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 o ...

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 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 generally t ...

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

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

and asserts moreover a

homotopy invariance In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

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