**Proof.**
It is clear that (2) implies (1) and that (1) is equivalent to (6). Assume (1). Let $I^\bullet $ be a K-injective complex of $A$-modules representing $K$. Condition (1) signifies that $\mathop{\mathrm{Hom}}\nolimits _ A(A_ f, I^\bullet )$ is acyclic. Let $M^\bullet $ be a complex of $A_ f$-modules representing $E$. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(A)}(E, K) = \mathop{\mathrm{Hom}}\nolimits _{K(A)}(M^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(A_ f)}(M^\bullet , \mathop{\mathrm{Hom}}\nolimits _ A(A_ f, I^\bullet )) \]

by Algebra, Lemma 10.14.4. As $\mathop{\mathrm{Hom}}\nolimits _ A(A_ f, I^\bullet )$ is a K-injective complex of $A_ f$-modules by Lemma 15.56.3 the fact that it is acyclic implies that it is homotopy equivalent to zero (Derived Categories, Lemma 13.31.2). Thus we get (2).

A free resolution of the $A$-module $A_ f$ is given by

\[ 0 \to \bigoplus \nolimits _{n \in \mathbf{N}} A \to \bigoplus \nolimits _{n \in \mathbf{N}} A \to A_ f \to 0 \]

where the first map sends the $(a_0, a_1, a_2, \ldots )$ to $(a_0, a_1 - fa_0, a_2 - fa_1, \ldots )$ and the second map sends $(a_0, a_1, a_2, \ldots )$ to $a_0 + a_1/f + a_2/f^2 + \ldots $. Applying $\mathop{\mathrm{Hom}}\nolimits _ A(-, I^\bullet )$ we get

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ A(A_ f, I^\bullet ) \to \prod I^\bullet \to \prod I^\bullet \to 0 \]

Since $\prod I^\bullet $ represents $\prod _{n \geq 0} K$ this proves the equivalence of (1) and (7). On the other hand, by construction of derived limits in Derived Categories, Section 13.34 the displayed exact sequence shows the object $T(K, f)$ is a representative of $R\mathop{\mathrm{Hom}}\nolimits _ A(A_ f, K)$ in $D(A)$. Thus the equivalence of (1) and (3).

There is a spectral sequence

\[ E_2^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^ q_ A(A_ f, H^ p(K)) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{p + q}_ A(A_ f, K) \]

(details omitted). This spectral sequence degenerates at $E_2$ because $A_ f$ has a length $1$ resolution by projective $A$-modules (see above) hence the $E_2$-page has only 2 nonzero rows. Thus we obtain short exact sequences

\[ 0 \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(A_ f, H^{p - 1}(K)) \to \mathop{\mathrm{Ext}}\nolimits ^ p_ A(A_ f, K) \to \mathop{\mathrm{Hom}}\nolimits _ A(A_ f, H^ p(K)) \to 0 \]

This proves (4) and (5) are equivalent to (1).
$\square$

## Comments (2)

Comment #3060 by Noah Olander on

Comment #3164 by Johan on

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