projective dimension from last nonzero betti number

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

betti number is rank in minimal resolution: For every minimal degreewise finite free resolution $F_{\bullet }\to M$ of a finite module over a nonzero Noetherian local ring, ${\beta }_i^R(M)={\mathrm{rank}}_R{F}_i$ for all $i\ge 0$.

Projective dimension of an object: Assume projective resolutions are supplied or exist in the relevant class. The projective dimension of $M$ is $\mathrm{pd}(M)=\inf \{d\ge 0:M\text{ has a projective resolution of length }d\},$ with value $\mathrm{\infty }$ if this set is empty. A length-zero projective resolution exists exactly when $M$ is projective.

Projective dimension at most n iff the nth syzygy is projective: Let $\mathcal{A}$ be an abelian category with enough projectives, fix a projective resolution $P_{\bullet }\to M$, and let $n\ge 1$. Then $\mathrm{pd}(M)\le n⟺{\mathrm{\Omega }}_P^n(M)\text{ is projective}\mathrm{.}$ In particular, the condition is independent of the chosen projective resolution.

A finite flat module over a local ring is free: The standard theorem holds over arbitrary local rings; the proof written here is the Noetherian local case. Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finite flat $R$-module. Then $M$ is free.

Projective left and right modules are flat over an arbitrary ring: Every projective left or right module over an arbitrary ring is flat on its appropriate side.

Assuming the Axiom of Choice, Nakayama's lemma: Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I⊴R$ satisfy $I\subseteq J(R)$, and let $M$ be a finitely generated left $R$-module. If $IM=M$, then $M=0$.

The balanced Tor bifunctor: For a right $R$-module $N$, a left $R$-module $M$, and $i\ge 0$, define ${\mathrm{Tor}}_i^R(N,M)$ to be either $H_i(N{\otimes }_R{P}_{\bullet })$ for a projective resolution of $M$ or $H_i(Q_{\bullet }{\otimes }_{R}M)$ for a projective resolution of $N$, identified by the preceding natural balance isomorphism. On maps it uses the homology maps induced by comparison maps; coherence makes this a well-defined covariant bifunctor.

minimal free resolution reduces to zero differential: Reducing a minimal degreewise finite free resolution modulo $\mathfrak{m}$ gives the zero differential, so ${\mathrm{Tor}}_i^R(k,M)\cong k{\otimes }_R{F}_i$.

finite local modules admit minimal free resolutions: Every finite module $M$ over a nonzero Noetherian local ring $(R,\mathfrak{m},k)$ has an augmented resolution $\cdots \to F_1\to F_0\to M\to 0$ by finite-rank free modules, with $d_i(F_i)\subseteq \mathfrak{m}{F}_{i-1}$ for $i>0$. Such a resolution is called minimal; it need not be bounded. This extends the bounded terminology without changing it.