Graded modules with degree-zero maps form an abelian category

Pointwise addition makes ${\mathrm{Hom}}_{{\mathrm{GrMod}}_0(A)}(M,N)$ an abelian group: a sum of degree-zero $A$-linear maps is degree-zero and $A$-linear, and composition is additive in each variable, so ${\mathrm{GrMod}}_0(A)$ is preadditive.

The zero module $0$, with all homogeneous pieces zero, is a zero object of ${\mathrm{GrMod}}_0(A)$: for every graded $M$ the unique maps $0\to M$ and $M\to 0$ are $A$-linear and degree-zero, since the only element of $0$ lies in the zero piece $0_d$ for every $d$.

For graded modules $M,N$ put $(M\oplus N{)}_d:=M_d\oplus N_d$; then $M\oplus N={⨁}_d(M\oplus N{)}_d$ and $A_i(M_d\oplus N_d)\subseteq M_{i+d}\oplus N_{i+d}$, so $M\oplus N$ is a graded $A$-module, and the coordinate inclusions ${\mathrm{ȷ}}_M,{\mathrm{ȷ}}_N$ and projections ${\pi }_M,{\pi }_N$ are degree-zero $A$-linear and satisfy ${\pi }_M{\mathrm{ȷ}}_M=1_M$, ${\pi }_N{\mathrm{ȷ}}_N=1_N$, ${\pi }_M{\mathrm{ȷ}}_N=0$, ${\pi }_N{\mathrm{ȷ}}_M=0$ and ${\mathrm{ȷ}}_M{\pi }_M+{\mathrm{ȷ}}_N{\pi }_N=1_{M\oplus N}$, because the sum of a summand in $M_d$ and one in $N_d$ is the unique decomposition of its sum in $(M\oplus N{)}_d$.

Let $f:M\to N$ be degree-zero and let $f_d:M_d\to N_d$ be its restriction. Then $\ker f={⨁}_d\ker f_d$: if $f(m)=0$ and $m={\sum }_d{m}_d$ is the finite decomposition of $m$ into homogeneous components, then $0=f(m)={\sum }_{d}f(m_d)$ with $f(m_d)\in N_d$, so every $f(m_d)=0$ by uniqueness of homogeneous decomposition, and conversely each $m_d\in \ker f_d$ lies in $\ker f$. Hence $\ker f$ is a graded submodule, its inclusion into $M$ is degree-zero, and any degree-zero $h:L\to M$ with $fh=0$ takes values in $\ker f$, so the inclusion is a kernel in ${\mathrm{GrMod}}_0(A)$.

The module $M\oplus N$ is a coproduct and a product in ${\mathrm{GrMod}}_0(A)$. Given degree-zero maps $h:M\to L$ and $k:N\to L$, [L6] produces the unique additive map $(m,n)\mapsto h(m)+k(n)$, which satisfies both composite identities, is $A$-linear and sends $(M\oplus N{)}_d$ into $L_d$, hence is degree-zero; given degree-zero maps $h^{\mathrm{\prime }}:L\to M$ and $k^{\mathrm{\prime }}:L\to N$, the map $l\mapsto (h^{\mathrm{\prime }}(l),k^{\mathrm{\prime }}(l))$ is degree-zero $A$-linear and is the unique map with the two required composites. Thus $M\oplus N$ is a binary biproduct.

Similarly $\mathrm{im}f={⨁}_d\mathrm{im}{f}_d$: the image of $f$ is the sum of the images of the restrictions, and every $f(m_d)$ lies in $N_d$, so these pieces are the homogeneous pieces of a graded submodule of $N$.

Steps 1.1, 1.2 and 2.1 give an abelian-group enrichment with bilinear composition, a zero object and binary biproducts, so ${\mathrm{GrMod}}_0(A)$ is an additive category.

Put $\mathrm{coker}f:=N\mathrm{/}\mathrm{im}f$ with pieces $(N\mathrm{/}\mathrm{im}f{)}_d:=N_d\mathrm{/}\mathrm{im}{f}_d$. This is a graded $A$-module, the quotient map $q:N\to \mathrm{coker}f$ is degree-zero $A$-linear, and $\ker q=\mathrm{im}f$. Any degree-zero $g:N\to P$ with $gf=0$ kills $\mathrm{im}f$ and so factors uniquely through $q$ by [L4]; the resulting map is degree-zero because $q$ is surjective in each degree. Hence $q$ is a cokernel and $\mathrm{im}f$, being $\ker q$, is the categorical image of $f$.

For degree-zero maps $M^{\mathrm{\prime }}\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{\mathrm{\prime }\mathrm{\prime }}$ with $gf=0$ one has $\mathrm{im}f={⨁}_d\mathrm{im}{f}_d$ and $\ker g={⨁}_d\ker g_d$ by steps 1.4 and 2.2. Since a graded submodule is determined by its homogeneous pieces, $\mathrm{im}f=\ker g$ holds if and only if $\mathrm{im}(f_d)=\ker (g_d)$ for every $d$; applied at each position of a sequence, exactness is equivalent to exactness degreewise.

The coimage is $\mathrm{coim}f=M\mathrm{/}\ker f$ with pieces $M_d\mathrm{/}\ker f_d$, a graded module by the same argument as step 3.2, and the canonical comparison $\mathrm{coim}f\to \mathrm{im}f$ sends $m+\ker f$ to $f(m)$. On degree $d$ it is the map $M_d\mathrm{/}\ker f_d\to \mathrm{im}{f}_d$ of [L5], an isomorphism of $k$-modules; it is $A$-linear and degree-zero, and a degree-zero bijection of graded modules has degree-zero inverse, so the comparison is an isomorphism in ${\mathrm{GrMod}}_0(A)$.

Steps 3.1, 1.4, 3.2 and 4.1 exhibit an additive category in which every morphism has a kernel and a cokernel and the canonical coimage-to-image comparison is an isomorphism; by [L2] the category ${\mathrm{GrMod}}_0(A)$ is abelian.

Steps 5.1 and 3.3 give both assertions: ${\mathrm{GrMod}}_0(A)$ is abelian, and its kernels, images, cokernels, finite biproducts and exactness are computed degreewise.