4,432 research outputs found
Witt's theorem for noncommutative conics
Let k be a field. We show that all homogeneous noncommutative curves of genus
zero over k are noncommutative P^1-bundles over a (possibly) noncommutative
base. Using this result, we compute complete isomorphism invariants of
homogeneous noncommutative curves of genus zero, allowing us to generalize a
theorem of Witt.Comment: Section two generalize
Noncommutative Tsen's theorem in dimension one
Let k be a field. In this paper, we find necessary and sufficient conditions
for a noncommutative curve of genus zero over k to be a noncommutative
P^1-bundle. This result can be considered a noncommutative, one-dimensional
version of Tsen's theorem. By specializing this theorem, we show that every
arithmetic noncommutative projective line is a noncommutative curve, and
conversely we characterize exactly those noncommutative curves of genus zero
which are arithmetic. We then use this characterization, together with results
regarding arithmetic noncommutative projective lines, to address some problems
posed by D. Kussin.Comment: Error in proof of Lemma 3.8 correcte
The Grothendieck Group of a Quantum Projective Space Bundle
We compute the Grothendieck group K_0 of non-commutative analogues of quantum
projective space bundles. Our results specialize to give the Grothendieck
groups of non-commutative analogues of projective spaces, and specialize to
recover the Grothendieck group of a usual projective space bundle over a
regular noetherian separated scheme. As an application we develop an
intersection theory for the quantum ruled surfaces defined by Van den Bergh.Comment: This paper is being replaced so I can correct the metadata, the
title! I (Paul) spelled Grothendieck's name incorrectly. The paper is being
reposted with the journal reference and doi added to the metadat
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