Associative graded algebras, bimodules, and internal shifts

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.