In mathematics, Weibel's conjecture gives a criterion for vanishing of negative
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
groups. The conjecture was proposed by and proven in full generality by using methods from
derived algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative ...
. Previously partial cases had been proven by
,
,
,
,
, and
.
Statement of the conjecture
Weibel's conjecture asserts that for a Noetherian
scheme ''X'' of finite
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
''d'', the ''K''-groups vanish in degrees < −''d'':
:
and asserts moreover a
homotopy invariance property for negative ''K''-groups
:
Generalization
Recently, have generalized Weibel's conjecture to arbitrary quasi-compact quasi-separated derived schemes. In this formulation the Krull dimension is replaced by the valuative dimension (that is, maximum of the Krull dimension of all blow-ups). In the case of Noetherian schemes, the Krull dimension is equal to the valuative dimension.
References
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Algebraic geometry
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K-theory
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