Hochschild cohomology was introduced in a 1945 paper by Hochschild, and Grothendieck

duality dates back to the early 1960s. The fact that the two have some relation with each

other is very new - it came up in papers by Avramov and Iyengar [2008], Avramov, Iyengar,

and Lipman [2010] and Avramov, Iyengar, Lipman and Nayak [2011]. We will review this

history, and the surprising formulas that come out.

We will then discuss more recent progress. The remarkable feature of all this is the role

played by Hochschild homology. One example, which we will discuss in some detail,

comes about as follows. The new techniques permit us to write formulas giving trace

and residue maps in Grothendieck duality in terms of expressions that are very

'Hochschild-homological - Alonso, Jeremias and Lipman gave such a formula, but

couldn't prove that it agrees with the usual formula dating back to Verdier in the

1960s. The proof that these two agree, due to Lipman and the speaker, turns out to

hinge on considering the action of ordinary Hochschild homology on the various

objects in the formula.

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