process

Torsion classes of finite type and spectra 407induce bijections between the set of all open subsets V Proj A and the set of allt-filters of finite type containing fA n C g n1.Let F be such a t-filter. Then the set ƒ F of finitely generated graded ideals Ibelonging to F is a basis for F. Clearly V F D S I 2ƒ FV.I/,soV F is open in Proj A.Now let V be an open subset of Proj A. Let ƒ be the set of finitely generatedhomogeneous ideals I such that V.I/ V . Then V D S I 2ƒ V.I/and I 1 I n 2 ƒfor any I 1 ;:::;I n 2 ƒ. We denote by F 0 Vthe set of homogeneous ideals I A suchthat I J for some J 2 ƒ. By Proposition 5.4(2) F 0 Vis a t-filter of finite type.Clearly, F 0 V F V . Suppose I 2 F V n F 0 V .We can use Zorn’s lemma to find an ideal J I which is maximal with respect toJ … F 0 V (we use the fact that F0 Vhas a basis of finitely generated ideals). We claim thatJ is prime. Indeed, suppose a; b 2 A are two homogeneous elements not belonging toJ . Then J CaA and J CbA must be members of F 0 V , and also .J CaA/.J CbA/ 2 F0 Vby Proposition 5.4(1). But .J C aA/.J C bA/ J C abA, and therefore ab … J .We see that J 2 V.I/ V , and hence J 2 V.I 0 / for some I 0 2 ƒ. But this impliesJ 2 F 0 V , a contradiction. Thus F0 V D F V . Clearly, V D V FV for every open subsetV Proj A. Let F be a t-filter of finite type and I 2 F. Then I J for someJ 2 ƒ F , and hence V.I/ V.J/ V F . It follows that F F VF . As above, there isno ideal belonging to F VF n F. We have shown the desired bijection between the setsof all t-filters of finite type and all open subsets in Proj A.7 The prime spectrum of an ideal latticeInspired by recent work of Balmer [4], Buan, Krause, and Solberg [5] introduce thenotion of an ideal lattice and study its prime ideal spectrum. Applications arise fromabelian or triangulated tensor categories.Definition (Buan, Krause, Solberg [5]). An ideal lattice is by definition a partiallyordered set L D .L; /, together with an associative multiplication L L ! L, suchthat the following holds.(L1) The poset L is a complete lattice, that is,sup A D _ a and inf A D ^aa2Aa2Aexist in L for every subset A L.(L2) The lattice L is compactly generated, that is, every element in L is the supremumof a set of compact elements. (An element a 2 L is compact, if for all A Lwith a sup A there exists some finite A 0 A with a sup A 0 .)(L3) We have for all a; b; c 2 La.b _ c/ D ab _ ac and .a _ b/c D ac _ bc: