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Sur la thĂ©orie des complexes parfaits. (French) Zbl 0554.13005
Commutative algebra, Symp. Durham 1981, Lond. Math. Soc. Lect. Note Ser. 72, 83-90 (1982).
[For the entire collection see Zbl 0489.00008.]
With a perfect complex over a commutative ring A, is meant a finite complex of free A-modules of finite rank. - Let $$A=\oplus^{\infty}_{n=0}A_ n$$ be a graded ring where $$A_ 0$$ is Artinian and A is finitely generated over $$A_ 0$$. One denotes by A(n) the graded A-module with $$A(n)_ k=A_{n+k}$$. A homogeneous perfect complex over A has the form: $\begin{split} L_ 0:0 \to \oplus^{r_ s}_{j=1}A(-n_{sj})\to...\\ \to \to \oplus^{r_ i}_{j=1} A(- n_{ij})\to^{\phi_ i}\oplus^{r_{i-1}}_{j=1}A(-n_{i- 1,j})\to...\to \oplus^{r_ 0}_{j=1}A(-n_{0j}) \to 0 \end{split}$ where $$\phi_ i$$ are homogeneous homomorphisms of degree 0. Define $$\rho_ k=(1/k!)\sum^{s}_{i=0}(-1)^ i\sum^{r_ i}_{j=!}n^ k_{ij}.$$ The following ”theorem on the degree” is proved for the graded case. Let M be a finitely generated graded A-module of finite projective dimension, L. its projective resolution and $$g=\inf \{k:\rho_ k\neq 0\}$$ then: (1) if N is a finitely generated graded A-module for which $$\dim (M\otimes N)=0,$$ then $$\dim N\leq g;$$ $$(2)\quad g=\text{grad}e M=\dim A- \dim M.$$
Let A be a Noetherian local ring, L. be a perfect complex over A, $$X=Spec A$$ and $$Y=Supp(H.(L.))$$. A ”Riemann-Roch hypothesis” is formulated which asserts the existence of a functorial commutative diagram: $$K.(X)\to CH.(X)\otimes {\mathbb{Q}}\to^{\kappa}K.(Y); CH.(X)\otimes {\mathbb{Q}}\to K.(Y)\to CH.(Y)\otimes {\mathbb{Q}},$$ where K.( ) denotes the Grothendieck group of finitely generated modules, CH.( ) denotes the Chow group; and $$\kappa (N)=\sum (-1)^ i[H_ i(L.\otimes N)].$$ In the case where A is a localization of a finitely generated algebra over a field, this hypothesis has been proved by Fulton (not published).

MSC:
 13D25 Complexes (MSC2000) 14C40 Riemann-Roch theorems 16W50 Graded rings and modules (associative rings and algebras)