Author: Eiko

Tags: Homological Algebra, Resolution, Derived Functor, Derived Category

Acyclic Resolution

Acyclic Object

We say an object $X\in \mathcal{C}$ is acyclic (for $F$) if the higher derived functors $R^{i}F(X)=H^i(RF(X))=0$ for all $i>0$.

Acyclic Resolution

An acyclic resolution of an object $M\in \mathcal{C}$ is an exact sequence

$$0\to M\to X^0\to X^1\to \cdots$$

or equivalently, a quasi-isomorphism $M\to X^{\bullet }$, where all $X^i$ are acyclic.

Warning: confusing terminology

• Acyclic object means $R^{i}F(X)=0$ for all $i>0$.

• A complex is acyclic if it is exact, i.e. $H^i(X^{\bullet })=0$ for all $i>0$.

• An acyclic resolution is a complex of acyclic objects, it is not necessarily an acyclic complex.

• A cyclic complex is a complex with differentials all being zero.

The use of acyclicity

Consider first the simplified principle where an acyclic object sits in the middle of a short exact sequence,

$$0\to A\stackrel{d^{-1}}{\to }{X}^0\to X^0/A\to 0.$$

or written in fancy way

$$0\to \mathrm{Coim}(d^{-1})\to X^0\to \mathrm{Coker}(d^{-1})\to 0.$$

Then the long exact sequence of cohomology tells us

$$\begin{array}{rl}0& \to F(A)\to F(X^0)\to F(X^0/A)\\ & \to R^{1}F(A)\to 0\to R^{1}F(X^0/A)\\ & \to R^{2}F(A)\to 0\to R^{2}F(X^0/A)\\ & \to R^{3}F(A)\to 0\to \cdots \end{array}$$

which gives

$$F(A)=\ker (F(X^0)\to F(X^0/A))$$ $$R^{1}F(A)=\mathrm{Coker}(F(X^0)\to F(X^0/A))=\mathrm{Coker}(F(X^0)\to F(\mathrm{Coker}(d^{-1})))$$ $$R^{i+1}F\mathrm{Coim}(d^{-1})=R^{i}F(X^0/A)=R^i(F\mathrm{Coim}(d^0)),i\ge 1$$

Principle of Acyclic Resolution

The fundamental principle of acyclic resolution is, it can be used to compute cohomology or derived functors.

$$R^{i}F(M)=H^i(F(X^{\bullet })).$$

Proof

To see why this is the case, let us assume that

$$0\to M\stackrel{d^{-1}}{\to }{X}^0\stackrel{d^0}{\to }{X}^1\stackrel{d^1}{\to }\cdots$$

is an acyclic resolution, then we can break it into short exact sequences

$$0\to \mathrm{Coim}(d^{i-1})\stackrel{d^{i-1}}{\to }{X}^i\stackrel{d^i}{\to }\mathrm{Im}(d^i)\to 0$$

i.e.

$$\mathrm{Coim}(d^{i-1})\cong \ker (d^i)=\mathrm{Im}(d^{i-1}),\mathrm{Coker}(d^{i-1})\cong \mathrm{Im}(d^i)=\ker (d^{i+1})$$

$$F(\mathrm{Coim}(d^{i-1}))\cong F(\ker (d^i))=\ker (F(d^i))$$

as a special case $i=0$ it gives $F(M)\cong \ker (F(d^0))=H^0(F(X^{\bullet }))$. For the first cohomology, we have by the use of acyclicity,

$$\begin{array}{rl}{R}^{1}F(\mathrm{Coim}(d^{i-1}))& \cong \mathrm{Coker}(F(X^i)\to F(\mathrm{Coker}(d^{i-1})))\\ & =\mathrm{Coker}(F(X^i)\to F(\mathrm{Im}(d^i)))\\ & =\mathrm{Coker}(F(X^i)\to F(\ker (d^{i+1})))\\ & =\mathrm{Coker}(F(X^i)\to \ker (F{d}^{i+1}))\\ & =\frac{\ker (F{d}^{i+1})}{\mathrm{Im}(F{d}^i)}\\ & =H^{i+1}(F(X^{\bullet })).\end{array}$$

The use of acyclicity also brings us an isomorphism to allow us to use induction

$$R^{q+1}F(\mathrm{Coim}(d^{i-1}))\cong R^q(F\mathrm{Coim}(d^i)),q\ge 1.$$

Thus for higher cohomology groups we also have

$$\begin{array}{rl}{R}^{q+1}F(\mathrm{Coim}(d^{i-1}))& \cong R^q(F\mathrm{Coim}(d^i))\\ & \cong R^{q-1}(F\mathrm{Coim}(d^{i+1}))\\ & \cong \cdots \\ & \cong R^1(F\mathrm{Coim}(d^{i+q-1}))\\ & \cong H^{i+q+1}(F(X^{\bullet })).\end{array}$$

Letting $i=0$ gives the principle

$$R^{q}F(M)=H^q(F(X^{\bullet })).$$

Remark

Although acyclic resolution can be used to compute derived functors, we still need injective resolutions (or projective resolutions for right derived functors) to define derived functors, since acyclic objects can not be defined without defining derived functors in the first place. Note that the concept of injectivity only depends on the category, while the concept of acyclicity depends on the base functor $F$.

Example

For affine scheme $X$, any quasi-coherent sheaf of ${\mathcal{O}}_X$-modules is acyclic for the global section functor ${\mathrm{\Gamma }}_X:{\mathcal{O}}_X\text{-mod}\to \mathbf{Ab}$.