings. (English) Zbl 1274.13031
Let \(R\) be a Cohen-Macaulay ring and \(\text{MCM}(R)\) be the category of maximal Cohen-Macaulay modules over \(R\). An \(R\)-module \(M \in \text{MCM}(R)\) is called cluster tilting if: \[ \text{add}(M) = \{X\in \text{MCM}(R)~|~\text{Ext}_R^1(X,M)=0\} = \{X\in \text{MCM}(R)~|~\text{Ext}_R^1(M,X)=0\} \] where \(\text{add}(M)\) consists of direct summands of direct sums of copies of \(M\).
In [I. Burban, et al., Adv. Math. 217, No. 6, 2443–2484 (2008; Zbl 1143.13014)], the following result was obtained:
Theorem. Let \(k\) be an algebraically closed field of characteristic \(0\) and \(R=k[[x,y]]/(f)\) be a one-dimensional reduced hypersurface singularity. Then \(\text{MCM}(R)\) has a cluster–tilting object if and only if \(f= f_1\cdots f_n\) such that \(f_i\notin (x,y)^2\) for each \(i\).
The initial goal for this paper was to understand how one can obtain the above theorem from a pure homological approach, with a view towards proving it in more general situations. The authors prove the following:
Theorem A Let \(k\) be an algebraically closed field of characteristic not \(2\) and \(R=k[[x,y]]/(f)\) be a one-dimensional reduced hypersurface singularity. Then \(\text{MCM}(R)\) has a cluster-tilting object if and only if \(f= f_1\cdots f_n\) such that \(f_i\notin (x,y)^2\) for each \(i\).
Theorem B Let \(R\) be a Cohen-Macaulay ring of dimension \(d\geq 3\). Let \(M \in \text{MCM}(R)\) such that \(M\) has a free summand and \(A=\text{\operatorname{Hom}}_R(M,M)\) is \(\text{MCM}\). For an integer \(n>0\) let: \[ M^{\perp_n} = \{X\in \text{MCM}(R) | \text{Ext}_R^i(M,X)=0 \;\text{for} \;1\leq i \leq n\} \] Consider the following: (1) There exist an integer \(n\) such that \(1\leq n\leq d-2\) and \( M^{\perp_n} = \text{add}(M) \). (2) \(\text{gldim} A\leq d\). (3) \(\text{gldim} A=d\). (4) \(\text{Ext}_R^i(M,M)=0 \;\text{for} \;1\leq i \leq d-2\) (i.e., \(M\in M^{\perp_{d-2}}\)). (5) \( M^{\perp_{d-2}} = \text{add}(M)\).
Then \((5)\Rightarrow (1) \Rightarrow (2) \Leftrightarrow (3)\). If in addition \(R\) is locally Gorenstein on the non-maximal primes of \(\text{Spec}R\) then \((3)+(4) \Rightarrow (5)\). If \(R\) is also locally regular on the non-maximal primes of \(\text{Spec} R\) (i.e. \(\text{Spec} R\) has isolated singularities) then \((3) \Rightarrow (5)\).

MSC:
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
16G30 Representations of orders, lattices, algebras over commutative rings
16G50 Cohen-Macaulay modules in associative algebras
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