# zbMATH — the first resource for mathematics

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
Full Text: