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Local Chern characters and intersection multiplicities. (English) Zbl 0651.13011

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 389-400 (1987).
[For the entire collection see Zbl 0626.00011.]
If M and N are finitely generated modules over a regular local ring A, and if the supports of M and N meet only at the maximal ideal of A, Serre defined the intersection multiplicity \(\chi\) (M,N) and proved that: \((1)\quad \dim (M)+\dim (N)\leq \dim (A).\) In addition, if A contains a field and is unramified he proved that: \((2)\quad If\) \(\dim (M)+\dim (N)<\dim (A)\), then \(\chi (M,N)=0\); and \((3)\quad If\) \(\dim (M)+\dim (N)=\dim (A)\), then \(\chi (M,N)>0\); and he conjectured that these statements hold over arbitrary regular local rings.
The author shows in this paper that the vanishing conjecture holds not only for regular local rings, but also when M and N are modules of finite projective dimension over a complete intersection or a local ring whose singular locus has dimension at most one. The main tool in the proof is the theory of local Chern characters. Connections with related results are presented.

MSC:

13H15 Multiplicity theory and related topics
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13D05 Homological dimension and commutative rings
13D15 Grothendieck groups, \(K\)-theory and commutative rings

Citations:

Zbl 0626.00011