Holmstrom Shukla homology

Shukla homology

Mentioned in Weibel, p. 32. In certain situation, it gives Hochschild or MacLane homology.

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Shukla homology

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Shukla homology

NCG (Algebra and noncommutative geometry)?

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Shukla homology

Speaker: L. Avramov

Title: ‘’Derived Hochschild cohomology and Grothendieck duality’‘

ABSTRACT

We study the derived Hochschild cohomology, or Shukla cohomology, of a commutative algebra $S$ essentially of finite type and of finite flat dimension over a commutative noetherian ring $K$. We construct a complex of $S$-modules $D^{\sigma}$ and canonical isomorphisms \mathrm{SH}^{*}(S\var K;M\otimes_{K}^{\mathsf L}N)\simeq \mathrm{Ext}^{*}_{S}(\mathrm{\mathsf{R}Hom}_{S}(M,D^{\sigma}),N) for all complexes of $\mathrm{SH}^{*}(S\var K;M\otimes_{K}^{\mathsf L}N)\simeq \mathrm{Ext}^{*}_{S}(\mathrm{\S$-modules $M$ and $N$ with $\mathrm{H}(M)$ finitely generated and of finite flat dimension over $K$. Such a complex is unique up to isomorphism. It is used to establish basic results about relative dualizing complexes for the $K$-algebra $S$. Conversely, by using the machinery of Grothendieck duality theory we establish a version of the isomorphism above for noetherian schemes that are essentially of finite type and flat over $K$ . This is joint work with Srikanth Iyengar and Joseph Lipman.

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Shukla homology

Baues and Pirashvili: Comparison of Mac Lane, Shukla and Hochschild cohomologies. J. Reine Angew. Math. 598 (2006)

nLab page on Shukla homology