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It concerns Hochschild *homology*, not cohomology; fixed typos
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edit approved Jul 23, 2020 at 8:02
pbelmans
edit approved Jul 23, 2020 at 8:02
pbelmans
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A conceptual answer to "exactly how and where Todd classes will appear:" You don't have to use Todd classes if you stick to Hochschild Cohomologyhomology. This might not be the best reference, but I learned this from 1.2.1 in https://arxiv.org/pdf/1804.00879.pdf You can stick to just the chernChern character and factor through HH(Perf(X)); the toddTodd class corrects the fact that the HKR isomorphisms $HH(Perf(X)) \simeq H^*(X, \bigoplus \Omega_X^p)$ are NOT natural in $X$ until correction by Todd.

My apologies -- I'm sure there's a more down-to-earth reference of which I'm not aware.

A conceptual answer to "exactly how and where Todd classes will appear:" You don't have to use Todd classes if you stick to Hochschild Cohomology. This might not be the best reference, but I learned this from 1.2.1 in https://arxiv.org/pdf/1804.00879.pdf You can stick to just the chern character and factor through HH(Perf(X)); the todd class corrects the fact that the HKR isomorphisms $HH(Perf(X)) \simeq H^*(X, \bigoplus \Omega_X^p)$ are NOT natural in $X$ until correction by Todd.

My apologies -- I'm sure there's a more down-to-earth reference of which I'm not aware.

A conceptual answer to "exactly how and where Todd classes will appear:" You don't have to use Todd classes if you stick to Hochschild homology. This might not be the best reference, but I learned this from 1.2.1 in https://arxiv.org/pdf/1804.00879.pdf You can stick to just the Chern character and factor through HH(Perf(X)); the Todd class corrects the fact that the HKR isomorphisms $HH(Perf(X)) \simeq H^*(X, \bigoplus \Omega_X^p)$ are NOT natural in $X$ until correction by Todd.

My apologies -- I'm sure there's a more down-to-earth reference of which I'm not aware.

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Leo Herr
Leo Herr
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A conceptual answer to "exactly how and where Todd classes will appear:" You don't have to use Todd classes if you stick to Hochschild Cohomology. This might not be the best reference, but I learned this from 1.2.1 in https://arxiv.org/pdf/1804.00879.pdf You can stick to just the chern character and factor through HH(Perf(X)); the todd class corrects the fact that the HKR isomorphisms $HH(Perf(X)) \simeq H^*(X, \bigoplus \Omega_X^p)$ are NOT natural in $X$ until correction by Todd.

My apologies -- I'm sure there's a more down-to-earth reference of which I'm not aware.