Restriction and extension along a graded algebra map

The right action $b\cdot a=bf(a)$ makes $B$ a graded $(B,A)$-bimodule: it is additive in $b$ and in $a$, satisfies $(b{b}^{\mathrm{\prime }})\cdot a=b(b^{\mathrm{\prime }}\cdot a)$, $b\cdot (a{a}^{\mathrm{\prime }})=bf(a{a}^{\mathrm{\prime }})=(bf(a))f(a^{\mathrm{\prime }})=(b\cdot a)\cdot a^{\mathrm{\prime }}$ and $b\cdot 1_A=b$, and it is homogeneous because $B_i{B}_j\subseteq B_{i+j}$ and $f(A_j)\subseteq B_j$ give $B_i\cdot A_j\subseteq B_{i+j}$.

Evaluation $\mathrm{e}\mathrm{v}:{\mathrm{HOM}}_B(B,Y)\to Y{\mathrm{\mid }}_A$, $g\mapsto g(1_B)$, is a degree-zero isomorphism of graded $A$-modules, where ${\mathrm{HOM}}_B(B,Y)$ carries $(a\cdot g)(b)=g(ba)=g(bf(a))$. For $g\in {\mathrm{Hom}}_{B,j}(B,Y)$ one has $g(1_B)\in Y_j$, so $\mathrm{e}\mathrm{v}$ is degree-zero and $A$-linear, since $(a\cdot g)(1_B)=g(f(a))=f(a)g(1_B)=a\cdot g(1_B)$; it is injective because $g$ is $B$-linear and hence $g(b)=bg(1_B)$, and surjective because for $y\in Y_j$ the map $b\mapsto by$ is $B$-linear, homogeneous of degree $j$, and has value $y$ at $1_B$.

Restriction is a functor: for a graded left $B$-module $Y$ the formula $a\cdot y=f(a)y$ makes $Y$ a graded left $A$-module, since $A_i\cdot Y_j=f(A_i)Y_j\subseteq B_i{Y}_j\subseteq Y_{i+j}$; a degree-zero $B$-linear map $u:Y\to Z$ is degree-zero $A$-linear because $u(a\cdot y)=u(f(a)y)=f(a)u(y)=a\cdot u(y)$.

Extension is left adjoint to restriction: applying [L3] to the graded $(B,A)$-bimodule $B$ of step 1.1 gives a natural bijection ${\mathrm{Hom}}_{B,0}(B{\otimes }_{A}X,Y)\cong {\mathrm{Hom}}_{A,0}(X,{\mathrm{HOM}}_B(B,Y))$, and composing with the natural isomorphism of step 1.2 gives the displayed natural bijection ${\mathrm{Hom}}_{B,0}(B{\otimes }_{A}X,Y)\cong {\mathrm{Hom}}_{A,0}(X,Y{\mathrm{\mid }}_A)$.

If $B$ is flat as a right $A$-module, then $B{\otimes }_A-$ is exact by [L4] applied to the graded $(B,A)$-bimodule $B$.

$B{\otimes }_A-$ always preserves finite graded projectives: $B=B\{0\}$ is a finite direct sum of shifts of $B$, hence finite graded projective as a left $B$-module by [L5], so [L4] applied to $M=B$ gives the claim for every finite graded projective left $A$-module.

Restriction is exact. For a degree-zero $B$-linear $u:Y\to Z$, [L6] computes $\ker u$ and $\mathrm{coker}u$ degreewise on the underlying $k$-modules, and the underlying graded submodule $\ker u$ and quotient $Z\mathrm{/}u(Y)$ carry the $A$-action induced by $f$; with these actions they are the kernel and cokernel of $u$ in ${\mathrm{GrMod}}_0(A)$, because the universal properties of the kernel and quotient are those of the underlying modules. Hence restriction preserves kernels and cokernels, and a sequence is exact in ${\mathrm{GrMod}}_0(B)$ exactly when its restriction is exact in ${\mathrm{GrMod}}_0(A)$.

Assume $B$ is finite graded projective as a left $A$-module, and let $Y$ be a finite graded projective left $B$-module. By [L5] there are degree-zero maps $i:Y\to F$, $p:F\to Y$ with $pi=1_Y$, where $F=B\{s_1\}\oplus \cdots \oplus B\{s_n\}$; restricting the same underlying maps and the same shifts makes $Y{\mathrm{\mid }}_A$ a degree-zero direct summand of $F{\mathrm{\mid }}_A=B\{s_1\}{\mathrm{\mid }}_A\oplus \cdots \oplus B\{s_n\}{\mathrm{\mid }}_A$. Each $B\{s_j\}{\mathrm{\mid }}_A$ is finite graded projective over $A$, being a shift of the finite graded projective left $A$-module $B$ by hypothesis; by [L5] their finite direct sum is finite graded projective, and again by [L5] its degree-zero direct summand $Y{\mathrm{\mid }}_A$ is finite graded projective.

Steps 3.1, 2.2, 2.3, 2.4 and 3.2 give the four clauses: restriction is exact and right adjoint to extension, extension is exact when $B$ is right $A$-flat and always preserves finite graded projectives, and restriction preserves finite graded projectives when $B$ is finite graded projective over $A$. ∎