The spectrum of a twisted commutative algebra

Let $A$ be a tca. One can then formulate equivariant analogs of many familiar concepts from commutative algebra¹¹1In fact, one can do this for commutative algebras in any tensor category.:

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  A $\mathbf{GL}$-ideal of $A$ is an ideal that is $\mathbf{GL}$-stable.

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  A $\mathbf{GL}$-prime is a $\mathbf{GL}$-ideal $\mathfrak{p}$ such that $VW\subset\mathfrak{p}$ implies $V\subset\mathfrak{p}$ or $W\subset\mathfrak{p}$, for subrepresentations $V,W\subset A$. Here $VW$ denotes the image of the map $V\otimes W\to A$.

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  The $\mathbf{GL}$-radical of a $\mathbf{GL}$-ideal $I$, denoted $\operatorname{rad}_{\mathbf{GL}}{I}$, is the sum of all subrepresentations $V$ of $A$ such that $V^{n}\subset I$ for some $n$. Here $V^{n}$ denotes the image of the map $V^{\otimes n}\to A$. This is equal to the intersection of the $\mathbf{GL}$-primes containing $I$ (Proposition 2.5).

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  The $\mathbf{GL}$-spectrum of $A$, denoted $\operatorname{Spec}_{\mathbf{GL}}(A)$, is the set of $\mathbf{GL}$-primes, endowed with the usual Zariski topology. The closed subsets of $\operatorname{Spec}_{\mathbf{GL}}(A)$ correspond bijectively to $\mathbf{GL}$-radical ideals.

One can keep going, but this is all we need for the moment.

In ordinary commutative algebra, prime ideals are of central importance; this is no less true of $\mathbf{GL}$-primes in twisted commutative algebra. For instance: the support of an equivariant module is most naturally a subset of the $\mathbf{GL}$-spectrum; the $\mathbf{GL}$-primes can be used to generate the Grothendieck group of equivariant modules; and, under suitable hypotheses, one has an equivariant version of primary decomposition for $\mathbf{GL}$-ideals. Therefore, to understand tca’s it is of crucial importance to understand their $\mathbf{GL}$-primes.

Every $\mathbf{GL}$-stable prime ideal of a tca is a $\mathbf{GL}$-prime. However, the converse is not true. The following is an instructive example:

The category $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$ of polynomial representations of $\mathbf{GL}_{\infty}$ is equivalent to the category $\mathbf{Pol}$ of polynomial functors of rational vector spaces; the equivalence is obtained by evaluating a functor on $\mathbf{Q}^{\infty}$. We can thus view a tca $A$ as an algebra object of $\mathbf{Pol}$. From this perspective, $A(\mathbf{Q}^{\infty})$ can be seen as the “incarnation” of $A$ in the category $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$. However, polynomial functors can be evaluated on objects in any $\mathbf{Q}$-linear tensor category. We can thus form the “super incarnation” $A(\mathbf{Q}^{\infty|\infty})$ of $A$ by evaluating on the super vector space $\mathbf{Q}^{\infty|\infty}$. This is an algebra object of $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty|\infty})$. We can now succinctly express the point of this paper:

Key principle. The geometry of the super incarnation of a tca is sufficiently rich to detect its $\mathbf{GL}$-primes.

We now state several precise theorems. In what follows, $A$ is a tca (considered as a polynomial functor) and $I$ and $J$ are $\mathbf{GL}$-ideals of $A$.

In other words, the theorem says that $\operatorname{rad}_{\mathbf{GL}}{I}\subset\operatorname{rad}_{\mathbf{GL}}{J}$ if and only if $\mathcal{V}(J(\mathbf{Q}^{\infty|\infty}))\subset\mathcal{V}(I(\mathbf{Q}^{\infty|\infty}))$; the latter condition is equivalent to $\mathcal{V}(J(\mathbf{Q}^{r|s}))\subset\mathcal{V}(I(\mathbf{Q}^{r|s}))$ for all $r$ and $s$ and thus (usually) reduces to a condition about finite dimensional algebraic varieties. Note that since we are only concerned with vanishing loci here, we can pass to the reduced quotient of $A(\mathbf{Q}^{r|s})$, which is an ordinary (non-super) commutative ring. We emphasize that the theorem is false if one uses only ordinary vector spaces, as Example 1.1 shows.

Once again, we can verify the latter condition on finite dimensional spaces. The theorem therefore reduces the problem of showing that $\operatorname{rad}_{\mathbf{GL}}(I)$ is $\mathbf{GL}$-prime to showing that the algebraic varieties $\mathcal{V}(I(\mathbf{Q}^{r|s}))$ are irreducible.

In the two remaining theorems, we require a finiteness condition: we assume that $A_{0}$ is noetherian and that $A$ is finitely generated over $A_{0}$.

This theorem gives a useful way to find the minimal $\mathbf{GL}$-primes, at least up to $\mathbf{GL}$-radical, and thus the irreducible components of the $\mathbf{GL}$-spectrum.

This theorem is a strengthening of Draisma’s topological noetherianity theorem: indeed, Draisma’s theorem only encompasses the $\mathbf{GL}$-stable prime ideals, while this theorem accommodates all $\mathbf{GL}$-primes. In fact, this theorem is easily deduced by combining Draisma’s theorem with our other theorems; we do not have a new proof of Draisma’s result.

In §5, we examine the tca $A=\operatorname{Sym}(\operatorname{Sym}^{2}(\mathbf{C}^{\infty}))$. Using our main theorems, we classify the $\mathbf{GL}$-primes of $A$ (they are the rectangular ideals) and the $\mathbf{GL}$-radical ideals of $A$ (they are the ideals generated by a single irreducible representation). This example provides a good illustration of how our main theorems allow one to understand the equivariant commutative algebra of $A$ through standard geometric means. It also provides a reconceptualization of sorts for the rectangular ideals: they constitute the equivariant spectrum of $A$. These ideals have long been of interest, as they are symbolic powers of determinantal ideals.

The notions of $\mathbf{GL}$-prime and the $\mathbf{GL}$-spectrum were discussed in [SS4, §3] (with slightly different terminology). However, that paper only works with so-called bounded tca’s, and for these $\mathbf{GL}$-primes are the same as $\mathbf{GL}$-stable primes, so the chief difficulties disapper. (Note that the tca in Example 1.1 is not bounded.)

In forthcoming joint work with Rohit Nagpal [NS1, NS2], we study the $\mathfrak{S}_{\infty}$-primes in the ring $\mathbf{C}[x_{1},x_{2},\ldots]$ and manage to completely classify them. (Here $\mathfrak{S}_{\infty}$ denotes the infinite symmetric group.) However, there is no general theory of $\mathfrak{S}_{\infty}$-primes yet.

One can, and usually does, define twisted commutative algebras without any reference to super vector spaces. However, our results show that one is essentially forced to use super vector spaces to fully understand tca’s. This is reminscent of Deligne’s theorem [De, Theorem 0.6], in which a natural class of tensor categories (defined without any reference to super vector spaces) is characterized using super objects. It would be interesting to find a direct connection between these results.

In §2, we review relevant background material. In §3, we prove the key theorem, which is a certain special case of Theorem A. In §4, we deduce the main theorems stated above. Finally, in §5, we work out the $\operatorname{Sym}(\operatorname{Sym}^{2})$ case in detail.

We thank Steven Sam for helpful discussions, and allowing us to include some jointly conceived ideas in §5.

We recall some background material on polynomial representations. We refer to [SS2] and [NSS2, §2.2] for more details.

Regard $\mathbf{GL}_{n}$ as an algebraic group over $\mathbf{Q}$. By an algebraic representation of $\mathbf{GL}_{n}$, we mean a comodule over $\mathbf{Q}[\mathbf{GL}_{n}]$; the dimension is not required to be finite. We regard $\mathbf{GL}_{\infty}$ as the inductive system of algebraic groups $\bigcup_{n\geq 1}\mathbf{GL}_{n}$. An algebraic representation of $\mathbf{GL}_{\infty}$ is a vector space equipped with compatible algebraic representations of $\mathbf{GL}_{n}$ for all $n$. The basic example of such a representation is the standard representation $\mathbf{Q}^{\infty}=\bigcup_{n\geq 1}\mathbf{Q}^{n}$. A polynomial representation of $\mathbf{GL}_{\infty}$ is one that occurs as a subquotient of a (possibly infinite) direct sum of tensor powers of the standard representation. We denote the category of such representations by $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$. It is a $\mathbf{Q}$-linear abelian category and closed under tensor products. The structure of this category is well-understood: it is semi-simple, and the simple objects are given by $\mathbf{S}_{\lambda}(\mathbf{Q}^{\infty})$, where $\mathbf{S}_{\lambda}$ denotes the Schur functor associated to the partition $\lambda$. All simple objects are absolutely simple. Every polynomial representation $V$ admits a canonical decomposition $V=\bigoplus_{\lambda}V_{\lambda}\otimes\mathbf{S}_{\lambda}(\mathbf{Q}^{\infty})$ where the $V_{\lambda}$ are multiplicity spaces. We endow $V$ with a grading by declaring the elements of $\mathbf{S}_{\lambda}(\mathbf{Q}^{\infty})$ to have degree $|\lambda|$.

By an algebraic representation of $\mathbf{GL}_{r|s}$, we mean a comodule over the Hopf superalgebra $\mathbf{Q}[\mathbf{GL}_{r|s}]$. We then define polynomial representations of $\mathbf{GL}_{\infty|\infty}$ just as before, and denote the category by $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty|\infty})$. It is again semi-simple abelian and closed under tensor product, and the simple objects have the form $\mathbf{S}_{\lambda}(\mathbf{Q}^{\infty|\infty})$. Warning: if $V$ is a polynomial representation of $\mathbf{GL}_{\infty|\infty}$ then $V[1]$ (shift in super grading) is typically not a polynomial representation, according to our definition. This disagrees with the convention used in [NSS2].

Consider the category $\operatorname{Fun}(\mathrm{Vec},\mathrm{Vec})$ of endofunctors of the category $\mathrm{Vec}$ of $\mathbf{Q}$-vector spaces. Let $T_{n}$ be the functor given by $T_{n}(V)=V^{\otimes n}$. We say that an object of $\operatorname{Fun}(\mathrm{Vec},\mathrm{Vec})$ is polynomial if it is a subquotient of a (possibly infinite) direct sum of $T_{n}$’s. We let $\mathbf{Pol}$ be the category of polynomial functors. It is semi-simple abelian and closed under tensor products. The simple objects are the Schur functors $\mathbf{S}_{\lambda}$. We have equivalences of categories

These equivalences are compatible with tensor products, and respect algebras, modules, and ideals within the categories.

We will at times need to evaluate polynomial functors on finite dimensional spaces. We recall the relevant result.

We will require one additional simple result on polynomial representations. Define the width of a partition $\lambda$, denote $w(\lambda)$, to be $\lambda_{1}$. Define the width of a polynomial functor $F$, denoted $w(F)$, to be the supremum of $w(\lambda)$ over those $\lambda$ for which $\mathbf{S}_{\lambda}$ occurs in $F$.

A twisted commutative algebra (tca) is a commutative algebra object in one of the three equivalent categories $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$, $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty|\infty})$, or $\mathbf{Pol}$. For the moment, we work in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$ to be definite.

Let $A$ be a tca in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$. Every polynomial representation carries a natural grading, and so $A$ is canonically graded. This is compatible with the ring structure, i.e., $A$ is a graded ring. In particular, we can regard $A$ as an algebra over is degree 0 piece $A_{0}$, which is an ordinary commutative ring.

By a “subrepresentation” of $A$, we mean a $\mathbf{Q}$-subspace that is stable by $\mathbf{GL}_{\infty}$. In practice, $A$ will often be a $\mathbf{C}$-algebra, but we use $\mathbf{Q}$-subrepresentations nonetheless. One should think of a finite length subrepresentations of $A$ as providing an equivariant substitute for the concept of element (or perhaps finite sets of elements).

Suppose $A\to B$ is a homomorphism of tca’s. We say that $B$ is finitely $\mathbf{GL}$-generated over $A$ if there is some finite length subrepresentation $E$ of $B$ such that the natural map $A\otimes\operatorname{Sym}(E)\to B$ is surjective. We typically apply this in the case $A=B_{0}$.

Let $A$ be as above, i.e., a tca in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$. A $\mathbf{GL}$-ideal of $A$ is a $\mathbf{GL}$-stable ideal of $A$. We say that a $\mathbf{GL}$-ideal $I$ is finitely $\mathbf{GL}$-generated if there is a finite length subrepresentation $E$ of $A$ that generates $I$ as an ideal. The sum and product of two finitely $\mathbf{GL}$-generated ideals is again finitely $\mathbf{GL}$-generated; for products, this relies on the fact that the tensor product of two finite length polynomial representations is again finite length.

We say that a $\mathbf{GL}$-ideal $\mathfrak{p}$ is $\mathbf{GL}$-prime if $VW\subset\mathfrak{p}$ implies $V\subset\mathfrak{p}$ or $W\subset\mathfrak{p}$, for subrepresentations $V$ and $W$ of $A$; here $VW$ denotes the image of the map $V\otimes W\to A$. It is equivalent to ask the same condition with $V$ and $W$ finite length representations, or cyclic representations, or $\mathbf{GL}$-idelas, or finitely generated $\mathbf{GL}$-ideals. We define the $\mathbf{GL}$-radical of a $\mathbf{GL}$-ideal $I$, denoted $\operatorname{rad}_{\mathbf{GL}}{I}$ to be the sum of all subrepresentations $V$ or $A$ such that $V^{n}\subset I$ for some $n$; again, one can use ideals in place of subrepresentations. We say that $I$ is $\mathbf{GL}$-radical if $I=\operatorname{rad}_{\mathbf{GL}}{I}$. We note that every $\mathbf{GL}$-prime is $\mathbf{GL}$-radical.

We now establish some properties of the above definitions that are analogous to the classical situation.

The above concepts ($\mathbf{GL}$-prime, $\mathbf{GL}$-radical, etc.) are defined using only the language of the tensor category $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL})$. It follows that the same definitions can be made in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty|\infty})$ and $\mathbf{Pol}$, and that the definitions agree on objects that correspond under the equivalences. Thus the above propositions also hold in all three settings. In fact, one can formulate and prove these results for commutative algebra objects in quite general tensor categories.

One important construction that cannot be formulated using only the language of the tensor category is the ordinary radical. Suppose $A$ is a tca in $\mathbf{Pol}$ and $I$ is a $\mathbf{GL}$-ideal in it. We can then consider their incarnations $I(\mathbf{Q}^{\infty})\subset A(\mathbf{Q}^{\infty})$ in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty})$, and form $\operatorname{rad}(I(\mathbf{Q}^{\infty}))$. Similarly, we can consider their incarnations in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}_{\infty|\infty})$ and form $\operatorname{rad}(I(\mathbf{Q}^{\infty|\infty}))$. There is no reason to expect these two radicals to be comparable in any way (except in tautological ways, e.g., both contain $I$). In fact, the point of this paper is that they really are not comparable, and the construction is better behaved on the super side.

We require the following result.

We require the following result on $\mathbf{GL}$-primes. See [SS2, §8.6] for some similar results.

We require some preliminary work before proving the proposition. Let $\{e_{i}\}_{i\in\mathcal{I}}$ be a homogeneous basis for $\mathbf{V}=\mathbf{Q}^{\infty|\infty}$. Given an element $x$ in a polynomial representation $V$ of $\mathbf{GL}_{\infty|\infty}$ and a subset $S$ of $\mathcal{I}$, we say that $x$ has support contained in $S$ if $V$ can be embedded into a direct sum of tensor powers of $\mathbf{Q}^{\infty|\infty}$ such that $x$ can be expressed using the basis elements in $\mathcal{I}$. One can define this more canonically by looking at the weight decomposition of $x$. We define the support of an element $X$ of $\mathfrak{gl}_{\infty|\infty}$ to be the set of indices $i$ such that $X$ has a non-zero entry in row $i$ or column $i$. We say that an element of $\mathcal{U}(\mathfrak{gl}_{\infty|\infty})$ has support contained in $S$ if it can be expressed in terms of elements of $\mathfrak{gl}_{\infty|\infty}$ having this property. We say that elements of representations or $\mathcal{U}(\mathfrak{gl}_{\infty|\infty})$ are disjoint if they have disjoint supports (or if they have supports contained in disjoint sets). For an element $x\in A$, we let $\langle x\rangle$ be the $\mathbf{GL}$-ideal it generates.

Suppose that a group $G$ acts on a topological space $X$. We say that $X$ is $G$-noetherian if every descending chain of closed $G$-stable subsets stabilizes. Draisma [Dr, Corollary 3] proved the following important theorem in this context:

In fact, Draisma only states this theorem when $A_{0}$ is finitely generated over a field, but a slight modification in his proof yields the above statement. Since it is not critical for this paper, we do not include details. We give a few corollaries of the theorem.