Bimodule tensor exactness and preservation of finite projectives have separate hypotheses

Let $f:N\to N^{\mathrm{\prime }}$ be a degree-zero $A$-linear map. The pairing $(m,n)\mapsto m\otimes f(n)$ is balanced and additive in each variable, so [L6] gives a unique additive map $M{\otimes }_{A}f:M{\otimes }_{A}N\to M{\otimes }_A{N}^{\mathrm{\prime }}$ with $(M{\otimes }_{A}f)(m\otimes n)=m\otimes f(n)$. It is left $B$-linear, since $b(m\otimes f(n))=(bm)\otimes f(n)$ by the outer action of [L1], and degree-zero, since $f(n)$ has the degree of $n$ and the tensor grading is total degree. Identities and composites are inherited from those of $f$, so ${\mathrm{\Phi }}_M$ is a functor ${\mathrm{GrMod}}_0(A)\to {\mathrm{GrMod}}_0(B)$.

For every $r$ the map ${\theta }_r:M{\otimes }_{A}A\{r\}\to M\{r\}$, $m\otimes a\mapsto ma$, is a degree-zero isomorphism of graded left $B$-modules. It is well defined and additive by [L6], since the pairing is balanced; for homogeneous $m\in M_i$ and $a\in A\{r{\}}_j=A_{j-r}$ one has $ma\in M_{i+j-r}=(M\{r\}{)}_{i+j}$, so ${\theta }_r$ is degree-zero, and it is left $B$-linear because $(bm)a=b(ma)$. The inverse $m\mapsto m\otimes 1_A$ is the published unit isomorphism on $M{\otimes }_{A}A$, transported along the shift; hence ${\theta }_r$ is bijective.

Assume $M$ is flat as a right $A$-module and let $0\to N^{\mathrm{\prime }}\stackrel{i}{\to }N\stackrel{p}{\to }{N}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence in ${\mathrm{GrMod}}_0(A)$. By [L5] its underlying sequence of $A$-modules is exact, so flatness [L4] makes $0\to M{\otimes }_A{N}^{\mathrm{\prime }}\stackrel{1\otimes i}{\to }M{\otimes }_{A}N\stackrel{1\otimes p}{\to }M{\otimes }_A{N}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ exact as a sequence of abelian groups, with the degree-zero $B$-linear maps of step 1.1. The maps are degree-zero, so this ungraded exactness restricts to exactness of the degree-$d$ part at every $d$: a preimage can be replaced by its degree-$d$ component, and an element of degree $d$ killed by $1\otimes p$ is the image of an element of degree $d$ because $1\otimes i$ is injective on homogeneous components. By [L5] the graded sequence is exact in ${\mathrm{GrMod}}_0(B)$, so ${\mathrm{\Phi }}_M$ is exact.

For graded left $A$-modules $N_1,\dots ,N_n$, the coordinate inclusions induce a degree-zero isomorphism ${⨁}_j(M{\otimes }_A{N}_j)\cong M{\otimes }_A(N_1\oplus \cdots \oplus N_n)$ of graded left $B$-modules: the pairing $(m,(n_j))\mapsto {\sum }_{j}m\otimes n_j$ is balanced, its finite sum being a finite sum of elementary tensors, so [L6] gives a map $\psi$ out of the tensor product, while the maps $1\otimes {\mathrm{ȷ}}_j$ assemble by the biproduct property of [L5] into $\phi$; both composites fix elementary tensors and therefore are identities, and every map involved is degree-zero and $B$-linear.

If $M$ is finite graded projective as a left $B$-module, then so is each shift $M\{r\}$. By [L3] there is a degree-zero splitting of $M$ inside a finite direct sum $F=B\{s_1\}\oplus \cdots \oplus B\{s_n\}$; the same underlying maps, read with the gradings shifted by $r$, give a degree-zero splitting of $M\{r\}$ inside $F\{r\}=B\{s_1+r\}\oplus \cdots \oplus B\{s_n+r\}$, because shifting changes no underlying map and translates every degree by $r$. Hence $M\{r\}$ is a degree-zero direct summand of a finite direct sum of shifts, so finite graded projective by [L3].

Finite direct sums of finite graded projectives are finite graded projective, and degree-zero direct summands of finite graded projectives are finite graded projective. For the first claim, write each summand as a degree-zero direct summand of a finite direct sum of shifts using [L3] and take the direct sum of the splittings, the direct sum of finitely many finite shifted-free modules being finite shifted-free. For the second, compose the two splittings: a degree-zero direct summand of a degree-zero direct summand is a degree-zero direct summand. Both closures then follow from [L3].

Assume now that $M$ is finite graded projective as a left $B$-module and let $X$ be a finite graded projective left $A$-module. By [L3] there are degree-zero maps $i:X\to F$ and $p:F\to X$ with $pi=1_X$ for some finite direct sum $F=A\{r_1\}\oplus \cdots \oplus A\{r_n\}$. Applying the functor of step 1.1 gives $1\otimes i$ and $1\otimes p$ with $(1\otimes p)(1\otimes i)=1\otimes 1_X=1_{M{\otimes }_{A}X}$, so $M{\otimes }_{A}X$ is a degree-zero direct summand of $M{\otimes }_{A}F$. By steps 1.2 and 2.2, $M{\otimes }_{A}F\cong {⨁}_{j}M{\otimes }_{A}A\{r_j\}\cong {⨁}_{j}M\{r_j\}$, which is finite graded projective by steps 2.3 and 3.1; by step 3.1 again, its degree-zero direct summand $M{\otimes }_{A}X$ is finite graded projective as a left $B$-module.

Step 2.1 proves the exactness clause under right $A$-flatness and step 4.1 proves the preservation of finite graded projectives under finite graded projectivity of $M$ over $B$. The two hypotheses are used separately and neither is derived from the other. ∎