The isomorphism in Equation (2.3) of [1] is incorrect; take for instance X = A2, S = A1 and N the ideal sheaf of the point (0, 0, 0). Then the left-hand side of (2.3) is the stalk of N at (0, 0, 0), i.e. the maximal ideal of the local ring of A3 at (0, 0, 0), while the right-hand side is the ideal of the point (0, 0) of A2. As a result, the proofs of Lemma 2.1 and Proposition 2.3 are not valid. We provide here a different proof of Lemma 2.1 (with slightly strengthened hypotheses) and prove a modified Proposition 2.3, which is enough for our purposes. Lemma 2.1 will now imply a modified Lemma 2.2. Lemma 2.4 is correct, while Lemmas 2.5 and 2.6 do not hold (but are no longer needed). All results stated in Sections 3 to 6 - in particular, the Main Theorem 3.4 - hold true, and their proofs remain unchanged (with the exception of that of Proposition 6.4, which can be easily fixed as we show at the end of this Erratum). If not stated otherwise, the notation is the same as in [1]. In particular, T is a product T = X × S, where X is a smooth connected projective variety over C, and S a noetherian reduced scheme of finite type over C; we denote by ti, i =1, 2 the canonical projections onto the first and second factor, respectively. If is an OT-module, we denote by s its restriction to the fibre of T over s ∈ S.

Erratum: Monads for framed sheaves on Hirzebruch surfaces / Bartocci, Claudio; Bruzzo, Ugo; Rava, Claudio L. S.. - In: ADVANCES IN GEOMETRY. - ISSN 1615-715X. - 16:4(2016), pp. 531-534. [10.1515/advgeom-2016-0023]

Erratum: Monads for framed sheaves on Hirzebruch surfaces

Bruzzo, Ugo;
2016

Abstract

The isomorphism in Equation (2.3) of [1] is incorrect; take for instance X = A2, S = A1 and N the ideal sheaf of the point (0, 0, 0). Then the left-hand side of (2.3) is the stalk of N at (0, 0, 0), i.e. the maximal ideal of the local ring of A3 at (0, 0, 0), while the right-hand side is the ideal of the point (0, 0) of A2. As a result, the proofs of Lemma 2.1 and Proposition 2.3 are not valid. We provide here a different proof of Lemma 2.1 (with slightly strengthened hypotheses) and prove a modified Proposition 2.3, which is enough for our purposes. Lemma 2.1 will now imply a modified Lemma 2.2. Lemma 2.4 is correct, while Lemmas 2.5 and 2.6 do not hold (but are no longer needed). All results stated in Sections 3 to 6 - in particular, the Main Theorem 3.4 - hold true, and their proofs remain unchanged (with the exception of that of Proposition 6.4, which can be easily fixed as we show at the end of this Erratum). If not stated otherwise, the notation is the same as in [1]. In particular, T is a product T = X × S, where X is a smooth connected projective variety over C, and S a noetherian reduced scheme of finite type over C; we denote by ti, i =1, 2 the canonical projections onto the first and second factor, respectively. If is an OT-module, we denote by s its restriction to the fibre of T over s ∈ S.
16
4
531
534
Bartocci, Claudio; Bruzzo, Ugo; Rava, Claudio L. S.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/118146
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