[Paper Review] Derived Hochschild functors over commutative adic algebras

Liran Shaul|arXiv (Cornell University)|Jul 22, 2013

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper develops complete and torsion versions of derived Hochschild homology and cohomology for adic algebras over a noetherian base ring, using weak proregularity and DG-affine formal schemes. The key result shows that when the base ring is noetherian and the algebra is essentially of finite type over it, these derived functors admit explicit formulas within the noetherian category, recovering classical isomorphisms in the case of a field base.

ABSTRACT

Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian and $(A,\mfrak{a})$ is essentially of finite type (in the adic sense) over $\k$, then there are formulas to compute them that stay inside the noetherian category. In the classical case, where $\k$ is a field, we deduce that topological Hochschild cohomology and discrete Hochschild cohomology are isomorphic.

Motivation & Objective

• To extend derived Hochschild homology and cohomology to the setting of adic algebras over a commutative ring.
• To address the challenge that $A \otimes_{\Bbbk} A$ is typically non-noetherian when $\Bbbk \to A$ is not flat, especially in adic settings.
• To define complete and torsion variants of derived Hochschild functors that remain within the noetherian category under suitable conditions.
• To establish a duality and MGM equivalence in the context of DG-affine formal schemes.
• To show that for a field base, topological and discrete Hochschild cohomology are isomorphic.

Proposed method

• Prove that the ideal $\mathfrak{a}^e = \mathfrak{a} \otimes_{\Bbbk} A + A \otimes_{\Bbbk} \mathfrak{a}$ in $A \otimes_{\Bbbk} A$ is weakly proregular when $\Bbbk$ is noetherian and $(A,\mathfrak{a})$ is essentially of finite type over $\Bbbk$.
• Develop a theory of DG-affine formal schemes to extend Greenlees-May duality and the MGM equivalence to this setting.
• Define the complete derived Hochschild homology as $\mathrm{L}\Lambda_{\mathfrak{a}}(A \otimes^{\mathrm{L}}_{A \otimes^{\mathrm{L}}_{\Bbbk} A} \mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-))$ with values in the $\mathfrak{a}$-complete derived category.
• Define the torsion derived Hochschild homology as $\mathrm{R}\Gamma_{\mathfrak{a}}(A \otimes^{\mathrm{L}}_{A \otimes^{\mathrm{L}}_{\Bbbk} A} \mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-))$ with values in the $\mathfrak{a}$-torsion derived category.
• Use K-injective resolutions and derived completion functors to relate the torsion functor to the derived torsion over the completed ring $\widehat{A \otimes_{\Bbbk} A}$.
• Establish an isomorphism $\mathrm{R}\Gamma_{\mathfrak{a}}(A \otimes^{\mathrm{L}}_{A \otimes^{\mathrm{L}}_{\Bbbk} A} \mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-)) \cong A \otimes^{\mathrm{L}}_{\Lambda_I(\widetilde{A} \otimes_{\Bbbk} \widetilde{A})} \mathrm{R}\Gamma_I(\mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-))$ under weak proregularity assumptions.

Experimental results

Research questions

• RQ1 Can derived Hochschild functors be meaningfully extended to adic algebras over a noetherian base ring?
• RQ2 Under what conditions does the derived torsion or completion of $A \otimes_{\Bbbk} A$ preserve finiteness and allow explicit computation?
• RQ3 Is there a duality or equivalence analogous to Greenlees-May duality in the setting of DG-affine formal schemes?
• RQ4 Does the derived Hochschild cohomology over a formal power series ring over a field coincide with the topological version?
• RQ5 Can the torsion and complete derived Hochschild functors be computed using formulas within the noetherian category?

Key findings

• The ideal $\mathfrak{a}^e \subseteq A \otimes_{\Bbbk} A$ is weakly proregular when $\Bbbk$ is noetherian and $(A,\mathfrak{a})$ is essentially of finite type over $\Bbbk$, enabling the use of Greenlees-May theory.
• The torsion derived Hochschild homology functor admits an explicit formula: $\mathrm{R}\Gamma_{\mathfrak{a}}(A \otimes^{\mathrm{L}}_{A \otimes^{\mathrm{L}}_{\Bbbk} A} \mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-)) \cong A \otimes^{\mathrm{L}}_{\Lambda_I(\widetilde{A} \otimes_{\Bbbk} \widetilde{A})} \mathrm{R}\Gamma_I(\mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-))$.
• When $\Bbbk$ is a field, the derived Hochschild cohomology and topological Hochschild cohomology are isomorphic, recovering a classical result in the topological setting.
• The complete derived Hochschild homology functor is defined as $\mathrm{L}\Lambda_{\mathfrak{a}}(A \otimes^{\mathrm{L}}_{A \otimes^{\mathrm{L}}_{\Bbbk} A} \mathrm{R}\operatorname{Hom}_{\Bbbk}(-,-))$, with values in the $\mathfrak{a}$-complete derived category.
• The derived torsion and completion functors behave well over $A \otimes_{\Bbbk} A$ due to weak proregularity, allowing explicit computation via infinite dual Koszul complexes.
• The theory of DG-affine formal schemes is developed to extend the MGM equivalence and Greenlees-May duality to the adic setting.