A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology

In this paper, we introduce a notion of derived involutive algebras in $C_2$-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt–Mathew and Raksit. We show that the $\mathrm{\infty }$-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using $C_2$-$\mathrm{\infty }$-categories of Barwick–Dotto–Glasman–Nardin–Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.