On the stratification of noncommutative prime spectra

We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. The G-action on R induces a stratification of the prime spectrum of R which was introduced by Goodearl and Letzter. For a noetherian algebra R, Goodearl and Letzter showed that the strata of the spectrum of R are isomorphic to the spectra of certain commutative Laurent polynomial algebras. The purpose of this note is to give a new proof of this result which works for arbitrary algebras R.


INTRODUCTION
Let G be an affine algebraic group and let R be an associative algebra on which G acts rationally by algebra automorphisms. The induced G-action on the set Spec R of all prime ideals of R leads to a stratification of Spec R which was pioneered by Goodearl and Letzter [3]. For the special case of an algebraic torus G and a noetherian algebra R, Goodearl and Letzter have given a description of the strata of Spec R in terms of the spectra of certain (commutative) Laurent polynomial algebras. Later, a different description of the strata was given in [6], for any algebra R and any connected affine algebraic group G. The purpose of this short note is to consolidate the description of [6], for the case of an algebraic torus G, with the earlier one due to Goodearl and Letzter, still working with a general algebra R.
Throughout, k will be an algebraically closed base field of arbitrary characteristic. The largest Gstable ideal of the algebra R that is contained in a given ideal I of R, called the G-core of I, will be denoted by I:G; so I:G = g∈G g.I. It is easy to see [5,Proposition 8(b)] that, for any rational action of an affine algebraic group G on R, the collection of all G-cores of prime ideals of R coincides with the set of all G-prime ideals of R; the latter set will be denoted by G-Spec R. If the algebraic group G is connected, then G-Spec R is simply the set of all G-stable prime ideals of R [5, Proposition 19(a)]. The Goodearl-Letzter stratification of Spec R is the partition For an algebraic torus G, we will give a description of each stratum Spec I R in terms of the spectrum of a suitable affine commutative algebra Z I . The construction of the algebra Z I and the precise statement of the main result will be given in Section 1 while the proof will occupy Section 2.

THE STRATIFICATION THEOREM
Let G be a connected affine algebraic group over k and let R be an associative k-algebra with a rational G-action by k-algebra automorphism. For a given I ∈ G-Spec R, let C(R/I) denote the extended centroid of the algebra R/I; this is a k-field, called the heart of I, on which G acts via its action on R/I [5, 2.3]. We put Clearly, Z I is a G-stable k-subalgebra of C(R/I) and Z I contains the subfield of G-invariants, C(R/I) G . Moreover, the G-action on Z I is rational by [5,Lemma 18(b)].
We now focus on the case of an algebraic torus G. As usual, X(G) will denote the lattice of rational characters of G. The following theorem, under the additional assumption that the algebra R is noetherian, is originally due to Goodearl and Letzter [3]; see also [2], [4] and [1, II.2.13].
Stratification Theorem. Let G be an algebraic torus over k that acts rationally by algebra automorphisms on the k-algebra R, and let I ∈ G-Spec R. Then: (a) There is an isomorphism the group algebra over the field C(R/I) G of the sublattice

Reductions and preliminaries.
We begin with some remarks that hold for an arbitrary connected affine algebraic group G over k. In order to describe the G-stratum Spec I R for a given I ∈ G-Spec R, we may replace R by R/I and thus assume that I = 0. In particular, R is a prime ring. We will write C = C(R) and Z = Z 0 for brevity; so C is a commutative k-field on which G acts by automorphisms and Our goal is to give a description of Z and to establish a suitable order isomorphism We will need to consider various G-actions; they will usually be indicated by a simple dot as in the foregoing. When more precision is necessary, the G-action on C will be denoted by ρ. The group G also acts on k[G], the algebra of regular functions of G, via the right and left regular representations 2.2. The algebra Z. Consider the Hopf C-algebra this is an algebra of C-valued functions on G via ( i c i ⊗ f i )(g) = i c i f i (g). The group G acts on the ring S via ρ ⊗ ρ r . If g ∈ G and s = i c i ⊗ f i ∈ S then g.s ∈ S is the function G → C that is given by Let S G denote the subring of G-invariants in S. Thus, s ∈ S G if and only if g −1 .s(1) = s(g) for all g ∈ G.
Next, we show that there is an isomorphism holds for all g ∈ G. Indeed, by [5, Lemma 18(b)], the G-action on Z is rational: it arises from a map of k-algebras ∆ Z : (17) and (18)], the k[G]-linear extension of ∆ Z , which will also be denoted by ∆ Z , is an isomorphism of k[G]-algebras that satisfies the "intertwining formula" Since the latter algebra is clearly Z, the isomorphism (2) follows. It is easy to see that the isomorphism (2) is just the restriction to S G of the Hopf counit S → C, s → s(1). In particular, for any s ∈ S G , we have proving (3).
2.3. The case of an algebraic torus. Now let G ∼ = (k × ) d be an algebraic torus over k and let Λ = X(G) ∼ = Z d be its lattice of rational characters. Then k[G] = kΛ, the group algebra of Λ over k. As it is customary to use additive notation for the lattice Λ, we will write the standard k-basis of k[G] as {x λ | λ ∈ Λ}; so x λ x λ ′ = x λ+λ ′ and x λ (g) = λ, g ∈ k × for g ∈ G. Then ρ r (g)x λ = λ, g x λ and the group algebra of Λ over the field C. Consider an element s = λ s λ ⊗ x λ ∈ S with s λ ∈ C. Then g.s = λ g.s λ ⊗ λ, g x λ for g ∈ G. Hence, s ∈ S G if and only if g.s λ = −λ, g s λ for all g, λ.
Putting C λ = {c ∈ C | g.c = λ, g c for all g ∈ G} and noting that each nonzero C λ is 1-dimensional over the fixed field C G , we have the group algebra of the sublattice Γ = {λ ∈ Λ | C λ = 0} over C G . We remark that Γ = X(G/N ) is the character lattice of the torus G/N , where N is the kernel of the action of G on Z; so Z = λ∈Γ C λ . The isomorphisms (4) and (2) prove part (a) of the Theorem. Note also the S is free over S G . We claim that each G-stable ideal a of S is generated by its intersection with S G : For the nontrivial inclusion ⊆, let s = λ s λ ⊗ x λ ∈ a be given. In order to show that s ∈ (a ∩ S G )S, we argue by induction on the size of Supp(s) = {λ ∈ Λ | s λ = 0}, the length of s. Our claim being clear for s = 0, assume that s = 0. Suppose there exists an element 0 = t ∈ a with Supp(t) Supp(s). Multiplying t and s with suitable units of the form c ⊗ x µ , we may assume that 0 ∈ Supp(t) and t 0 = s 0 = 1. Since t and s − t are shorter than s, they both belong to (a ∩ S G )S and hence s ∈ (a∩S G )S as well. Therefore, we may assume that if t ∈ a satisfies Supp(t) Supp(s) then t = 0. Continuing to assume that s 0 = 1, this holds in particular for t = s−g.s = 0 =λ (s λ − λ, g g.s λ )⊗x λ for each g ∈ G. Therefore, we must have s ∈ S G and (5) is proved. Now let b be an ideal of S G and let a denote sum of all ideals of S that contract to b. Since S is free over S G , we have a ∩ S G = b. Moreover, a is clearly G-stable and so (5) gives that a = bS. Thus, bS is the unique largest ideal of S that contracts to b.
2.4. The prime correspondence. We start with some reminders from [6]. For now, let G again be an arbitrary connected affine algebraic group over k and let k(G) = Fract k[G] be the field of rational functions of G. The G-action on S via ρ ⊗ ρ r extends uniquely to an action of G on the following localization of S: Let Spec G (T ) denote the collection of all G-stable prime ideals of T . Then [6, Theorem 9] establishes an order isomorphism c : Spec 0 R ∼ −→ Spec G (T ) (6) with the following G-equivariance property, for P ∈ Spec 0 R and g ∈ G: c(g.P ) = (1 C ⊗ ρ ℓ (g))(c(P )) .
Since T is the localization S at the nonzero elements of k[G], contraction and extension yields a G-equivariant order isomorphism Spec T ∼ −→ {p ∈ Spec S | p ∩ k[G] = 0}. Note that k[G] is a Gsimple ring, because G acts transitively on itself by right multiplication. Therefore, each p ∈ Spec G (S) satisfies p ∩ k[G] = 0, and hence the above bijection restricts to a bijection given by contraction and extension. We now return to the case of an algebraic torus G. Then the map p → p ∩ S G injects Spec G (S) into Spec(S G ) by (5). Moreover, for any q ∈ Spec(S G ), we know that p = qS is the unique largest ideal of S that contracts to q, which implies that p ∈ Spec G (S). Thus we obtain a bijection that is again given by contraction and extension. From (6) -(9) in conjunction with the isomorphism (2) we obtain the desired order isomorphism γ : Spec 0 R