Graded ring

A graded ring is a direct sum $R={⨁}_{n\ge 0}{R}_n$ with $R_m{R}_n\subseteq R_{m+n}$.

A graded module over a graded ring $R={⨁}_i{R}_i$ is an $R$-module with a direct sum decomposition $M={⨁}_d{M}_d$ satisfying $R_i{M}_d\subseteq M_{i+d}$. A graded homomorphism preserves these degrees. Over a polynomial ring with positive variable degrees, a finitely generated module of this kind is bounded below and has finite-dimensional graded components over the coefficient field.

If $f$ is homogeneous of degree one, multiplication by powers of the invertible element $f$ identifies every homogeneous component of $M_f$ with $(M_f{)}_0$. A homogeneous element $m$ of degree $a$ is written $f^a(f^{-a}m)$. Thus the displayed map is an isomorphism of graded modules. In particular, the degree-zero part of a tensor product of localized graded modules is the tensor product of their degree-zero parts over $(S_f{)}_0$. This proves tensor compatibility of graded sheafification on degree-one-generated Proj.

The graded shift $M(-a)$ is the same underlying module with $M(-a{)}_d=M_{d-a}$. Thus $S(-a)$ has a free generator in degree $a$. Its Hilbert series is $t^a{H}_M(t)$, and multiplication by a homogeneous element of degree $a$ becomes a degree-preserving map $M(-a)\to M$.

A graded algebra over a field $k$ is a $k$-algebra $A={⨁}_{n\ge 0}{A}_n$ whose multiplication satisfies $A_i{A}_j\subseteq A_{i+j}$.

The bracket of homogeneous elements in a parity-graded associative algebra. It is a commutator if at least one element is even and an anticommutator if both are odd. This is the natural bracket for the action of an odd BRST charge on quantum operators.

A graded algebra equipped with a degree-one differential satisfying $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$ and $d^2=0$. Its cohomology inherits a graded algebra structure. In rational homotopy, a Sullivan minimal model is a free graded-commutative differential graded algebra with decomposable generator differentials.

For an even-degree generator, take a free graded $R$-module with basis $a_k$, $k\ge 0$, and multiplication $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$, with $a_0=1$ and $|a_k|=k|a|$. Associativity follows from the multinomial identity. Over $\mathbb{Z}$, $a_1^k=k!a_k$, so the algebra need not be polynomial on $a_1$.

An odd graded derivation satisfies $Q(ab)=(Qa)b+(-1{)}^{|a|}a(Qb)$ for homogeneous $a$. Its square is an even derivation; consequently checking $Q^2=0$ on algebra generators establishes nilpotence everywhere.

A graded algebra is a graded commutative algebra if $uv=(-1{)}^{pq}vu$ for homogeneous elements of a graded algebra of degrees $p,q$. In characteristic not equal to two, every odd-degree element then squares to zero. The exterior algebra and Hochschild cohomology with the Hochschild cup product are examples.

An element of a single graded component $A_n$ of a graded algebra is homogeneous of degree $n$. The degree sign rules for an exterior product or a graded Lie bracket apply first to homogeneous elements, then extend by linearity.

A Gerstenhaber algebra has an associative product making it a graded commutative algebra and a degree-minus-one graded Lie bracket. In the left convention, $[u,vw]=[u,v]w+(-1{)}^{(|u|-1)|v|}v[u,w]$. The shifted degrees $|u|-1$ determine the signs of antisymmetry and the Jacobi identity. Hochschild cohomology and the exterior algebra of polynomial derivations are examples.

On a graded algebra, a left derivation of degree $r$ satisfies the displayed rule for homogeneous $a$. A right derivation of degree $r$ satisfies $R(ab)=aR(b)+(-1{)}^{r|b|}R(a)b$. For odd $r$, these reduce to the familiar parity sign rules. The left Gerstenhaber bracket makes $[u,-]$ a left derivation of degree $|u|-1$ on Hochschild cohomology; this rule holds for the induced cohomology operations, not necessarily for their cochain representatives.

A standard graded algebra over a field $k$ has $A_0=k$ and is generated as a $k$-algebra by finitely many elements of degree one. Its Hilbert function agrees eventually with a polynomial.

For an ideal $I$, the associated graded ring is

For an increasing filtration of a module $M_n$ compatible with a filtered algebra $A$, set $\mathrm{gr}M={⨁}_n{M}_n/M_{n-1}$. Multiplication on the quotients makes it a graded module over $\mathrm{gr}A$. Its cumulative homogeneous dimensions equal $\dim M_n$ for a finite-dimensional exhaustive nonnegative filtration of a module.

For a Noetherian local ring $(A,\mathfrak{m})$, the pole dimension $d({\mathrm{gr}}_{\mathfrak{m}}(A))$ is the order of the pole at $t=1$ ofThe Hilbert-Serre theorem makes this series rational, so the order is defined.

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, $\mathrm{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.

For a Noetherian local ring $(R,\mathfrak{m})$, let $H_R(t)=\mathrm{length}(R/{\mathfrak{m}}^t)$. Its eventual Hilbert-Samuel polynomial has degree $d(R)$, equivalently the pole order at $1$ of the Hilbert series of ${\mathrm{gr}}_{\mathfrak{m}}R$. This is the local length-growth invariant, not the embedding dimension. The prime-chain lower bound for local length proves $\dim R\le d(R)$; the full local dimension theorem gives equality.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.

For an ideal of definition $I$ of a Noetherian local ring and a finitely generated module $M$, the Hilbert–Samuel function isFor sufficiently large $n$, it equals the Hilbert-Samuel polynomial.

If $M$ has Krull dimension $d$, the coefficient of $n^d$ in its Hilbert-Samuel polynomial is $e_I(M)/d!$. The positive integer $e_I(M)$ is the Hilbert–Samuel multiplicity of $M$ with respect to $I$.

For a graded module $M={⨁}_{n\ge 0}{M}_n$ whose pieces have finite length, its Hilbert series is

The Hilbert function of a graded module $M={⨁}_n{M}_n$ over a field is $n\mapsto {\dim }_k{M}_n$. Its generating function is the Hilbert series.

A quasipolynomial is a function $f:\mathbb{Z}\to \mathbb{C}$ for which there are polynomials $P_0,\dots ,P_{m-1}$ such that $f(n)=P_r(n)$ whenever $n\equiv r(\mathrm{mod}m)$. The least possible positive $m$ is its period.

Let $S=k[x_1,\dots ,x_r]$ be graded with positive degrees $d_i=\mathrm{deg}{x}_i$, and let $M$ be a finitely generated graded $S$-module with finite-dimensional graded pieces. Thenfor a Laurent polynomial $P(t)\in \mathbb{Z}[t,t^{-1}]$. For a standard grading, ${\dim }_k{M}_n$ consequently agrees with a polynomial for all sufficiently large $n$.

The proof is by induction on $r$. For $x=x_r$ of degree $d$, the exact sequence of graded modulesgivesBoth modules on the right are finitely generated over $k[x_1,\dots ,x_{r-1}]$, so induction supplies the asserted denominator.

For a homogeneous element $x$ of degree $e>0$ acting on a finite graded module, set $N=(0{:}_{M}x)$ and $C=M/xM$. The multiplication exact sequence $0\to N(-e)\to M(-e)\to M\to C\to 0$ and additivity of component lengths give $(1-t^e)P_M=P_C-t^e{P}_N$. This is the inductive step in the Hilbert-Serre theorem; the annihilator term must be kept unless multiplication by $x$ is injective.

Let a commutative $k$-algebra $R$ be generated by $x_1,\dots ,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The associated graded ring of this filtration is a standard graded algebra, andThe Hilbert-Serre theorem therefore makes ${\dim }_k{R}_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same Krull dimension.

Forwith the four displayed generators, $R_j$ has basis $Y_1^a{Y}_2^b$ with $|a|+|b|\le j$. The number of integer lattice points in this diamond isIts quadratic growth agrees with $\dim R=2$.

For a chosen additive function such as module length, the Poincare series of a graded module is the generating function of that function on its homogeneous components. In this usage it is a Hilbert series.