Right hyperderived functor of a complex

Let $F:\mathcal{A}\to \mathcal{B}$ be additive and left exact between abelian categories. For a bounded-below complex $K$ with supplied Cartan–Eilenberg resolution $I$, define the right hyperderived object relative to $I$ by ${\mathbb{R}}_I^{n}F(K)=H^n(\mathrm{Tot}(FI)),(FI{)}^{p,q}=F(I^{p,q}),D=F(h)+(-1{)}^{p}F(v)\mathrm{.}$ Additivity preserves the two square-zero equations and the commuting square, so the total differential squares to zero. Each diagonal is finite, and the canonical finite-biproduct comparison identifies $\mathrm{Tot}(FI)$ with $F(\mathrm{Tot}I)$.

The totalization lemma gives a bounded-below termwise injective replacement $K\to \mathrm{Tot}I$. With its DC or supplied homotopy-extension qualification, this is a K-injective model, so the same formula is $H^n(RF(K))$. Without comparison data the subscript $I$ is retained: the definition alone asserts neither independence nor a choice of a resolution for every complex. This extends Right derived objects relative to supplied injective resolution data: an object in degree zero resolved in one column gives precisely $R_I^{n}F$ of that object.

If $K$ and the supplied resolution are zero, every value is zero. For lower bound $b$, the values vanish for $n<b$, and the total diagonal at $n=b$ has one term. Translating the first index to $p-b$ changes the total degree to $n-b$, not the original degree $n$ in the formula.