Graded Bimodules and Tensor Functors

Ground ring and graded algebras. Fix a commutative ring $k$. A graded $k$-algebra is a $k$-algebra $A$ in the sense of Algebras over a commutative ring, central structure maps, and algebra homomorphisms, together with a direct sum decomposition of $k$-modules

$A={⨁}_{i\in \mathbb{Z}}{A}_i$

such that $A_i{A}_j\subseteq A_{i+j}$ for all $i,j\in \mathbb{Z}$ and $1_A\in A_0$. It is unital and associative when its underlying ring is; every graded algebra on this page is unital and associative. Since ${\eta }_A(r)=r{1}_A$ and $A_0$ is a $k$-submodule containing $1_A$, the structure map has image in $A_0$, so each $A_i$ is a $k$-submodule and the scalars act centrally on $A$.

Graded modules and bimodules. A graded left $A$-module is a left $A$-module $M$ with a decomposition $M={⨁}_{d\in \mathbb{Z}}{M}_d$ of $k$-modules such that

$A_i{M}_d\subseteq M_{i+d}\text{for all }i,d\in \mathbb{Z}\mathrm{.}$

A graded right $A$-module is a right $A$-module with a decomposition $M={⨁}_d{M}_d$ such that $M_d{A}_i\subseteq M_{d+i}$ for all $i,d$. A graded $(B,A)$-bimodule is a $(B,A)$-bimodule ($(S,R)$-bimodules and commuting left and right scalar actions) that is graded as a $k$-module and homogeneous under both actions, the two actions continuing to commute. Both induced $k$-actions must agree with the given $k$-module structure: ${\eta }_B(t)m=tm=m{\eta }_A(t)$ for $t\in k$ and $m\in M$. Elements of $M_d$ are homogeneous of degree $d$, and the decomposition expresses every $m\in M$ uniquely as a finite sum of nonzero homogeneous components.

Degree-zero maps. A map $f:M\to N$ of graded left $A$-modules is degree-zero, or a graded map, when it is $A$-linear and $f(M_d)\subseteq N_d$ for every $d$. The category ${\mathrm{GrMod}}_0(A)$ has the graded left $A$-modules as objects and the degree-zero maps as morphisms, with composition of maps. Every $M$ is an object of ${\mathrm{GrMod}}_0(A)$ by its own grading, and $A$ is a graded $(A,A)$-bimodule under left and right multiplication.

Graded submodules. A graded submodule of a graded left $A$-module $M$ is a submodule $S\le M$ with $S={⨁}_{d\in \mathbb{Z}}(S\cap M_d)$. Such an $S$ is itself a graded left $A$-module with homogeneous pieces $S_d:=S\cap M_d$, because $A_i{S}_d\subseteq A_i{M}_d\cap S\subseteq M_{i+d}\cap S=S_{i+d}$; equivalently, $S$ is generated by its homogeneous elements. In particular a graded submodule is determined by its homogeneous pieces: if $S={⨁}_d{S}_d$ and $T={⨁}_d{T}_d$ are graded submodules with $S_d=T_d$ for every $d$, then $S=T$.

Internal shift. For $r\in \mathbb{Z}$ and a graded module $M$, the internal shift $M\{r\}$ is the module with

$(M\{r\}{)}_d:=M_{d-r}(d\in \mathbb{Z}),$

carrying the same scalar action as $M$. Its components are additive subgroups of $M$ whose sum is direct, so $M\{r\}$ is a $k$-module; and $M\{r\}$ is graded because

$A_i(M\{r\}{)}_d=A_i{M}_{d-r}\subseteq M_{i+d-r}=(M\{r\}{)}_{i+d}$

for all $i,d$. For a graded bimodule both actions are unchanged, remain homogeneous and still commute, so $M\{r\}$ is again a graded bimodule. The shift is invertible: $(M\{r\})\{-r\}=M$, and $M\{0\}=M$. For a graded algebra $A$, the shifts $A\{r\}$ of the regular bimodule are the modules denoted $A\{r\}$ elsewhere on this page.

Dictionary with the published twist. In the published nonnegative commutative convention (Nonnegatively graded rings and modules, homogeneous elements, and twists) the twist $M(a)$ is the graded module with $M(a{)}_d=M_{d+a}$ for every integer $a$. Substituting $a=-r$ gives

$(M\{r\}{)}_d=M_{d-r}=M(-r{)}_d,$

so $M\{r\}=M(-r)$ as graded modules: the internal shift reverses the sign of the published twist parameter. For $r=0$ both conventions agree with $M$.

No super signs. The internal degree is a genuine $\mathbb{Z}$-grading, not a $\mathbb{Z}\mathrm{/}2$-parity. No sign is inserted into the multiplication, the scalar action or a degree-zero map merely because elements have nonzero degree; the balancing relation of the tensor product below is likewise signed by nothing. These are the unsigned associative conventions for this page; no differential or signed symmetry on tensor products is part of this definition. They do not exclude Koszul sign conventions in other categories of graded objects.

For any unital associative $\mathbb{Z}$-graded algebra $A$, the category ${\mathrm{GrMod}}_0(A)$ of graded left $A$-modules and degree-zero maps is abelian. Kernels, images, cokernels and finite biproducts are computed in each homogeneous degree; a sequence is exact precisely when it is exact degreewise.

Total internal degree on the balanced tensor product. Let $A$ be a graded $k$-algebra, $M$ a graded right $A$-module and $N$ a graded left $A$-module (Associative graded algebras, bimodules, and internal shifts). First use the canonical decomposition of the ordinary tensor product of the underlying abelian groups $M{\otimes }_{\mathbb{Z}}N\cong {⨁}_{i,j\in \mathbb{Z}}{M}_i{\otimes }_{\mathbb{Z}}{N}_j\mathrm{.}$ Its inverse sends $m\otimes n$ to the finite sum ${\sum }_{i,j}{m}_i\otimes n_j$, where $m={\sum }_i{m}_i$ and $n={\sum }_j{n}_j$ are their homogeneous decompositions; the two maps are inverse because they agree with the identity on homogeneous elementary tensors. Give the summand $M_i{\otimes }_{\mathbb{Z}}{N}_j$ total degree $i+j$, and write $G_d:={⨁}_{i+j=d}{M}_i{\otimes }_{\mathbb{Z}}{N}_j$.

The balanced tensor product $M{\otimes }_{A}N=(M{\otimes }_{\mathbb{Z}}N)\mathrm{/}{\mathcal{R}}_A$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums), where ${\mathcal{R}}_A$ is generated by the balance relations $(ma)\otimes n-m\otimes (an)$. These relations are generated by the homogeneous ones: for $m={\sum }_i{m}_i$, $a={\sum }_{\mathrm{\ell }}{a}_{\mathrm{\ell }}$ and $n={\sum }_j{n}_j$, the relation is the finite sum of $(m_i{a}_{\mathrm{\ell }})\otimes n_j-m_i\otimes (a_{\mathrm{\ell }}{n}_j)$ over $i,\mathrm{\ell },j$. Each such homogeneous relation belongs to $G_{i+\mathrm{\ell }+j}$, since the two module actions preserve total degree. Thus ${\mathcal{R}}_A={⨁}_d({\mathcal{R}}_A\cap G_d)$, and

$M{\otimes }_{A}N={⨁}_{d\in \mathbb{Z}}(M{\otimes }_{A}N{)}_d,(M{\otimes }_{A}N{)}_d:=(G_d+{\mathcal{R}}_A)\mathrm{/}{\mathcal{R}}_A,$

is a graded abelian group. For homogeneous $m\in M_i$ and $n\in N_j$ the elementary tensor $m\otimes n$ has total internal degree $i+j$; every tensor is a finite sum of homogeneous elementary tensors, and the balancing relations used in the quotient are homogeneous. This grading is the only grading used below, and it introduces no sign.

Outer actions. If $M$ is a graded $(B,A)$-bimodule and $N$ a graded left $A$-module, then the action $b(m\otimes n):=(bm)\otimes n$ of A commuting outer scalar action descends to a tensor product makes $M{\otimes }_{A}N$ a graded left $B$-module: the action is homogeneous because homogeneous $b\in B_l$ sends $m\otimes n$ of degree $i+j$ to an element of degree $l+i+j$. Symmetrically, if $N$ is a graded $(A,C)$-bimodule, the action $(m\otimes n)c:=m\otimes (nc)$ makes $M{\otimes }_{A}N$ a graded right $C$-module, and the two outer actions commute when both are present.

Homogeneous Hom. For graded left $A$-modules $P,Q$, with ${\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)$ the abelian group of all $A$-linear maps $P\to Q$, put

${\mathrm{Hom}}_{A,d}(P,Q):=\{f\in {\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)\mid f(P_i)\subseteq Q_{i+d}\text{ for all }i\in \mathbb{Z}\},{\mathrm{HOM}}_A(P,Q):={⨁}_{d\in \mathbb{Z}}{\mathrm{Hom}}_{A,d}(P,Q)\mathrm{.}$

Each ${\mathrm{Hom}}_{A,d}(P,Q)$ is a subgroup of ${\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)$, and the sum is direct: if $f={\sum }_d{f}_d={\sum }_d{f}_d^{\mathrm{\prime }}$ with $f_d,f_d^{\mathrm{\prime }}\in {\mathrm{Hom}}_{A,d}$, then for $p\in P_i$ the element $f(p)$ has homogeneous components $f_d(p)\in Q_{i+d}$, and comparison of the graded components in $Q$ gives $f_d(p)=f_d^{\mathrm{\prime }}(p)$ for every $d$ and $p$, so $f_d=f_d^{\mathrm{\prime }}$. Thus ${\mathrm{HOM}}_A(P,Q)$ is a graded abelian group whose degree-$d$ part is ${\mathrm{Hom}}_{A,d}(P,Q)$, and the degree-zero morphisms are ${\mathrm{Hom}}_{A,0}(P,Q)$. By definition ${\mathrm{HOM}}_A(P,Q)$ consists of the finite sums of homogeneous $A$-linear maps, so ${\mathrm{HOM}}_A(P,Q)\subseteq {\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)$ with equality only when every $A$-linear map has bounded spread in degree.

The associative left action. If $M$ is a graded $(B,A)$-bimodule and $Y$ a graded left $B$-module, then ${\mathrm{HOM}}_B(M,Y)$ carries a left $A$-module structure

$(a\cdot f)(m):=f(ma)(a\in A,m\in M),$

which is homogeneous: for $a\in A_i$ and $f\in {\mathrm{Hom}}_{B,d}(M,Y)$ one has $ma\in M_{l+i}$ for $m\in M_l$, hence $(a\cdot f)(m)\in Y_{l+i+d}$ and $a\cdot f\in {\mathrm{Hom}}_{B,d+i}(M,Y)$. The action is associative and unital by the module axioms of $M$, so ${\mathrm{HOM}}_B(M,Y)$ is a graded left $A$-module and an object of ${\mathrm{GrMod}}_0(A)$. The same formula defines an action on the full ungraded $\mathrm{Hom}$ group, where it is denoted by the same symbol.

${\mathrm{HOM}}_A(P,Q)$ need not be all of ${\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)$. Let $k$ be a field, $A=k[x]$ with $\mathrm{deg}x=1$, and $Q=A$; let $P={⨁}_{j\ge 0}A\{-j\}$ with generator $u_j:=1_A\in A\{-j\}$, an element of degree $-j$, and define the $A$-linear map

$f:P⟶A,f(u_j):=x^{2j}(j\ge 0)\mathrm{.}$

For each $j$ the map $f_j$ that sends $u_j$ to $x^{2j}$ and every other $u_l$ to $0$ is $A$-linear and homogeneous of degree $3j$: it kills all pieces $A\{-l{\}}_d$ with $l\mathrm{\ne }j$, and on $A\{-j{\}}_d=A_{d+j}$ it sends $a{u}_j$ to $a{x}^{2j}\in A_{d+3j}$. The pointwise sum ${\sum }_j{f}_j$ equals $f$, because every element of $P$ has finite support, but it is not a finite sum: if $f={\sum }_{t\in T}{g}_t$ with $T$ finite and $g_t$ homogeneous of degree $t$, then evaluating at $u_j$ gives $x^{2j}={\sum }_t{g}_t(u_j)$ with $g_t(u_j)\in A_{t-j}$, and since $x^{2j}$ is a nonzero homogeneous element of $A$ of degree $2j$ the only contribution comes from $t-j=2j$. Hence $g_{3j}(u_j)=x^{2j}\mathrm{\ne }0$ for every $j\ge 0$, so $T$ would have to contain the infinitely many distinct degrees $3j$, contradicting finiteness. Therefore $f\mathrm{\notin }{\mathrm{HOM}}_A(P,A)$, and the inclusion ${\mathrm{HOM}}_A(P,Q)\subseteq {\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(P,Q)$ is proper in general.

Let $R,S$ be graded $k$-algebras and let $B,C$ be graded $k$-algebras; let $M$ be a graded $(B,R)$-bimodule, $N$ a graded $(R,S)$-bimodule and $P$ a graded $(S,C)$-bimodule.

1. The balanced associator ${\alpha }_{M,N,P}:(M{\otimes }_{R}N){\otimes }_{S}P⟶M{\otimes }_R(N{\otimes }_{S}P),{\alpha }_{M,N,P}((m\otimes n)\otimes p)=m\otimes (n\otimes p),$ is an isomorphism of graded abelian groups, natural in $M,N,P$ and compatible with the outer actions that make both sides graded $(B,C)$-bimodules.

2. For every graded left $R$-module $N$ and graded right $R$-module $M$ the tensor-unit maps ${\lambda }_N:R{\otimes }_{R}N\to N$, $r\otimes n\mapsto rn$, and ${\rho }_M:M{\otimes }_{R}R\to M$, $m\otimes r\mapsto mr$, are degree-zero isomorphisms, compatible with outer actions.

3. For all $r,s\in \mathbb{Z}$, the identity on elementary tensors induces a degree-zero isomorphism $M\{r\}{\otimes }_{R}N\{s\}\cong (M{\otimes }_{R}N)\{r+s\},$ natural in $M$ and $N$ and compatible with outer actions.

Fix a graded $k$-algebra $A$ and let $P$ be a graded left $A$-module (Associative graded algebras, bimodules, and internal shifts). The category ${\mathrm{GrMod}}_0(A)$ of graded left $A$-modules and degree-zero maps is abelian (Graded modules with degree-zero maps form an abelian category).

Graded projective. $P$ is graded projective when it is a projective object of ${\mathrm{GrMod}}_0(A)$ (Projective object): for every degree-zero epimorphism $q:E\twoheadrightarrow M$ and every degree-zero $f:P\to M$ there exists a degree-zero $\tilde{f}:P\to E$ with $q\tilde{f}=f$. Thus projectivity is tested only against degree-zero epimorphisms and degree-zero maps, and the lift need not be unique.

Finitely generated. $P$ is finitely generated as a graded left $A$-module, or generated by finitely many homogeneous elements, when for some $n\ge 0$ there are homogeneous elements $p_1,\dots ,p_n\in P$ with

$P=A{p}_1+\cdots +A{p}_n\mathrm{.}$

By Generated submodule, cyclic and finitely generated modules, module basis and free module this is the same notion as finite generation of the underlying $A$-module: a finite homogeneous family is a finite family, and conversely, if $P=⟨S{⟩}_A$ for a finite set $S$, then writing each $s={\sum }_e{s}_e$ as its finite sum of nonzero homogeneous components gives, for every $x={\sum }_{s\in S}{a}_{s}s\in P$,

$x={\sum }_{s\in S}{a}_s{\sum }_e{s}_e={\sum }_{s,e}{a}_s{s}_e,$

a finite $A$-linear combination of the homogeneous elements $s_e\in P$; so the finitely many components of the members of $S$ generate $P$. In particular the notion does not depend on the chosen finite generating set.

Finite graded projective. $P$ is finite graded projective when it is graded projective and generated by finitely many homogeneous elements, that is, when it is graded projective and finitely generated as an $A$-module. The family may be empty: $n=0$ gives $P=A{p}_1+\cdots +A{p}_n=0$, so the zero module is generated by the empty homogeneous family, and the corresponding finite direct sum of shifts in the characterization below is the empty direct sum. Its finite shifted-free characterization is the next result (Finite graded projectives are finite shifted-free summands), and it uses this same notion of finite generation throughout: a finite homogeneous generating family there is exactly a family as above. No choice principle is used in the equivalence above, which only rewrites a given finite expression.

Let $A$ be a graded $k$-algebra and let $P$ be a graded left $A$-module. Then $P$ is finite graded projective (Finite graded projective modules) if and only if $P$ is a degree-zero direct summand of a finite direct sum of internal shifts $A\{r_1\}\oplus \cdots \oplus A\{r_n\}$. In particular every finite direct sum of shifts $A\{r_j\}$ is a projective object of ${\mathrm{GrMod}}_0(A)$, and this conclusion uses no assumption of arbitrary-index choice.

Let $A$ and $B$ be graded $k$-algebras and $M$ a graded $(B,A)$-bimodule. Write ${\mathrm{\Phi }}_M:=M{\otimes }_A-$ for the functor that sends a graded left $A$-module $N$ to the graded left $B$-module $M{\otimes }_{A}N$ of the total-degree grading.

1. Exactness. If $M$ is flat as an underlying right $A$-module, then ${\mathrm{\Phi }}_M$ is exact on graded left $A$-modules.

2. Projectives. If $M$ is finite graded projective as a left $B$-module, then ${\mathrm{\Phi }}_M$ carries every finite graded projective left $A$-module to a finite graded projective left $B$-module.

Neither hypothesis is asserted to imply the other; the companion page exhibits a right-flat $M$ whose output is not projective and a left-projective $M$ whose tensor functor is not exact.