Abstract
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
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Acknowledgements
The author thanks Jörg Schürmann for interesting discussions and Sönke Rollenske for comments on the text. He also thanks the referee for helpful comments.
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Communicated by Vasudevan Srinivas.
Appendix A. Computations with power series
Appendix A. Computations with power series
Given a power series F(Q), we denote by \(F(Q)_{\mid Q^n}\) the coefficient of \(Q^n\). The verification that the two formulae of Corollary 4.6 are equivalent comes down to the following
Proposition A.1
For the proof, we use two simple auxiliary lemmas.
Lemma A.2
Proof
One way to see this is to apply the logarithm to both sides. \(\square \)
Lemma A.3
For \(k\in \mathbb {N}\) and \(\chi \in \mathbb {C}\), we have
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(i)
\(s^k\chi =(-1)^k \lambda ^k(-\chi )\) ,
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(ii)
\((1+Q)^\chi _{\mid Q^k}=\lambda ^k\chi \) ,
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(iii)
\((1-Q)^{-\chi }_{\mid Q^k}=s^k\chi \) .
Proof
The verification of (i) is a direct computation using Definition 4.5 of the numbers \(s^k\chi \) and \(\lambda ^k\chi \). Part (ii) is the binomial coefficient theorem. Part (iii) follows from (i) and (ii). \(\square \)
Proof of Proposition A.1
We have
Hence, by Lemma A.2, we get
Now, the assertion follows using Lemma A.3. \(\square \)
The verification of Remark 5.6 is very similar.
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Krug, A. Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles. Math. Ann. 371, 461–486 (2018). https://doi.org/10.1007/s00208-018-1660-5
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DOI: https://doi.org/10.1007/s00208-018-1660-5