Realizations of pairs and Oka families in tensor triangulated categories
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Abstract
In this paper, we apply some methods from ring theory to the framework of prime ideals in tensor triangulated categories developed by Balmer. Given a thick tensor ideal \({\mathscr {A}}\) and a multiplicatively closed family \({\mathscr {S}}\) of objects in a tensor triangulated category \(({\mathscr {C}}, \otimes ,1)\), we say that a prime ideal \({\mathscr {P}}\) realizes \(({\mathscr {A}},{\mathscr {S}})\) if \({\mathscr {P}}\supseteq {\mathscr {A}}\) and \({\mathscr {P}}\cap {\mathscr {S}}=\varnothing \). Analogously to the results of Bergman with ordinary rings, we show how to construct a realization of a family \(\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}\) of such pairs indexed by a finite chain I, i.e., a collection \(\{{\mathscr {P}}_i\}_{i\in I}\) of prime ideals such that each \({\mathscr {P}}_i\) realizes \(({\mathscr {A}}_i,{\mathscr {S}}_i)\) and Open image in new window for each \(i\leqslant j\) in I. Thereafter we obtain conditions on a family \({\mathfrak {F}} \) of thick tensor ideals of \(({\mathscr {C}},\otimes ,1)\) so that any ideal that is maximal with respect to not being contained in \({\mathfrak {F}}\) must be prime. This extends the Prime Ideal Principle of Lam and Reyes from commutative algebra. We also combine these methods to consider realizations of templates \(\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}\), where each \({\mathscr {A}}_i\) is a thick tensor ideal and each \({\mathfrak {F}}_i\) is a family of thick tensor ideals that is also a monoidal semifilter.
Keywords
Tensor triangulated categories Oka familiesMathematics Subject Classification
13A15 18E301 Introduction
Triangulated categories were introduced by Verdier [37] and have since assumed an increasing significance in several fields of modern mathematics; from algebraic geometry to motives and homotopy theory, modular representation theory and noncommutative geometry. Additionally, a triangulated category appearing in these areas is often gifted with a tensor structure, making it into a tensor triangulated category, i.e., a symmetric monoidal category \(({\mathscr {C}},\otimes ,1)\) such that \({\mathscr {C}}\) is triangulated and Open image in new window is exact in both variables. The classification of thick subcategories is a common theme that runs through the work of Devinatz et al. [17] in homotopy theory, that of Thomason [36] in algebraic geometry, the work of Benson et al. [11] in modular representation theory and that of Friedlander and Pevtsova with finite group schemes [19]. Tensor triangular geometry developed by Balmer (see [1, 2, 3, 4, 5, 8]) unites these classifications in terms of classification of thick tensor ideal subcategories in a tensor triangulated category (also see further work by Balmer and Favi [9, 10]).
Given a tensor triangulated category \(({\mathscr {C}},\otimes ,1)\), Balmer [1] associates to it a spectrum Open image in new window consisting of prime ideals of \({\mathscr {C}}\). A thick tensor ideal \({\mathscr {P}}\) (see Definition 2.1) in \(({\mathscr {C}},\otimes ,1)\) is called prime if Open image in new window for any objects \(a,b \in {\mathscr {C}}\) implies that at least one of a, b lies in \({\mathscr {P}}\). Then Open image in new window is equipped with a Zariski topology and the support theory obtained by associating to each object \(a\in {\mathscr {C}}\) the closed subset Open image in new window unites various support theories in algebraic geometry, modular representation theory and homotopy theory. Further, support theory for a tensor triangulated category \(({\mathscr {C}},\otimes ,1)\) acting on a triangulated category \({\mathscr {M}}\) has been developed by Stevenson [34]. Stevenson’s work in [34] may be viewed as categorification of some of the work of Benson, Iyengar and Krause [12, 13, 14] in the case of actions of the unbounded derived category Open image in new window for a commutative noetherian ring R. Tensor triangular geometry has further emerged as an object of study in its own right: for instance, Balmer [6] introduced Chow groups of rigid tensor triangulated categories and properties of these Chow groups have been studied in detail by Klein [23, 24]. For further work in tensor triangular geometry, we refer the reader, for example, to Dell’Ambrogio and Stevenson [16], Peter [28], Sanders [31], Stevenson [35] and Xu [38].
The purpose of this paper is to bring some methods in ring theory to the framework of tensor triangulated categories. Let R be an ordinary commutative ring, A be an ideal in R and let \(S\subseteq R\) be a multiplicatively closed subset. Then, Bergman [15] refers to a prime ideal P such that \(P\supseteq A\) and \(P\cap S=\varnothing \) as a realization of the pair (A, S) and says that Open image in new window . More generally, if \((I,\leqslant )\) is a partially ordered set, Bergman [15] has studied conditions under which a template \(\{(A_i,S_i)\}_{i\in I}\) of such pairs indexed by I admits a realization, i.e., a family \(\{P_i\}_{i\in I}\) of prime ideals such that \(P_i\) realizes \((A_i,S_i)\) and \(P_i\subseteq P_j\) whenever \(i\leqslant j\) in I (see also further work by Sharma [33]). In [1, Lemma 2.2], Balmer shows that if \({\mathscr {A}}\) is a thick tensor ideal in \(({\mathscr {C}}, \otimes ,1)\) and \({\mathscr {S}}\) is a multiplicatively closed family of objects of \({\mathscr {C}}\) such that \({\mathscr {A}}\cap {\mathscr {S}}=\varnothing \), there exists a prime ideal \({\mathscr {P}}\supseteq {\mathscr {A}}\) satisfying \({\mathscr {P}} \cap {\mathscr {S}}=\varnothing \). Our first main aim in this paper is to formulate conditions analogous to those of Bergman [15] for construction of realizations of certain templates \(\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}\), where each \({\mathscr {A}}_i\) is a thick tensor ideal in \(({\mathscr {C}},\otimes ,1)\) and each \({\mathscr {S}}_i\) is a multiplicatively closed family of objects of \({\mathscr {C}}\).
We start in Sect. 2 by defining a relation \(\preccurlyeq \) among pairs such that Open image in new window if every prime ideal Open image in new window realizing Open image in new window contains a prime ideal \({\mathscr {P}}\) realizing the pair \(({\mathscr {A}}, {\mathscr {S}})\). Thereafter, given a template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}\) indexed by a finite decreasing chain I, we show that its realizations can be described in terms of realizations of a template Open image in new window satisfying Open image in new window for each \(i\leqslant j\) in I. Under certain finiteness conditions on the pairs in T, we construct a template \({\mathscr {D}}(T)=\{({\mathscr {B}}_i,{\mathscr {T}}_i)\}_{i\in I}\) such that for any fixed \(i_0\in I\), we can start with a prime ideal \({\mathscr {P}} \) realizing the pair \(({\mathscr {B}}_{i_0},{\mathscr {T}}_{i_0})\) and obtain a realization \(\{{\mathscr {P}}_i\}_{i\in I}\) of T with \({\mathscr {P}}_{i_0}={\mathscr {P}}\). Further, looking at the subsets of the form Open image in new window themselves, we show that these are exactly the convex subsets of Open image in new window , i.e., subsets Open image in new window satisfying the property that if \(\bigcup _{{\mathscr {P}} \in {\mathfrak {X}}}{\mathscr {P}}\subseteq {\mathscr {Q}} \subseteq \bigcup _{{\mathscr {P}}\in {\mathfrak {X}}}{\mathscr {P}}\), then \({\mathscr {Q}}\in {\mathfrak {X}}\). We also give necessary and sufficient conditions for a family of finite chains of prime ideals to be a collection of realizations of such template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}\).
In Sect. 4, we combine the methods of Sects. 2 and 3. We consider pairs \(({\mathscr {A}},{\mathfrak {F}})\) such that \({\mathscr {A}}\) is a thick tensor ideal and \({\mathfrak {F}}\) is a monoidal semifilter. Further, we assume that any nonempty increasing chain of ideals in the complement Open image in new window of \({\mathfrak {F}}\) has an upper bound in Open image in new window (see Definition 4.1), whence it follows that if \({\mathscr {A}}\notin {\mathfrak {F}}\), there always exists a prime ideal \({\mathscr {P}}\supseteq {\mathscr {A}}\) such that \({\mathscr {P}}\notin {\mathfrak {F}}\). We refer to such prime ideal \({\mathscr {P}}\) as a realization of \(({\mathscr {A}},{\mathfrak {F}})\). Accordingly, we can define realizations of templates \(\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}\) indexed by a partially ordered set I. As in Sect. 2, we define a relation \(\preccurlyeq \) among pairs such that Open image in new window if every prime ideal Open image in new window realizing Open image in new window contains a prime ideal \({\mathscr {P}}\) realizing \(({\mathscr {A}},{\mathfrak {F}})\). We show how to construct realizations of a template \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}\) indexed by a finite decreasing chain I by replacing it with a template Open image in new window that satisfies Open image in new window for each \(i\leqslant j\) in I. Again, under certain finiteness conditions, we construct a template \({\mathscr {D}}(T)=\{({\mathscr {B}}_i, {\mathfrak {G}}_i)\}_{i\in I}\) such that for any chosen \(i_0\in I\), we can start with a prime ideal \({\mathscr {P}}\) realizing \(({\mathscr {B}}_{i_0}, {\mathfrak {G}}_{i_0})\) and obtain a realization \(\{{\mathscr {P}}_i\}_{i\in I}\) of T such that \({\mathscr {P}}_{i_0}={\mathscr {P}}\). Towards the end of Sect. 4, we also construct realizations for templates indexed by finite descending trees.
Finally, in Sect. 5, we assume that all thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\) are radical, i.e., Open image in new window for all thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\). In fact, this property is quite common in examples of tensor triangulated categories (see [1, Remark 4.3]). For us, the key consequence of this assumption is that it implies Open image in new window for all thick tensor ideals \({\mathscr {I}}\) and \({\mathscr {J}}\). We then show that all thick tensor ideals being radical, every monoidal family \({\mathfrak {F}}\) of thick tensor ideals satisfies the Prime Ideal Principle, i.e., any ideal that is maximal with respect to not being in \({\mathfrak {F}}\) must be prime. Accordingly, we show that any ideal in \(({\mathscr {C}},\otimes ,1)\) that is maximal with respect to not being principal is prime. An analogous result holds for ideals that are maximal with respect to not being generated by a set of cardinality \(\leqslant \alpha \) for some infinite cardinal \(\alpha \). Thereafter, we formulate conditions for the construction of realizations of certain templates \(\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}\) indexed by an infinite decreasing chain I. We conclude by showing how we can construct families \({\mathfrak {F}}^*\) of closed subspaces of Open image in new window that are not closed under finite unions such that any closed subspace that is minimal with respect to not being in \({\mathfrak {F}}^*\) is irreducible. This is done with the help of Ako families of thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\).
We mention here that in [8], Balmer has proved a GoingUp Theorem in tensor triangular geometry with profound connections to Quillen stratification in modular representation theory. When \({\mathscr {C}}\) is idempotentcomplete, Balmer’s result (see [8, Section 1.5]) gives goingup and incomparability results for prime ideals in the spectra of categories of modules over ttrings in \(({\mathscr {C}},\otimes ,1)\). The ttrings are commutative ring objects in \(({\mathscr {C}},\otimes ,1)\) that are also separable in a suitable sense (see [8, Section 2] for details). As such, it is hoped that the methods in this paper can be developed in the future to study prime ideals in the spectra of categories of modules over ttrings, thus using tensor triangular geometry to obtain further connections between classical commutative algebra and modular representation theory. For more on ttrings in \(({\mathscr {C}},\otimes ,1)\), we also refer the reader to Balmer [7].
In this paper, we will always assume that our categories are essentially small. Further, by abuse of notation, for any category \({\mathscr {D}}\), we will always write \(x\in {\mathscr {D}}\) to mean that x is an object of \({\mathscr {D}}\).
2 Prime ideals in \(({\mathscr {C}},\otimes ,1)\) and realizations of pairs
Throughout this section and the rest of this paper, \(({\mathscr {C}},\otimes ,1)\) will be a symmetric monoidal category with \({\mathscr {C}}\) also having the structure of a triangulated category (see [37]). Further, we will always assume that the symmetric monoidal tensor product Open image in new window is exact in each variable and the category \({\mathscr {C}}\) contains all finite direct sums. We will say that \(({\mathscr {C}},\otimes ,1)\) is a tensor triangulated category. Further, a tensor triangulated functor \(F:({\mathscr {C}},\otimes ,1)\rightarrow ({\mathscr {D}},\otimes ,1)\) between tensor triangulated categories \({\mathscr {C}}\) and \({\mathscr {D}}\) will be an exact functor \(F:{\mathscr {C}}\rightarrow {\mathscr {D}}\) that preserves the symmetric monoidal product and carries the unit object in \({\mathscr {C}}\) to the unit object in \({\mathscr {D}}\). Unless otherwise mentioned, we do not require our tensor triangulated categories to satisfy the additional axioms due to May [27].
Given an object a in the triangulated category \({\mathscr {C}}\), we will denote the translation of a in \({\mathscr {C}}\) by \(\mathrm{T}a\). We now recall from Balmer [1] the notion of a prime ideal in the tensor triangulated category \(({\mathscr {C}},\otimes ,1)\).
Definition 2.1

The subcategory \({\mathscr {A}}\) is triangulated, i.e. if \(a\rightarrow b\rightarrow c\rightarrow \mathrm{T}a\) is a distinguished triangle in \({\mathscr {C}}\) and any two of a, b and c lie in \({\mathscr {A}}\), so does the third.

The subcategory \({\mathscr {A}}\) is thick, i.e., if \(a\in {\mathscr {A}}\) splits in \({\mathscr {C}}\) as a direct sum Open image in new window , both direct summands b and c lie in \({\mathscr {A}}\).

The subcategory \({\mathscr {A}}\) is a tensor ideal, i.e., if \(a\in {\mathscr {A}}\) and \(b\in {\mathscr {C}}\), then we must have Open image in new window .
A family \({\mathscr {S}}\) of objects of \({\mathscr {C}}\) will be said to be multiplicatively closed if \(1\in {\mathscr {S}}\) and for any \(a,b\in {\mathscr {S}}\), we have Open image in new window . We will work with pairs \(({\mathscr {A}},{\mathscr {S}})\), where \({\mathscr {A}}\) is a thick tensor ideal and \({\mathscr {S}}\) is a multiplicatively closed family of objects of \({\mathscr {C}}\). From [1, Lemma 2.2], we know that if \(({\mathscr {A}},{\mathscr {S}})\) is such pair with \({\mathscr {A}}\cap {\mathscr {S}}=\varnothing \), there always exists a prime ideal \({\mathscr {P}}\) such that \({\mathscr {A}}\subseteq {\mathscr {P}}\) and \({\mathscr {P}}\cap {\mathscr {S}}=\varnothing \). We start with a prime avoidance result for the category \(({\mathscr {C}},\otimes ,1)\).
Proposition 2.2
Let \({\mathscr {A}}\) be a thick tensor ideal of \(({\mathscr {C}},\otimes ,1)\) that is contained in the union Open image in new window of finitely many prime ideals \({\mathscr {P}}_i\). Then, there exists some \(1\leqslant i\leqslant n\) such that \({\mathscr {A}}\subseteq {\mathscr {P}}_i\).
Proof
We proceed by induction on n. The result is obvious for \(n=1\). We suppose that the result holds for all integers up to \(n1\) and consider some Open image in new window . Suppose that we can choose some object \(a_j\in {\mathscr {A}}\) for each \(1\leqslant j\leqslant n\) such that Open image in new window and Open image in new window . We consider the object Open image in new window (since \({\mathscr {A}}\) is triangulated, it is easy to check that it contains direct sums).
Now suppose that Open image in new window . Then, since Open image in new window is thick, we must have Open image in new window . However, this is impossible, since Open image in new window is prime and we have chosen Open image in new window for all \(1\leqslant i\leqslant n1\). On the other hand, if \(a\in {\mathscr {P}}_i\) for some \(1\leqslant i\leqslant n1\), it follows that \(a_n\in {\mathscr {P}}_i\) which is also a contradiction. This contradicts the fact that Open image in new window . Hence, it follows that the thick tensor ideal \({\mathscr {A}}\) is already contained in the union of some proper subcollection of \(\{{\mathscr {P}}_i\}_{1\leqslant i\leqslant n}\). Using the induction assumption, the result follows. \(\square \)
Analogously to the terminology of Bergman [15, Definition 7], we now introduce the following definition.
Definition 2.3
Let \(({\mathscr {C}},\otimes ,1)\) be a tensor triangulated category as given above. Let \(({\mathscr {A}}, {\mathscr {S}})\) be a pair such that \({\mathscr {A}}\) is a thick tensor ideal and \({\mathscr {S}}\) is a multiplicatively closed family of objects of \({\mathscr {C}}\). Then, a prime ideal \({\mathscr {P}}\) in \({\mathscr {C}}\) is said to be a realization of the pair \(({\mathscr {A}},{\mathscr {S}})\) if \({\mathscr {A}}\subseteq {\mathscr {P}}\) and \({\mathscr {P}}\cap {\mathscr {S}}=\varnothing \). The collection of all realizations of such a pair \(({\mathscr {A}},{\mathscr {S}})\) will be denoted by Open image in new window . Further, we let \({\mathscr {M}}{({\mathscr {A}},{\mathscr {S}})}\) be the multiplicatively closed family given by the complement of Open image in new window .
More generally, let \((I,\leqslant )\) be a partially ordered set. By a template T indexed by I, we will mean a family \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}\) of pairs indexed by I. Then, we will say that a collection \(\{{\mathscr {P}}_i\}_{i\in I}\) of prime ideals in \({\mathscr {C}}\) is a realization of the template T if each \({\mathscr {P}}_i\) realizes the pair \(({\mathscr {A}}_i,{\mathscr {S}}_i)\) and Open image in new window for every \(i\leqslant j\) in I.
Lemma 2.4
Let \({\mathscr {A}}\) be a thick tensor ideal and \({\mathscr {S}}\) be a multiplicatively closed family. Then, the full subcategory Open image in new window as defined in (2) determines a thick tensor ideal in \(({\mathscr {C}},\otimes ,1)\) containing \({\mathscr {A}}\).
Proof
We note here that, by Definition 2.1, every thick tensor ideal is a full subcategory of \({\mathscr {C}}\). Hence, Open image in new window . Further, given any thick tensor ideal \({\mathscr {A}}\) and a multiplicatively closed family of objects \({\mathscr {S}}\), it is clear that Open image in new window is also a multiplicatively closed family of objects. Given multiplicatively closed families Open image in new window , we can also form the multiplicatively closed family Open image in new window . In order to understand realizations of chains, we will need the following result.
Proposition 2.5
 (a)
A prime ideal \({\mathscr {P}}\) contains Open image in new window if and only if it contains a prime ideal realizing \(({\mathscr {A}},{\mathscr {S}})\).
 (b)
If a prime ideal \({\mathscr {P}}\) is contained in a prime ideal that realizes \(({\mathscr {A}},{\mathscr {S}})\), then \({\mathscr {P}}\) must be disjoint from Open image in new window .
 (c)
Suppose that Open image in new window is finite. Then, a prime ideal \({\mathscr {P}}\) is contained in a prime ideal realizing \(({\mathscr {A}},{\mathscr {S}})\) if and only if \({\mathscr {P}}\) is disjoint from \({\mathscr {M}}({\mathscr {A}},{\mathscr {S}})\).
Proof
(a) From the definition of Open image in new window it is clear that any prime ideal \({\mathscr {Q}}\) realizing \(({\mathscr {A}},{\mathscr {S}})\) also contains Open image in new window . Hence, so does any prime ideal containing \({\mathscr {Q}}\). Conversely, if \({\mathscr {P}}\) is a prime ideal containing Open image in new window , it follows that Open image in new window whence it follows that \({\mathscr {S}}({\mathscr {C}}{\mathscr {P}})\cap {\mathscr {A}}=\varnothing \). Accordingly, we can choose a prime ideal \({\mathscr {Q}}\) such that \({\mathscr {Q}}\cap {\mathscr {S}}({\mathscr {C}}{\mathscr {P}})=\varnothing \) and \({\mathscr {A}}\subseteq {\mathscr {Q}}\). Since \({\mathscr {Q}}\cap {\mathscr {S}}\subseteq {\mathscr {Q}}\cap {\mathscr {S}}({\mathscr {C}}{\mathscr {P}})=\varnothing \), we know that \({\mathscr {Q}}\) realizes \(({\mathscr {A}},{\mathscr {S}})\). Finally, since \({\mathscr {Q}}\cap ({\mathscr {C}}{\mathscr {P}})\subseteq {\mathscr {Q}}\cap {\mathscr {S}}({\mathscr {C}}{\mathscr {P}})=\varnothing \), it follows that \({\mathscr {Q}}\subseteq {\mathscr {P}}\).
(b) Suppose that \({\mathscr {Q}}\) realizes \(({\mathscr {A}},{\mathscr {S}})\) and let \(a\in {\mathscr {A}}\), \(s\in {\mathscr {S}}\) be such that Open image in new window . Since \({\mathscr {Q}}\) is thick, this implies that \(s\in {\mathscr {Q}}\) which is a contradiction. Hence, Open image in new window and hence any prime ideal contained in \({\mathscr {Q}}\) is also disjoint from Open image in new window .
(c) Suppose that \({\mathscr {P}}\subseteq {\mathscr {Q}}\) for some Open image in new window . Then, Open image in new window and hence \({\mathscr {P}}\cap {\mathscr {M}}({\mathscr {A}},{\mathscr {S}})=\varnothing \). Conversely, if \({\mathscr {P}}\cap {\mathscr {M}}({\mathscr {A}},{\mathscr {S}})=\varnothing \), then Open image in new window . However, since Open image in new window is finite, it follows from the prime avoidance result in Proposition 2.2 that \({\mathscr {P}}\subseteq {\mathscr {Q}}\) for some Open image in new window . \(\square \)
Definition 2.6
Let \(({\mathscr {A}},{\mathscr {S}})\) and Open image in new window be two pairs as in Definition 2.3. Then, we will say that Open image in new window if every prime ideal Open image in new window realizing the pair Open image in new window contains a prime ideal \({\mathscr {P}}\) realizing \(({\mathscr {A}},{\mathscr {S}})\).
Proposition 2.7
 (a)
The pairs are related as Open image in new window , i.e., any prime ideal realizing Open image in new window contains a prime ideal realizing \(({\mathscr {A}},{\mathscr {S}})\).
 (b)
The radical of Open image in new window is contained in the radical of Open image in new window , i.e., Open image in new window .
Proof
(a) \(\Rightarrow \) (b). Consider any prime ideal Open image in new window such that Open image in new window . Then, from Proposition 2.5, we know that Open image in new window contains a prime ideal Open image in new window realizing Open image in new window . By assumption, there exists a prime ideal Open image in new window such that \({\mathscr {P}}\) realizes \(({\mathscr {A}},{\mathscr {S}})\). Hence, Open image in new window and therefore Open image in new window for any prime ideal Open image in new window containing Open image in new window . It now follows that Open image in new window .\(\square \)
Remark 2.8
We note that \(\preccurlyeq \) is not a partial order relation. In particular, if Open image in new window and Open image in new window , we do get Open image in new window but not necessarily that Open image in new window . However, it is clear that \(\preccurlyeq \) is reflexive and transitive.
Proposition 2.9
 (a)
A chain Open image in new window is a realization of the template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\) if and only if it is also a realization of the template Open image in new window .
 (b)The template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\) has a realization if and only if \({\mathscr {B}}_1\cap {\mathscr {S}}_1=\varnothing \), i.e.
Proof
(a) Let Open image in new window be a realization of the template T. We know that \({\mathscr {B}}_n={\mathscr {A}}_n\) and hence Open image in new window realizes \((\mathscr {B}_n,{\mathscr {S}}_n)\). Now suppose that \({\mathscr {P}}_i\) realizes \(({\mathscr {B}}_i,{\mathscr {S}}_i)\) for each \(n\geqslant i>j\) for some given j. Then, since Open image in new window and Open image in new window realizes Open image in new window , it follows from Proposition 2.5 that Open image in new window . Since Open image in new window realizes Open image in new window , we already know that Open image in new window and Open image in new window . From (4), it follows that Open image in new window and Open image in new window , i.e., Open image in new window realizes the pair Open image in new window . This proves the result by induction.
Conversely, let Open image in new window be a realization of the template Open image in new window . Then, for each \(1\leqslant i\leqslant n\), we know that \({\mathscr {P}}_i\) realizes the pair \(({\mathscr {B}}_i,{\mathscr {S}}_i)\). From (4), it is clear that \({\mathscr {A}}_i\subseteq {\mathscr {B}}_i\) and hence \({\mathscr {P}}_i\) realizes the pair \(({\mathscr {A}}_i,{\mathscr {S}}_i)\).
Proposition 2.10
 (a)
A chain \({\mathscr {Q}}_n\subseteq \dots \subseteq {\mathscr {Q}}_2\subseteq {\mathscr {Q}}_1\) of prime ideals is a realization of the template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\) if and only if it is also a realization of the template \({\mathscr {D}}(T)=\{({\mathscr {B}}_i,{\mathscr {T}}_i)\}_{i\in I^\mathrm{op}}\), i.e., the templates T and \({\mathscr {D}}(T)\) are equivalent.
 (b)
Fix any integer \(j\in \{1,2,\dots ,n\}\). Then, the template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\) has a realization if and only if Open image in new window .
Proof
Conversely, let \({\mathscr {Q}}_n\subseteq \cdots \subseteq {\mathscr {Q}}_2\subseteq {\mathscr {Q}}_1\) be a realization of the template \({\mathscr {D}}(T)\). Then, for each \(1\leqslant i\leqslant n\), we know that \({\mathscr {Q}}_i\) realizes the pair \(({\mathscr {B}}_i,{\mathscr {T}}_i)\). From (6) and (7), it is clear that each \({\mathscr {A}}_i\subseteq {\mathscr {B}}_i\) and \({\mathscr {S}}_i\subseteq {\mathscr {T}}_i\). Hence, each \({\mathscr {Q}}_i\) realizes the pair \(({\mathscr {A}}_i,{\mathscr {S}}_i)\).
The next result will explain what kinds of collections of prime ideals may arise as Open image in new window for some pair \(({\mathscr {A}},{\mathscr {S}})\).
Proposition 2.11
 (a)
The family Open image in new window for some thick tensor ideal \({\mathscr {A}}\) and some multiplicatively closed family of objects \({\mathscr {S}}\).
 (b)The family \({\mathfrak {X}}\) satisfies the following property: given a prime ideal \({\mathscr {Q}}\) such that then \({\mathscr {Q}}\in {\mathfrak {X}}\).
Proof
(a) \(\Rightarrow \) (b). Since each \({\mathscr {P}}\in {\mathfrak {X}}\) realizes the pair \(({\mathscr {A}}, {\mathscr {S}})\), we have \({\mathscr {A}}\subseteq \bigcap _{{\mathscr {P}}\in {\mathfrak {X}}}{\mathscr {P}}\) and \({\mathscr {S}}\cap \bigcup _{{\mathscr {P}}\in {\mathfrak {X}}}{\mathscr {P}}=\varnothing \). Hence, if a prime ideal \({\mathscr {Q}}\) satisfies the condition in (9), then \({\mathscr {Q}}\) realizes \(({\mathscr {A}},{\mathscr {S}})\), i.e., Open image in new window .
A collection \({\mathfrak {X}}\) of prime ideals of \(({\mathscr {C}},\otimes ,1)\) satisfying condition (b) in Proposition 2.11 will be referred to as a convex set. Under the finiteness conditions from Proposition 2.10, we will now characterize the families of chains of prime ideals that realize a given finite chain template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\).
Proposition 2.12
 (a)
\({\mathfrak {X}}\) is a collection of realizations of a finite chain template \(T=\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I^\mathrm{op}}\) indexed by the opposite \(I^\mathrm{op}\) of the ordered set \(I=\{1<2<\cdots <n\}\) such that each Open image in new window is a finite set.
 (b)For each \(1\leqslant j\leqslant n\), \({\mathfrak {X}}_j\) is a finite convex set of prime ideals of \(({\mathscr {C}},\otimes ,1)\). Further, \({\mathfrak {X}}\) consists of all chains of prime ideals whose ith element is in \({\mathfrak {X}}_i\) for each \(1\leqslant i\leqslant n\). In other words, we have
Proof
We will say that a multiplicatively closed family \({\mathscr {S}}\) is finitely generated if there exists a finite set Open image in new window of objects of \({\mathscr {C}}\) such that \({\mathscr {S}}\) is the smallest multiplicatively closed family containing all objects in Open image in new window .
Proposition 2.13
 (a)
Given a finitely generated (hence principal) thick tensor ideal \({\mathscr {A}}\) and a finitely generated multiplicatively closed family \({\mathscr {S}}\), Open image in new window is a constructible subset of Open image in new window .
 (b)
Every convex subset of Open image in new window is proconstructible, i.e., it may be expressed as an intersection of a family of constructible subsets of Open image in new window .
 (c)
For any thick tensor ideal \({\mathscr {A}}\) and any multiplicatively closed family \({\mathscr {S}}\), the subset Open image in new window is a spectral space, i.e., Open image in new window is quasicompact, quasiseparated, has a basis of quasicompact open subsets and every nonempty irreducible closed subset has a unique generic point.
Proof
(a) Since \({\mathscr {S}}\) is finitely generated, we can choose a finite set of objects Open image in new window such that \({\mathscr {S}}\) is the smallest multiplicatively closed family containing all objects in Open image in new window . Then, it is clear that for any object \(s\in {\mathscr {S}}\), we can choose nonnegative integers \(e_1,e_2, \dots , e_k\) such that Open image in new window . We now set \(s_0=\bigotimes _{i=1}^k s_i\) and see that \(Z({\mathscr {S}})=Z(s_0)\). Then, Open image in new window is quasicompact and open in Open image in new window . Further, if \({\mathscr {A}}\) is generated by the object \(a\in {\mathscr {C}}\), it is clear that we may express Open image in new window . Hence, Open image in new window is constructible.
(b) From Proposition 2.11, we know that each convex subset of Open image in new window is of the form Open image in new window for some thick tensor ideal \({\mathscr {A}}\) and some multiplicatively closed family \({\mathscr {S}}\). We now express Open image in new window . Since each \(U(a)\cap Z(s)\) is constructible, it follows that the intersection Open image in new window is proconstructible.
(c) follows from the fact that a proconstructible subspace of a spectral space is always spectral in the induced subspace topology (see, for instance, [32, Section 2]).\(\square \)
For the final result of this section, we will restrict ourselves to tensor triangulated categories that are topologically noetherian (see [2, Definition 3.13]), i.e., Open image in new window is a noetherian topological space. This happens, for instance, when X is a topologically noetherian scheme and Open image in new window , i.e., the derived category of perfect complexes over X (see [1, Corollary 5.6]). We denote by Open image in new window the space Open image in new window equipped with the constructible topology. From Proposition 2.13 we know that for any thick tensor ideal \({\mathscr {A}}\) and any multiplicatively closed family \({\mathscr {S}}\), Open image in new window is proconstructible. Equivalently, since Open image in new window is a spectral space, Open image in new window is closed in Open image in new window .
Corollary 2.14
 (a)
Given a finitely generated (hence principal) thick tensor ideal \({\mathscr {A}}\) and a multiplicatively closed family \({\mathscr {S}}\), Open image in new window is a constructible subset of Open image in new window . Further, such subsets form a basis for the constructible topology on Open image in new window .
 (b)
Any closed subset of Open image in new window may be expressed as a union Open image in new window with each \({\mathscr {A}}_i\) a thick tensor ideal and each \({\mathscr {S}}_i\) a multiplicatively closed family.
Proof
(b) Since Open image in new window is spectral, a subset of Open image in new window is closed if and only if it is proconstructible, i.e., it is the intersection of a family of constructible subsets. Let us consider a family Open image in new window , \(j\in J\), with each Open image in new window a constructible set. Let \({\mathscr {B}}=\sum _{j\in J} {\mathscr {A}}_j\) be the smallest ideal containing each of the ideals \({\mathscr {A}}_j\) and let \({\mathscr {T}}\) be the smallest multiplicatively closed family containing each of the families Open image in new window . Then, it is clear that Open image in new window . Combining with the fact (from part (a)) that any constructible subset may be expressed as a union of constructible sets of the form Open image in new window , we obtain the result. \(\square \)
For a topologically noetherian scheme, we know that there is a homeomorphism Open image in new window of X with the spectrum of the derived category of perfect complexes (see [1, Corollary 5.6]). Then, Corollary 2.14 gives us an understanding of the constructible topology on such a scheme in terms of the subsets Open image in new window for the tensor triangulated category Open image in new window .
3 Oka families and a Prime Ideal Principle for tensor triangulated categories
Accordingly, we turn to some methods from commutative algebra, where there are several well known results of the kind “maximal implies prime”. For example, given a commutative ring R and an Rmodule M, an ideal I that is maximal among annihilators of nonzero elements of M must be prime (see, for example, [20, Proposition 3.12]). In [25], Lam and Reyes gave a criterion that unifies these results, i.e., conditions on a family \({\mathscr {F}}\) of ideals in a ring such that any ideal that is maximal with respect to not being contained in \({\mathscr {F}}\) must be prime. They referred to this as the “Prime Ideal Principle”. Using the Prime Ideal Principle, the authors in [25] were also able to uncover several new results of a similar nature (see also further work in Lam and Reyes [26] and Reyes [29, 30]). The purpose of this section is to construct an analogous Prime Ideal Principle for thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\).
Further, given a thick tensor ideal \({\mathscr {I}}\) and an object \(a\in {\mathscr {C}}\), we will denote by Open image in new window the smallest thick tensor ideal containing all objects Open image in new window , where \(x\in {\mathscr {I}}\). Similarly, given thick tensor ideals \({\mathscr {I}}_1,{\mathscr {I}}_2\), we denote by Open image in new window the smallest thick tensor ideal containing all objects Open image in new window , where \(x_1\in {\mathscr {I}}_1\) and \(x_2\in {\mathscr {I}}_2\). In a manner analogous to [25], we will now define Oka families and Ako families of ideals in \(({\mathscr {C}},\otimes ,1)\).
Definition 3.1

\({\mathfrak {F}}\) is a semifilter if \({\mathscr {I}}\subseteq {\mathscr {J}}\) and \({\mathscr {I}}\in {\mathfrak {F}}\) implies that \({\mathscr {J}}\in {\mathfrak {F}}\).

\({\mathfrak {F}}\) is a filter if it is a semifilter and for any ideals \({\mathscr {I}},{\mathscr {J}}\in {\mathfrak {F}}\), the intersection \({\mathscr {I}}\cap {\mathscr {J}}\in {\mathfrak {F}}\).

\({\mathfrak {F}}\) is monoidal if \({\mathscr {I}}\), \({\mathscr {J}}\in {\mathfrak {F}}\) implies that Open image in new window .

\({\mathfrak {F}}\) is an Oka family (resp. a strongly Oka family) if Open image in new window for some object \(a\in {\mathscr {C}}\) (resp. Open image in new window , Open image in new window for some ideal Open image in new window ) implies that \({\mathscr {I}}\in {\mathfrak {F}}\).

\({\mathfrak {F}}\) is an Ako family (resp. a strongly Ako family) if Open image in new window for objects \(a,b\in {\mathscr {C}}\) (resp. Open image in new window for some object \(a\in {\mathscr {C}}\) and some ideal Open image in new window ) implies that Open image in new window (resp. Open image in new window ).
We will say that a family \({\mathfrak {F}}\) of thick tensor ideals satisfies the “Prime Ideal Principle” if any ideal that is maximal with respect to not being in \({\mathfrak {F}}\) is also prime. We will now prove the main Prime Ideal Principle for ideals in \(({\mathscr {C}},\otimes ,1)\).
Proposition 3.2
 (a)
If \({\mathfrak {F}}\) is an Oka family of ideals, then \({\mathscr {I}}\) is a prime ideal.
 (b)
If \({\mathfrak {F}}\) is an Ako family of ideals, then \({\mathscr {I}}\) is a prime ideal.
 (c)
In other words, if \({\mathfrak {F}}\) is either an Oka family or an Ako family, \({\mathfrak {F}}\) satisfies the “Prime Ideal Principle”, i.e., any ideal that is maximal with respect to not being in \({\mathfrak {F}}\) must be prime.
Proof
(a) We know that \({\mathfrak {F}}\) is an Oka family. Suppose that \({\mathscr {I}}\) is not a prime ideal, i.e., we can choose \(a,b\in {\mathscr {C}}\) such that Open image in new window but \(a\notin {\mathscr {I}}\) and \(b\notin {\mathscr {I}}\). Then, we note that Open image in new window because the latter contains b and Open image in new window because \(a\notin {\mathscr {I}}\). However, since \({\mathscr {I}}\) is maximal with respect to not being contained in \({\mathfrak {F}}\), we must have Open image in new window and Open image in new window . Since \({\mathfrak {F}}\) is an Oka family, we conclude that \({\mathscr {I}}\in {\mathfrak {F}}\), which is a contradiction.
(b) Again, we suppose that \({\mathscr {I}}\) is not prime and choose \(a,b\in {\mathscr {C}}\) such that Open image in new window but \(a\notin {\mathscr {I}}\) and \(b\notin {\mathscr {I}}\). As in part (a), we see that Open image in new window and Open image in new window because \(a,b\notin {\mathscr {I}}\). Then, \({\mathscr {I}}\) being maximal with respect to not being contained in \({\mathfrak {F}}\), we must have Open image in new window . Since \({\mathfrak {F}}\) is an Ako family, this implies that Open image in new window . But since Open image in new window , we have Open image in new window , which is a contradiction. \(\square \)
Proposition 3.3
Proof
We remark here that, in particular, if we apply the explicit description in Proposition 3.3 to the case of a thick tensor ideal generated by a single object, it follows from Definition 3.1 that a strongly Oka family is also an Oka family.
Lemma 3.4
Let \({\mathscr {I}}\) (resp. \({\mathscr {J}})\) be a thick tensor ideal of \(({\mathscr {C}},\otimes ,1)\) that is generated by Open image in new window (resp. \(Y=\{y_j\}_{j\in J})\). Then, the thick tensor ideal Open image in new window is generated by the collection Open image in new window .
Proof
We remark here that it follows from the proof of Lemma 3.4 and Definition 3.1 that a strongly Ako family is also an Ako family.
Theorem 3.5
 (P1)
\({\mathfrak {F}}\) is a monoidal filter.
 (P2)
\({\mathfrak {F}}\) is monoidal and, given thick tensor ideals \({\mathscr {I}},{\mathscr {J}}\) with \({\mathscr {J}}\in {\mathfrak {F}}\) and Open image in new window , we must have \({\mathscr {I}}\in {\mathfrak {F}}\).
 (P3)
For thick tensor ideals \({\mathscr {I}},{\mathscr {A}},{\mathscr {B}}\) such that Open image in new window , we must have Open image in new window .
 (Q1)
\({\mathfrak {F}}\) is a monoidal semifilter.
 (Q2)
\({\mathfrak {F}}\) is monoidal and, given thick tensor ideals \({\mathscr {I}},{\mathscr {J}}\) with \({\mathscr {J}}\in {\mathfrak {F}}\) and \({\mathscr {I}}\supseteq {\mathscr {J}}\supseteq {\mathscr {I}}^n\) for some \(n>1\), we must have \({\mathscr {I}}\in {\mathfrak {F}}\).
 (Q3)
If \({\mathscr {A}},{\mathscr {B}}\in {\mathfrak {F}}\) and \({\mathscr {I}}\) is a thick tensor ideal such that Open image in new window , we must have \({\mathscr {I}}\in {\mathfrak {F}}\).
 (a)The following chart of implications holds:
 (b)We have the following implications: In particular, a family \({\mathfrak {F}}\) satisfying condition (P3) also satisfies the Prime Ideal Principle, i.e., any ideal maximal with respect to not being in \({\mathfrak {F}}\) must be prime.
Proof
(a) For thick tensor ideals \({\mathscr {I}},{\mathscr {J}}\), it is clear that Open image in new window and hence it follows from Definition 3.1 that (P1) \(\Leftrightarrow \) (Q1). Further, it is obvious that (Q2) \(\Rightarrow \) (P2).
(Q1) \(\Rightarrow \) (Q2). Since \({\mathfrak {F}}\) is a semifilter, whenever we have \({\mathscr {I}}\supseteq {\mathscr {J}}\) with \({\mathscr {J}} \in {\mathfrak {F}}\), we see that \({\mathscr {I}}\in {\mathfrak {F}}\).
(P3) \(\Rightarrow \) (Q3). Let \({\mathscr {A}},{\mathscr {B}}\in {\mathfrak {F}}\) and suppose that Open image in new window . Then, Open image in new window and Open image in new window and hence it follows from (P3) that Open image in new window . But since Open image in new window , we have Open image in new window .
(Q3) \(\Rightarrow \) (P3). Suppose that Open image in new window . It is clear that we have Open image in new window . As in (17), we can show that Open image in new window . From condition (Q3), it now follows that Open image in new window .
(b) We assume condition (P3). To show that \({\mathfrak {F}}\) is strongly Ako, we consider \(a\in {\mathscr {C}}\) and thick tensor ideals \({\mathscr {I}},{\mathscr {B}}\) such that Open image in new window . Let \({\mathscr {A}}\) be the thick tensor ideal generated by a. Then, Open image in new window lies in \({\mathfrak {F}}\) and it follows from (P3) that Open image in new window . From Lemma 3.4, it is clear that Open image in new window . Hence, Open image in new window and \({\mathfrak {F}}\) is strongly Ako.
In order to show that \({\mathfrak {F}}\) is strongly Oka, we consider ideals \({\mathscr {I}},{\mathscr {A}}\) with Open image in new window , Open image in new window . We now set Open image in new window . Then, it is clear that Open image in new window and it follows from condition (P3) that Open image in new window . From the definition in (12), we know that if \(a\in {\mathscr {A}}\) and Open image in new window , we must have Open image in new window and hence Open image in new window . It follows that Open image in new window and \({\mathfrak {F}}\) is strongly Oka. We have also noted before that strongly Oka families are also Oka and hence it follows from Proposition 3.2 that \({\mathfrak {F}}\) satisfies the Prime Ideal Principle.
It remains to show that strongly Ako families are also Oka. Let \({\mathfrak {F}}\) be strongly Ako and suppose that Open image in new window . We set Open image in new window . Again since Open image in new window , we see that Open image in new window . But Open image in new window and hence Open image in new window and hence \({\mathfrak {F}}\) is Oka. \(\square \)
For the rest of this section, we shall construct various families of thick tensor ideals that satisfy the Prime Ideal Principle.
Proposition 3.6
Proof
We will show that \({\mathfrak {F}}\) satisfies condition (Q3) in Theorem 3.5. By Theorem 3.5 (b), condition (P3) which is equivalent to (Q3) will then imply that \({\mathfrak {F}}\) is a strongly Oka and a strongly Ako family.
Corollary 3.7
 (a)Let \({\mathscr {J}}\), \({\mathscr {K}}\) be thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\). Consider the following family of ideals: Then, \({\mathfrak {F}}\) satisfies the Prime Ideal Principle, i.e., any thick tensor ideal that is maximal with respect to not being in \({\mathfrak {F}}\) must be prime.
 (b)
Let \(\{{\mathscr {J}}_j\}_{j\in J}\) be a family of thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\). Then, any ideal that is maximal with respect to not containing a finite product of Open image in new window is prime.
Proof
(a) We set Open image in new window and \({\mathfrak {F}}_2=\{{\mathscr {K}}\}\). Then, \({\mathfrak {F}}_1\) is monoidal and \({\mathfrak {F}}_2\) is closed under finite intersections. Now applying Proposition 3.6, we see that \({\mathfrak {F}}\) satisfies the Prime Ideal Principle.
(b) We set Open image in new window and \({\mathfrak {F}}_2=\{{\mathscr {C}}\}\). It is clear that \({\mathfrak {F}}_1\) is monoidal and \({\mathfrak {F}}_2\) is closed under finite intersections. The result now follows from Proposition 3.6. \(\square \)
Proposition 3.8
 (a)
A thick tensor ideal that is maximal with respect to being disjoint from \({\mathscr {S}}\) is also prime.
 (b)
A thick tensor ideal that is maximal among ideals \({\mathscr {I}}\) satisfying Open image in new window is also prime.
 (c)
Let \({\mathfrak {F}}\) be a monoidal semifilter. Then, a thick tensor ideal that is maximal among ideals \({\mathscr {I}}\) satisfying Open image in new window is also prime.
Proof
(b) follows by applying the result of (c) with \({\mathfrak {F}}={\mathfrak {F}}_{{\mathscr {S}}}\). \(\square \)
Remark 3.9
We mention here that the result of part (a) of Proposition 3.8 is already known as a special case of [1, Lemma 2.2].
Given an object \(m\in {\mathscr {M}}\), we let the annihilator Open image in new window be the collection of all objects \(a\in {\mathscr {C}}\) such that Open image in new window . Given that the action Open image in new window in (19) is exact in both variables, it is clear that Open image in new window is actually a thick tensor ideal.
Proposition 3.10
 (a)Let \({\mathscr {S}}\) be a multiplicatively closed family of objects in \(({\mathscr {C}},\otimes ,1)\). Consider the following family of thick tensor ideals: Then, \({\mathfrak {F}}\) is a strongly Ako semifilter. In particular, the family \({\mathfrak {F}}\) satisfies the Prime Ideal Principle.
 (b)
A thick tensor ideal of \(({\mathscr {C}},\otimes ,1)\) that is maximal among the annihilators of nonzero objects of \({\mathscr {M}}\) is also prime.
Proof
(a) It is immediate from (20) that \({\mathfrak {F}}\) is a semifilter. To show that \({\mathfrak {F}}\) is strongly Ako, we choose thick tensor ideals \({\mathscr {I}}\), Open image in new window and some object \(a\in {\mathscr {C}}\) such that Open image in new window , Open image in new window . Suppose that Open image in new window for some \(m\in {\mathscr {M}}\). Then, Open image in new window and hence Open image in new window . Since Open image in new window , we conclude that there exists some \(s\in {\mathscr {S}}\) such that Open image in new window . It follows that Open image in new window .
On the other hand, since Open image in new window , we have Open image in new window . Then, Open image in new window . Since Open image in new window , it follows that there exists Open image in new window such that Open image in new window . Finally, since \({\mathscr {S}}\) is multiplicatively closed, we know that Open image in new window . This shows that Open image in new window and hence \({\mathfrak {F}}\) is strongly Ako. In particular, it now follows from Theorem 3.5 (b) that \({\mathfrak {F}}\) satisfies the Prime Ideal Principle.
Analogously to the usual definition in commutative algebra, we will say that a thick tensor ideal \({\mathscr {I}}\) in \(({\mathscr {C}},\otimes ,1)\) is essential if it has nontrivial intersection with every nonzero thick tensor ideal in \(({\mathscr {C}},\otimes ,1)\). Further, as in [1, Corollary 2.4], an object \(a\in {\mathscr {C}}\) will be called \(\otimes \)nilpotent if there exists an integer \(n>0\) such that Open image in new window . As such, we will say that \(({\mathscr {C}},\otimes ,1)\) is \(\otimes \)reduced if it has no nonzero \(\otimes \)nilpotent objects. We now have the following result.
Proposition 3.11
Let \(({\mathscr {C}},\otimes ,1)\) be a tensor triangulated category and let \({\mathfrak {F}}\) be the family of essential thick tensor ideals of \(({\mathscr {C}},\otimes ,1)\). Then, if \(({\mathscr {C}},\otimes ,1)\) is \(\otimes \)reduced, \({\mathfrak {F}}\) is a monoidal semifilter. In particular, a thick tensor ideal of \(({\mathscr {C}},\otimes ,1)\) that is maximal with respect to not being essential must be prime.
Proof
If \({\mathscr {I}}\in {\mathfrak {F}}\) is an essential ideal and \({\mathscr {J}}\) is a thick tensor ideal containing \({\mathscr {I}}\), it is clear that \({\mathscr {J}}\) is also essential. Hence, \({\mathfrak {F}}\) is a semifilter.
We now suppose that \({\mathscr {I}}_1,{\mathscr {I}}_2\in {\mathfrak {F}}\) and consider some nonzero thick tensor ideal \({\mathscr {A}}\). Since \({\mathscr {I}}_1\) is essential, we can choose \(0\ne x\in {\mathscr {I}}_1\cap {\mathscr {A}}\). We now consider the thick tensor ideal (x) generated by x. Now since \({\mathscr {I}}_2\) is essential, we may choose \(0\ne y\in (x)\cap {\mathscr {I}}_2\) and consider Open image in new window . Since \(({\mathscr {C}},\otimes ,1)\) is \(\otimes \)reduced, it follows that Open image in new window . This shows that Open image in new window is essential and hence \({\mathfrak {F}}\) is a monoidal semifilter. It now follows from Theorem 3.5 that any ideal that is maximal with respect to not being essential is also prime. \(\square \)
Proposition 3.12
Let \(({\mathscr {C}},\otimes ,1)\) be a tensor triangulated category. Then, an ideal that is maximal among thick tensor ideals \({\mathscr {I}}\) satisfying Open image in new window for each \(n\geqslant 0\) must be prime.
Proof
4 Monoidal semifilters of ideals and realizations of pairs
Definition 4.1

\({\mathscr {A}}\) is a thick tensor ideal in \(({\mathscr {C}},\otimes ,1)\).

\({\mathfrak {F}}\) is a family of thick tensor ideals that is a monoidal semifilter.

Let Open image in new window denote the collection of all thick tensor ideals of \(({\mathscr {C}},\otimes ,1)\) not contained in \({\mathfrak {F}}\). Then, every nonempty increasing chain of ideals in Open image in new window has an upper bound in Open image in new window .
More generally, let \((I,\leqslant )\) be a partially ordered set and let \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}\) be a collection of such pairs indexed by I. Then, we will say that a collection \(\{{\mathscr {P}}_i\}_{i\in I}\) of prime ideals in \({\mathscr {C}}\) is a realization of the template T if each \({\mathscr {P}}_i\) realizes the pair \(({\mathscr {A}}_i,{\mathfrak {F}}_i)\) and Open image in new window for every \(i\leqslant j\) in I.
Henceforth, we will only consider pairs \(({\mathscr {A}},{\mathfrak {F}})\) as in Definition 4.1. Further, since \({\mathfrak {F}}\) is a semifilter, it is clear that if there exists a prime ideal \({\mathscr {P}}\) realizing a pair \(({\mathscr {A}},{\mathfrak {F}})\), we must have \({\mathscr {A}}\notin {\mathfrak {F}}\).
Proposition 4.2
Let \(({\mathscr {A}},{\mathfrak {F}})\) be a pair as in Definition 4.1. Suppose that \({\mathscr {A}}\notin {\mathfrak {F}}\). Then, there always exists a prime ideal in \(({\mathscr {C}},\otimes ,1)\) that realizes the pair \(({\mathscr {A}},{\mathfrak {F}})\).
Proof
From Theorem 3.5, we know that a thick tensor ideal that is maximal with respect to not being in the monoidal semifilter \({\mathfrak {F}}\) must be prime. Further, since every increasing chain of ideals in Open image in new window has an upper bound in Open image in new window , given that \({\mathscr {A}}\notin {\mathfrak {F}}\), it follows from Zorn’s lemma that \({\mathscr {A}}\) must be contained in a thick tensor ideal \({\mathscr {P}}\) that is maximal with respect to not being in \({\mathfrak {F}}\). Then, \({\mathscr {P}}\) is a prime ideal realizing the pair \(({\mathscr {A}},{\mathfrak {F}})\).\(\square \)
Remark 4.3
In particular, let \({\mathscr {S}}\) be a multiplicatively closed family of objects of \({\mathscr {C}}\) and set Open image in new window as in (22). It is clear that the union of any chain of ideals in the complement Open image in new window of \({\mathfrak {F}}_{{\mathscr {S}}}\) lies in Open image in new window . Then, if \({\mathscr {A}}\) is a thick tensor ideal such that \({\mathscr {A}}\cap {\mathscr {S}}=\varnothing \), i.e., \({\mathscr {A}}\notin {\mathfrak {F}}_{{\mathscr {S}}}\), it follows from Proposition 4.2 that we can find a prime ideal \({\mathscr {P}}\) realizing the pair \(({\mathscr {A}},{\mathfrak {F}}_{{\mathscr {S}}})\). This allows us to recover [1, Lemma 2.2] as a special case.
Lemma 4.4
Proof
Lemma 4.5
Proof
Proposition 4.6
 (a)
Let \(({\mathscr {A}},{\mathfrak {F}})\) be a pair and let \({\mathscr {Q}}\) be any prime ideal in \(({\mathscr {C}},\otimes ,1)\). Then, \({\mathscr {Q}}\) contains a prime ideal realizing the pair \(({\mathscr {A}},{\mathfrak {F}})\) if and only if Open image in new window .
 (b)
Let \(({\mathscr {A}},{\mathfrak {F}})\) be a pair such that Open image in new window is finite. Then, a prime ideal \({\mathscr {P}}\) is contained in a prime ideal realizing \(({\mathscr {A}},{\mathfrak {F}})\) if and only if \({\mathscr {P}}\) is disjoint from \({\mathscr {M}}({\mathscr {A}},{\mathfrak {F}})\).
Proof
(a) Let \({\mathscr {P}}\) be a prime ideal realizing the pair \(({\mathscr {A}},{\mathfrak {F}})\). Choose any Open image in new window and some \({\mathscr {I}}\in {\mathfrak {F}}\) such that Open image in new window . Then, Open image in new window . Suppose that \(a\notin {\mathscr {P}}\). Then, for any \(x\in {\mathscr {I}}\), we have Open image in new window and hence \(x\in {\mathscr {P}}\). Hence, \({\mathscr {I}}\subseteq {\mathscr {P}}\). Since \({\mathfrak {F}}\) is a semifilter and \({\mathscr {I}}\in {\mathfrak {F}}\), it now follows that \({\mathscr {P}}\in {\mathfrak {F}}\) which is a contradiction. Hence, Open image in new window is contained in \({\mathscr {P}}\) and hence in any prime ideal \({\mathscr {Q}}\) containing \({\mathscr {P}}\).
Conversely, suppose that Open image in new window . We claim that Open image in new window . Otherwise, there exists \({\mathscr {I}}\in {\mathfrak {F}}\) and \(s\notin {\mathscr {Q}}\) such that Open image in new window , i.e., Open image in new window which is a contradiction. Further, from Lemma 4.5, we know that any increasing chain of ideals in Open image in new window has an upper bound in Open image in new window . Accordingly, we choose a prime ideal \({\mathscr {P}}\) realizing the pair Open image in new window . Now suppose that there exists some \(x\in {\mathscr {P}}\cap ({\mathscr {C}}{\mathscr {Q}})\) and take any Open image in new window . Then, Open image in new window and hence Open image in new window , which is a contradiction. Hence, \({\mathscr {P}}\subseteq {\mathscr {Q}}\).
(b) Suppose that we have prime ideals \({\mathscr {P}}\subseteq {\mathscr {Q}}\) such that \({\mathscr {Q}}\) realizes \(({\mathscr {A}},{\mathfrak {F}})\). Then, Open image in new window and hence \({\mathscr {P}}\cap {\mathscr {M}}({\mathscr {A}},{\mathfrak {F}})=\varnothing \). On the other hand, suppose that \({\mathscr {P}}\cap {\mathscr {M}}({\mathscr {A}},{\mathfrak {F}})=\varnothing \), i.e., Open image in new window . Since Open image in new window is finite, it follows from Proposition 2.2 that \({\mathscr {P}}\subseteq {\mathscr {Q}}\) for some Open image in new window . \(\square \)
Definition 4.7
Let \(({\mathscr {A}},{\mathfrak {F}})\) and Open image in new window be two pairs as in Definition 4.1. Then, we will say that Open image in new window if every prime ideal Open image in new window realizing the pair Open image in new window contains a prime ideal \({\mathscr {P}}\) that realizes \(({\mathscr {A}},{\mathfrak {F}})\).
Proposition 4.8
 (a)
We have Open image in new window , i.e., any prime ideal that realizes Open image in new window contains a prime ideal that realizes \(({\mathscr {A}},{\mathfrak {F}})\).
 (b)
The radical of Open image in new window is contained in the radical of Open image in new window , i.e., Open image in new window .
Proof
(a) \(\Rightarrow \) (b). Consider any prime ideal Open image in new window such that Open image in new window (if there is no such prime ideal Open image in new window , then Open image in new window and we are done). Then, from Proposition 4.6, we know that Open image in new window contains a prime ideal Open image in new window realizing Open image in new window . By assumption, there exists a prime ideal Open image in new window such that \({\mathscr {P}}\) realizes \(({\mathscr {A}},{\mathfrak {F}})\). Hence, Open image in new window and therefore Open image in new window for any prime ideal Open image in new window containing Open image in new window . It now follows that Open image in new window . \(\square \)
Proposition 4.9
 (a)
A chain Open image in new window is a realization of the template \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I^\mathrm{op}}\) if and only if it is also a realization of the template Open image in new window .
 (b)The template \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I^\mathrm{op}}\) has a realization if and only if \({\mathscr {B}}_1\notin {\mathfrak {F}}_1\), i.e.,
Proof
(a) Suppose that Open image in new window is a realization of the template Open image in new window . From (29), it is clear that each \( {\mathscr {A}}_i\subseteq {\mathscr {B}}_i\) and hence \({\mathscr {P}}_i\) realizes the pair \(({\mathscr {A}}_i,{\mathfrak {F}}_i)\). Hence, Open image in new window is a realization of T.
Conversely, suppose that Open image in new window is a realization of T. By definition, we know that \({\mathscr {B}}_n={\mathscr {A}}_n\) and hence Open image in new window . Suppose that \({\mathscr {B}}_i\subseteq {\mathscr {P}}_i\) for all \(n\geqslant i>j\) for some fixed j. Since Open image in new window and \({\mathscr {P}}_{j+1}\) realizes Open image in new window , it follows from Proposition 4.6 (a) that Open image in new window . Further, \({\mathscr {A}}_j\subseteq {\mathscr {P}}_{j}\) because Open image in new window realizes Open image in new window . Hence, Open image in new window and Open image in new window realizes the pair Open image in new window . Hence, Open image in new window becomes a realization of Open image in new window .
Conversely, suppose that the template \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I^\mathrm{op}}\) is realizable. From part (a), it follows that the template Open image in new window is also realizable and in particular this means that the pair \(({\mathscr {B}}_1,{\mathfrak {F}}_1)\) is realizable. Hence, we must have \({\mathscr {B}}_1\notin {\mathfrak {F}}_1\). \(\square \)
Let \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I^\mathrm{op}}\) be a template as above. We will now show that under certain finiteness conditions, we may construct a template \({\mathscr {D}}(T)=\{({\mathscr {B}}_i, {\mathfrak {G}}_i)\}_{i\in I^\mathrm{op}}\) equivalent to T such that if we start with an arbitrary realization \({\mathscr {Q}}_j\) of some pair Open image in new window , we can expand it in both directions to form a realization \({\mathscr {Q}}_n\subseteq \dots \subseteq {\mathscr {Q}}_j\subseteq \dots \subseteq {\mathscr {Q}}_1\) of \({\mathscr {D}}(T)\). We notice that since \({\mathscr {D}}(T)\) is equivalent to T, the latter becomes a realization of T.
Proposition 4.10
 (a)
The templates T and \({\mathscr {D}}(T)\) are equivalent.
 (b)
Choose any integer \(j\in \{1,2,\dots ,n\}\). Then, the template \(T=\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I^\mathrm{op}}\) has a realization if and only if Open image in new window .
Proof
(a) Suppose that \({\mathscr {Q}}_n\subseteq \dots \subseteq {\mathscr {Q}}_2\subseteq {\mathscr {Q}}_1\) is a realization of \({\mathscr {D}}(T)\). From (30) and (31), it is clear that each \({\mathscr {A}}_i\subseteq {\mathscr {B}}_i\) and \({\mathfrak {F}}_i\subseteq {\mathfrak {G}}_i\). Since each \({\mathscr {Q}}_i\) realizes the pair \(({\mathscr {B}}_i,{\mathfrak {G}}_i)\), we see that it also realizes \(({\mathscr {A}}_i,{\mathfrak {F}}_i)\).
Remark 4.11
In the proof above, we note that the finiteness condition is only used in part (b), i.e., the templates T and \({\mathscr {D}}(T)\) are always equivalent.
Proposition 4.12
Proof
“If part”: we claim that starting with any prime ideal realizing \(({\mathscr {B}}_{(1)},{\mathfrak {F}}_{(1)})\), we can obtain a realization of the entire template \(\{({\mathscr {A}}_I,{\mathfrak {F}}_I)\}_{I\in \mathbb T}\). We prove this by induction on \(\mathbb T\), the number of nodes in the tree \(\mathbb T\). This is obvious if \(\mathbb T=1\) and we assume that it holds for all trees with fewer than \(\mathbb T\) nodes. Since \({\mathscr {B}}_{(1)}\notin {\mathfrak {F}}_{(1)}\), we can choose a prime \({\mathscr {P}}_{(1)}\) realizing \(({\mathscr {B}}_{(1)},{\mathfrak {F}}_{(1)})\). From the definition in (34), we know that \({\mathscr {A}}_{(1)}\subseteq {\mathscr {B}}_{(1)}\) and each Open image in new window for each node (I, i) immediately below the root node. Hence, \({\mathscr {P}}_{(1)}\) realizes \(({\mathscr {A}}_{(1)},{\mathfrak {F}}_{(1)})\) and contains prime ideals \({\mathscr {P}}_{(1,i)}\) realizing \(({\mathscr {B}}_{(1,i)}, {\mathfrak {F}}_{(1,i)})\) for each \(1\leqslant i\leqslant n(1)\). Now, n(1) different subtrees \(\mathbb T_1\), ..., \(\mathbb T_{n(1)}\) obtained by cutting off the root node all have strictly less than \(\mathbb T\) nodes. By the induction assumption, it follows that starting from each \({\mathscr {P}}_{(1,i)}\), we may obtain a realization of the subtree \(\mathbb T_i\). This proves the result.
For the “only if” part, we can reverse our arguments and prove by induction the claim that if \(\{{\mathscr {P}}_I\}_{I\in \mathbb T}\) is a realization of \(\{({\mathscr {A}}_I,{\mathfrak {F}}_I)\}_{I\in \mathbb T}\), \({\mathscr {P}}_{(1)}\) must be a realization of \(({\mathscr {B}}_{(1)},{\mathfrak {F}}_{(1)})\). Hence, \({\mathscr {B}}_{(1)}\notin {\mathfrak {F}}_{(1)}\). \(\square \)
5 Monoidal families and the Prime Ideal Principle
In this final section of the paper, we shall assume that the tensor triangulated category \(({\mathscr {C}},\otimes ,1)\) has the additional property that all thick tensor ideals are radical, i.e., for any thick tensor ideal \({\mathscr {I}}\) in \(({\mathscr {C}},\otimes ,1)\), we have Open image in new window . This additional assumption is equivalent (see [1, Proposition 4.4]) to the assumption that for any object \(a\in {\mathscr {C}}\), a lies in the ideal generated by the object Open image in new window . In fact, it is very frequent for all thick tensor ideals to be radical (see [1, Remark 4.3] and [22, Lemma A.2.6]). In particular, this assumption holds in rigid tensor triangulated categories (see [34, Section 2]). For us, the key consequence of this assumption is the following fact.
Proposition 5.1
 (a)
Let \(({\mathscr {C}},\otimes ,1)\) be a tensor triangulated category such that every thick tensor ideal is a radical ideal. Then, for any thick tensor ideals \({\mathscr {I}}\) and \({\mathscr {J}}\), we have Open image in new window .
 (b)
Let \({\mathfrak {F}}\) be a family of thick tensor ideals such that \({\mathscr {C}}\in {\mathfrak {F}}\). Then, if \({\mathfrak {F}}\) is monoidal, the family \({\mathfrak {F}}\) is a strongly Oka and a strongly Ako family. In particular, \({\mathfrak {F}}\) satisfies the Prime Ideal Principle.
Proof
(a) It is clear that Open image in new window . We choose some object \(a\in {\mathscr {I}}\cap {\mathscr {J}}\). Then, Open image in new window . Since all thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\) are radical, it follows that a lies in the ideal generated by Open image in new window . We conclude that Open image in new window and hence Open image in new window .
(b) We will show that \({\mathfrak {F}}\) is strongly Oka and strongly Ako by showing that it satisfies condition (Q3) in Theorem 3.5. For this, we consider thick tensor ideals \({\mathscr {A}},{\mathscr {B}}\in {\mathfrak {F}}\) and some ideal \({\mathscr {I}}\) such that Open image in new window . From part (a), it follows that Open image in new window . By assumption, \({\mathfrak {F}}\) is monoidal and hence Open image in new window . Hence, the family \({\mathfrak {F}}\) satisfies condition (Q3) in Theorem 3.5. \(\square \)
Remark 5.2
Proposition 5.1 (b) may also be proved as follows: all thick tensor ideals being radical, [4, Theorem 14] now gives us a correspondence between closed subspaces of Open image in new window and all ideals in \({\mathscr {C}}\). As such, if \({\mathfrak {F}}\) is a monoidal family, then Open image in new window is closed under finite unions and hence any closed subspace of Open image in new window that is minimal with respect to not being in \({\mathfrak {F}}^*\) must be irreducible.
Proposition 5.3
 (a)
Let \({\mathscr {I}}\) be a thick tensor ideal that is maximal with respect to being nonprincipal. Then \({\mathscr {I}}\) is prime.
 (b)
Let \(\alpha \) be an infinite cardinal and let \({\mathfrak {F}}_\alpha \) denote the family of thick tensor ideals \({\mathscr {I}}\) having a generating set \(G_{{\mathscr {I}}}\) of cardinality \(G_{{\mathscr {I}}}\leqslant \alpha \). Then, any ideal that is maximal with respect to not being in \({\mathfrak {F}}_\alpha \) is prime.
Proof
(a) Using Proposition 5.1 (b), it is enough to show that the collection of principal ideals is monoidal. If we have principal ideals \({\mathscr {I}}=(x)\) and \({\mathscr {J}}=(y)\) generated by objects \(x,y\in {\mathscr {C}}\) respectively, it follows from Lemma 3.4 that Open image in new window is the principal ideal generated by Open image in new window . Hence, the family of principal ideals is monoidal and satisfies the Prime Ideal Principle. The result of part (b) follows similarly.\(\square \)
Using the same approach as in Proposition 5.3, we can prove the following more general result.
Proposition 5.4
Let \(({\mathscr {C}},\otimes ,1)\) be a tensor triangulated category such that every thick tensor ideal is radical. Let \({\mathscr {S}}\) be a multiplicatively closed family of objects of \({\mathscr {C}}\) containing 1 and also possibly 0. For any infinite cardinal \(\alpha \), let \({\mathfrak {G}}_{\leqslant \alpha }^{{\mathscr {S}}}\) (resp. \({\mathfrak {G}}_{<\alpha }^{{\mathscr {S}}})\) denote the family of thick tensor ideals \({\mathscr {I}}\) having a generating set \(G_{{\mathscr {I}}}\subseteq {\mathscr {S}}\) such that \(G_{{\mathscr {I}}}\leqslant \alpha \) (resp. \(G_{{\mathscr {I}}}<\alpha )\). Then, any ideal that is maximal with respect to not being in \({\mathfrak {G}}_{\leqslant \alpha }^{{\mathscr {S}}}\) (resp. \({\mathfrak {G}}_{<\alpha }^{{\mathscr {S}}})\) is prime.
Proof
We consider thick tensor ideals \({\mathscr {I}},{\mathscr {J}}\in {\mathfrak {G}}_{\leqslant \alpha }^{{\mathscr {S}}}\) with respective generating sets Open image in new window , \(G_{{\mathscr {J}}}=\{y_j\}_{j\in J}\subseteq {\mathscr {S}}\) of cardinality \(\leqslant \alpha \). Since \({\mathscr {S}}\) is multiplicatively closed, we see that each Open image in new window . It then follows from Lemma 3.4 that Open image in new window may be generated by the set Open image in new window of cardinality Open image in new window . Hence, \({\mathfrak {G}}_{\leqslant \alpha }^{{\mathscr {S}}}\) is a monoidal family and satisfies the Prime Ideal Principle. The case of \({\mathfrak {G}}_{<\alpha }^{{\mathscr {S}}}\) follows similarly. \(\square \)
Proposition 5.5
 (a)Let \({\mathfrak {F}}\) be a monoidal semifilter. Then, the family Open image in new window defined by is a strongly Oka family and satisfies the Prime Ideal Principle.
 (b)
Let \({\mathscr {S}}\) be a multiplicatively closed family of objects of \({\mathscr {C}}\) such that \(1\in {\mathscr {S}}\) and \(0\notin {\mathscr {S}}\). Then, the family is a strongly Oka family and satisfies the Prime Ideal Principle.
Proof
Proposition 5.6
 (a)
The family \({\mathfrak {F}}_{{\mathscr {I}}}\) of thick tensor ideals containing \({\mathscr {I}}\) is a monoidal semifilter. In particular, any ideal of \(({\mathscr {C}},\otimes ,1)\) that is maximal with respect to not containing \({\mathscr {I}}\) must be prime.
 (b)
Suppose that \({\mathscr {I}}\) is a finitely generated (hence principal) ideal. Then, any nonempty increasing chain of ideals in the complement Open image in new window of \({\mathfrak {F}}_{{\mathscr {I}}}\) has an upper bound in Open image in new window .
Proof
(a) It is clear that \({\mathfrak {F}}_{{\mathscr {I}}}\) is a semifilter. Further, if \({\mathscr {I}}\subseteq {\mathscr {A}}\) and \({\mathscr {I}}\subseteq {\mathscr {B}}\) for some \({\mathscr {A}},{\mathscr {B}}\in {\mathfrak {F}}_{{\mathscr {I}}}\), we have Open image in new window . Hence, \({\mathfrak {F}}_{{\mathscr {I}}}\) is also monoidal.
(b) Suppose that \({\mathscr {I}}\) is generated by an object x. We consider an increasing chain \(\{{\mathscr {J}}_j\}_{j\in N}\) of ideals in Open image in new window indexed by a totally ordered set \((N,\leqslant )\) and the union \({\mathscr {J}}=\bigcup _{j\in N}{\mathscr {J}}_j\). Now, if \({\mathscr {J}}\in {\mathfrak {F}}_{{\mathscr {I}}}\), there exists \(n \in N\) large enough such that \(x\in {\mathscr {J}}_n\). Hence, \({\mathscr {I}}\subseteq {\mathscr {J}}_n\) and \({\mathscr {J}}_n\in {\mathfrak {F}}_{{\mathscr {I}}}\), which is a contradiction. We conclude that \({\mathscr {J}}\notin {\mathfrak {F}}_{{\mathscr {I}}}\). \(\square \)
Lemma 5.7
In the notation above, for any \(m\geqslant 1\), we have Open image in new window .
Proof
Proposition 5.8
 (a)
A chain \(\cdots \subseteq {\mathscr {P}}_2\subseteq {\mathscr {P}}_1\) of thick prime ideals is a realization of the template T if and only if it is also a realization of the template Open image in new window .
 (b)
The template T has a realization if and only if \({\mathscr {B}}^1_\infty \notin {\mathfrak {F}}_1\).
 (c)
The template T is realizable if and only if each of the truncated templates \(T^m_n\) is realizable for \(1\leqslant m\leqslant n\).
Proof
(b) If T is realizable, it follows from part (a) that so is Open image in new window and hence in particular \({\mathscr {B}}_\infty ^1\notin {\mathfrak {F}}_1\). Conversely, if \({\mathscr {B}}_\infty ^1\notin {\mathfrak {F}}_1\), we can choose a prime ideal \({\mathscr {P}}_1\) realizing \(({\mathscr {B}}_\infty ^1,{\mathfrak {F}}_1)\). From Lemma 5.7, we know that Open image in new window for each \(m\geqslant 1\). It follows that we can choose a prime ideal \({\mathscr {P}}_2\subseteq {\mathscr {P}}_1\) realizing \(({\mathscr {B}}_\infty ^2,{\mathfrak {F}}_2)\) and so on to obtain a realization of Open image in new window . This gives a realization of T.
(c) The “only if” part is obvious. For the “if part”, we suppose that each truncated template \(T^m_n\) is realizable for \(1\leqslant m\leqslant n\). In particular, the truncated template \(T^1_n\) is realizable for each \(n\geqslant 1\). Hence, \({\mathscr {B}}_n^1\notin {\mathfrak {F}}_1\) for each \(n\geqslant 1\). We know that Open image in new window . From Proposition 5.6 (b), it follows that the increasing chain \({\mathscr {B}}_1^1\subseteq {\mathscr {B}}_2^1\subseteq {\mathscr {B}}_3^1\subseteq \cdots \) of ideals in Open image in new window must have some upper bound in Open image in new window , say \({\mathscr {B}}\). But then, \({\mathscr {B}}\supseteq {\mathscr {B}}^1_\infty \). Since \({\mathfrak {F}}_1\) is a semifilter and \({\mathscr {B}}\notin {\mathfrak {F}}_1\), we must have \({\mathscr {B}}_\infty ^1\notin {\mathfrak {F}}_1\). From part (b), it now follows that the template T is realizable. \(\square \)
Proposition 5.9
 (a)
 (b)
For any closed Open image in new window and any \(i,j\in I\) such that Open image in new window , we must have Open image in new window .
Proof
Clearly, any family \({\mathfrak {F}}^*\) of closed subspaces of Open image in new window containing \(\varnothing \) and closed under finite unions satisfies the conditions in Proposition 5.9. In that case, the corresponding family Open image in new window of thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\) is simply a monoidal family. We will conclude by showing how to construct a family \({\mathfrak {F}}^*\) that is not closed under finite unions but still satisfies the conditions in Proposition 5.9. This will be done with the help of Ako families of thick tensor ideals in \(({\mathscr {C}},\otimes ,1)\).
Notes
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