Liftings and weak liftings of modules
The context is mod \(\Lambda\), the category of all finitely generated (left) modules over a noetherian \(R\)-algebra \(\Lambda\) for a commutative local ring \(R\). Let \(\Lambda \to \Gamma\) be a homomorphism of rings and let \(M\) be in mod \(\Gamma\). \(L\) in mod \(\Lambda\) is called a lifting of \(M\) to \(\Lambda\) if (a) \(M \cong \Gamma \otimes_ \Lambda L\) and (b) \(\text{Tor}^ \Lambda{_ i(\Gamma,L)}=(0)\) for all \(i \geq 1\). When such a \(\Lambda\)-module \(L\) exists, \(M\) is said to be liftable to \(\Lambda\). If \(M\) is only a direct summand of \(\Gamma\otimes_ \Lambda L\) for some \(\Lambda\)-module \(L\) satisfying (b), then \(L\) is a weak lifting of \(M\) to \(\Lambda\) and \(M\) is said to be weakly liftable to \(\Lambda\). The main results follow. Section 1: Let \(\Lambda\) be a noetherian \(R\)-algebra with \(R\) a commutative complete local ring with maximal ideal \({\mathfrak m}\). Let \(\{x_ 1,x_ 2,\dots,x_ t\}\) be a \(\Lambda\)-regular sequence in \({\mathfrak m}\) and \(\Gamma = \Lambda/(x_ 1,x_ 2,\dots,x_ t)\Lambda\). If \(\text{Ext}^ 2_ \Gamma(M,M)=(0)\) for \(M\) in mod \(\Gamma\), then \(M\) is liftable to \(\Lambda\). Section 2: Every lifting of \(M\) to \(\Lambda\) is a factor module of \(\Omega_ \Lambda(M)\), the first syzygy of \(M\) over \(\Lambda\) given by the projective cover of \(M\) over \(\Lambda\). Section 3: \(R\) as in section 1 but not necessarily complete. Let \(\{x_ 1,x_ 2,\dots,x_ t\}\) be a \(\Lambda\)-regular sequence of central elements in \(\Lambda\) in \(I\), the ideal \((x_ 1,x_ 2,\dots,x_ t)\Lambda\) in \(\Lambda\). The following are equivalent for a \(\Gamma\)- module \(M\): (a) the module \(M\) is weakly liftable to \(\Lambda\), (b) the module \(M\) is liftable to \(\Gamma_ 2 = \Lambda/I^ 2 \Lambda\), and (c) the module \(M\) is isomorphic to a direct summand of \(\Omega^ t {_ \Lambda(M)}/I\Omega^ t{_ \Lambda(M)} \oplus Q/IQ\) for some projective \(\Lambda\)-module \(Q\) and for any given projective resolution defining a \(t\)-th syzygy \(\Omega^ t {_ \Lambda(M)}\). Section 4: The context is \(R \to R/(x)=R\), where \(R\) is a commutative local Gorenstein ring and \(x\) an \(R\)-regular element in the maximal ideal of \(R\). Let \(CM(R)\) denote the category of all finitely generated maximal Cohen-Macaulay modules over \(R\). Every lifting of a module \(C\) in CM\((R)\) is a submodule of the minimal Cohen-Macaulay approximation \(X_ C\) of \(C\) over \(R\). Section 5: Study of the properties of weakly liftable \(R\)-modules in terms of their minimal Cohen-Macaulay approximation over \(R\), where \(R\) is a commutative local Gorenstein ring.
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