Splitters and relative homological algebra (Q1204447)

Let \(K\subset\Delta\subset\Lambda\) be rings. The subring \(\Delta\) is called a splitter of \(\Lambda\) over \(K\) if the multiplication map \(m_ K: \Lambda\otimes_ K \Lambda\to \Lambda\) has a right inverse which is a homomorphism of \(\Delta\)-bimodules. \(\Delta\) is said to be non-splittable over \(K\) if \(K\) is the only splitter of \(\Lambda\) over \(K\). The authors give several characterizations of splitters -- one most commonly used being that \(\Delta\) is a splitter of \(\Lambda\) over \(K\) if and only if there exists an \(e\in \Lambda\otimes_ K \Lambda\) such that \(m_ K(e)=1\) and \(e\delta= \delta e\) for all \(\delta\in \Delta\). Among many general properties of splitters, it is proved that when \(K\) is a field and \(\Lambda\) is a \(K\)-algebra then every splitter is a finite- dimensional subalgebra of \(\Lambda\). If, in addition, \(\Lambda\) is local, it is shown that every splitter is imbedded in a division subalgebra of \(\Lambda\). When \(K\) is an arbitrary field, \(L\) an extension of \(K\), and \(F\) the algebraic closure of \(K\) in \(L\), then \(L\) is non-splittable over \(K\) if and only if \(F\) is a purely inseparable extension of \(K\). Using this it is established that a \(K\)-algebra \(\Lambda\) is algebraic non- splittable if and only if the Jacobson radical \(J(\Lambda)\) of \(\Lambda\) is nil and \(\Lambda/ J(\Lambda)\) is a purely inseparable field extension of \(K\). The question whether an algebraic non-splittable algebra remains non- splittable under an extension of the ground field is also investigated. This question is shown to be equivalent to the classical question of whether every nil algebra remains nil under extension of the ground field. A \(K\)-algebra \(\Lambda\), \(K\) a field, is said to be totally splittable if every finite-dimensional \(K\)-subalgebra of \(\Lambda\) is a splitter. It is then shown that an algebraic field extension \(L\) of a field \(K\) is separable if and only if \(L\) is a totally splittable \(K\)- algebra.