koszul euler characteristic and degree indexed multiplicity

Write ${\mathrm{\ell }}_R$ for module length. If $C_{\bullet }$ is a bounded homological complex of $R$-modules and every $H_i(C)$ has finite length, its Euler characteristic is $\chi (C)={\sum }_{i\in \mathbb{Z}}(-1{)}^i{\mathrm{\ell }}_R(H_i(C))\mathrm{.}$ Boundedness makes the sum finite. The terms of $C$ themselves need not have finite length. For a cochain complex use ${\sum }_n(-1{)}^n{\mathrm{\ell }}_R(H^n(C))$; reindexing $C^n=C_{-n}$ preserves this number.

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finite $R$-module. Call $I$ a module-relative ideal of definition if ${\mathrm{\ell }}_R(M\mathrm{/}IM)<\mathrm{\infty }$, allowing $I=R$. The unique eventual polynomial $P_{I,M}(n)={\mathrm{\ell }}_R(M\mathrm{/}{I}^{n+1}M)(n\gg 0)$ exists by The module-relative Hilbert–Samuel polynomial exists without a dimension theorem. For each integer $j\ge 0$ define the degree-indexed coefficient $e_j(I,M)=j![T^j]P_{I,M}(T)\mathrm{.}$ Here $[T^j]$ means the coefficient of $T^j$, and $0!=1$. Set $P_{I,0}=0$, $P_{R,M}=0$ and their coefficients equal to zero, consistently with that lemma. A coefficient above the degree is zero. Coefficients below the degree depend on the fixed $n+1$ convention; this definition does not identify the index with support dimension.

For a finite ordered sequence $\mathbf{f}$, $K(\mathbf{f};M)$ denotes Koszul Complex Of A Sequence With Coefficients. Its Euler characteristic is defined whenever its homology has finite length. For the empty sequence $I=0$ and $K(\mathrm{\varnothing };M)=M[0]$. The module-relative hypothesis then says $M$ has finite length, and $P_{0,M}$ is the constant ${\mathrm{\ell }}_R(M)$.

Source locators: Hochster, Math 615 (Winter 2012), printed pp.104–108; Stacks 43.15.1 and 43.15.6. Our definition extends coefficient indexing to every nonnegative integer and uses the fixed $n+1$ variable convention. Polynomial existence is an earlier prerequisite, so there is no circular well-definedness reference to the later bridge theorem.