Commutative Algebra with a View Toward Algebraic Geometry
《Commutative Algebra with a View Toward Algebraic Geometry》 · 第 472–473 页 · Section 19.1「Projective Dimension and Minimal Resolutions」
A complex $\mathbb F:\cdots\to F_n\xrightarrow{\varphi_n}F_{n-1}\to\cdots$ over a local ring $(R,P)$ is minimal if the maps in the complex $\mathbb F\otimes R/P$ are all $0$; that is, for each $n$, the image of $\varphi_n:F_n\to F_{n-1}$ is contained in $PF_{n-1}$. A free resolution over a local ring is a minimal complex if and only if for each $n$, a basis of $F_{n-1}$ maps onto a minimal set of generators of $\operatorname{coker}\varphi_n$.