Projective covers and minimal free resolutions (Q1907721)

Let \(R\) be a commutative ring with identity. A complex \(\mathbf C\) of \(R\)-modules is a sequence of \(R\)-module homomorphisms \(\dots C_n @>f_n>> C_{n-1} @>f_{n-1}>> \cdots\to C_1 @>f_1>> C_0\to\dots\), satisfying \(f_{n-1}f_n=0\). The maps \(f_n\) are called the boundary maps of \(\mathbf C\). A map of complexes \(g:{\mathbf C}\to{\mathbf D}\) is defined as a sequence of \(R\)-module homomorphisms \(g_i:C_i\to D_i\) which commutes with the boundary maps of the complexes \(\mathbf C\) and \(\mathbf D\). Generalizing the concept of the projective cover of a module, the author defines the projective cover of a complex. It is shown in the paper that the projective cover is a direct summand of every surjective free resolution and is the direct sum of the minimal free resolution and an exact complex. Necessary and sufficient conditions for the projective cover and minimal free resolution to be identical are discussed.