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 sense ...

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 commutat ...

. Previously partial cases had been proven by , , , , and .

Statement of the conjecture

Weibel's conjecture asserts that for a Noetherian

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

''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 generall ...

''d'', the ''K''-groups vanish in degrees < −''d'': :

K_i(X) = 0 \text i<-d

and asserts moreover a homotopy invariance property for negative ''K''-groups :

K_i(X) = K_i(X\times \mathbb A^r) \text i\le -d \text r.

References

* * Algebraic geometry K-theory {{Algebraic-geometry-stub