Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$

Question

Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms $$ A \to A/pA \to B. $$ Assume that we want to show that $B$ has flat dimension $\leq n$ (i.e., the $A$-module $B$ has Tor amplitude in $[-n,0]$). Then, is it enough to check the following condition (＊)?

(＊) $B$ has $p$-complete Tor amplitude in $[-n,0]$

Here (＊) means, by definition (ref. Bhatt-Morrow-Scholze "Topological Hochschild homology and integral $p$-adic Hodge theory"), that the derived tensor product $B\otimes_A^L A/pA \in D(A/pA)$ has Tor amplitude in $[-n,0]$. That is, $B\otimes_A N \in D^{[-n,0]}(A/pA)$ for any $A/pA$-module $N$. On the other hand, our aim is to show that $B\otimes_A M \in D^{[-n,0]}(A)$ for any $A$-module $M$. Is this true?

Answer 1

(You have to replace every usual tensor product $\otimes_A$ by derived tensor product $\otimes_A^L$ in the question.)

Yes. The key point is the $A/p$-module structure on $B$, which induces an $A/p$-structure on $B\otimes_A^LM\in D(A)$. In particular, the multiplication map by $p$ is zero on $B\otimes_A^L M$ in $D(A)$, and thus

$$B\otimes_A^L(M/^Lp)\simeq(B\otimes_A^LM)/^Lp\simeq B\otimes_A^LM\oplus B\otimes_A^LM[1]$$

in $D(A)$ (not in $D(A/p)$), where $N/^Lp:=\operatorname{cofib}(N\xrightarrow pN)$ for every $N\in D(A)$. By assumption, the left hand side is concentrated in (cohomological) degrees $[-(n+1),0]$ (since $M/^Lp\in D(A/p)$ is concentrated in degrees $[-1,0]$), and thus $B\otimes_A^LM$ is concentrated in degrees $[-(n+1),0]\cap[-n,1]=[-n,0]$.

The ring structure on $A$ is not used, and thus it can be extended to any $A/p$-module $B$, or even more generally, any $p$-torsion $A$-module of bounded $p^\infty$-torsion.