A version of Cartan’s Theorem A for coherent sheaves on real affine varieties

Abstract

We give a version of Cartan’s Theorem [Formula: see text] for nonsingular real affine varieties after blowing up.

Blown-up Čech cohomology and Cartan’s Theorem B on real algebraic varieties

“…In this section we provide an example (based on [3, Example 12.1.5]) of a quasi-coherent locally free sheaf F on R 2 of infinite rank such that for any multi-blowup α : X α → R 2 the pull-back sheaf α * F is not generated by global sections. This shows that the version of Cartan's Theorem A from our paper [14] cannot be generalized to quasi-coherent sheaves.…”

“…For any Zariski open set U ⊂ X we have the canonical and functorial homomorphism A crucial role in the proof of Theorem 3.1 is played by Lemmas 3.4 and 3.5. Before proving them, we recall three Corollaries 2.1, 2.2, 2.3 from our paper [14]. The first two will be used in the proofs of Lemmas 3.4 and 3.5, and the last (being a corollary to Cartan's Theorem A) in Section 5.…”

“…The case (k, l) = (1, 1) is treated in [14]. It is showed there that if σ : X σ → R 2 is a blowup of the plane at the origin, then the global section of σ * F 1,1 given by the pair s = ( s 1 , s 2 ) with s 1 = 1 (x 2 + y 2 ) σ and s 2 = P σ 1,1 (x, y) (x 2 + y 2 ) σ does not vanish on X σ .…”

“…It is known that Cartan's Theorems A and B do not hold in the real algebraic case. Nevertheless, in [14] we established a real version of Theorem A to the effect that for any coherent sheaf F on a nonsingular real affine variety X there exists a multi-blowup σ : X σ → X such that the pullback sheaf σ * F is generated by global sections on X σ .…”