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The Zariski Spectrum of the Category of Finitely Presented Modules Prest The Zariski spectrum of the category of finitely presented modules Prest, Mike 2006 MIMS EPrint: 2006.107 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 The Zariski spectrum of the category of ¯nitely presented modules Mike Prest School of Mathematics University of Manchester Manchester M13 9PL UK [email protected] May 21, 2006 Contents 1 Introduction 1 2 The Gabriel-Zariski and rep-Zariski spectra 5 2.1 The Zariski spectrum through representations . 5 2.2 Zariski-injective and Gabriel-Zariski spectra of categories . 8 2.3 Embedding the Zariski spectrum . 9 3 Dualities 11 4 Examples 13 4.1 The rep-Zariski spectrum of a PI Dedekind domain . 13 4.2 The rep-Zariski spectrum of a PI hereditary order . 15 4.3 The rep-Zariski spectrum of a tame hereditary artin algebra . 16 4.4 Other examples . 17 5 The Ziegler spectrum 18 5.1 rep-Zariski = dual-Ziegler . 19 6 Topological properties of ZarR 20 7 The presheaf structure 21 7.1 Rings of de¯nable scalars . 21 7.2 The sheaf of locally de¯nable scalars . 23 1 8 Examples 25 8.1 The sheaf of locally de¯nable scalars of a PI Dedekind domain . 26 8.2 The sheaf of locally de¯nable scalars of a PI hereditary order . 28 8.3 The sheaf of locally de¯nable scalars of the Kronecker algebra . 30 9 The spectrum of a commutative coherent ring 31 10 Appendix: pp conditions 37 1 Introduction Here is yet another \non-commutative geometry". It can be described briefly as follows: take a commutative noetherian ring R (e.g. the coordinate ring of an a±ne variety); describe Spec(R) in terms of the category, Mod-R, of representations of R; now use the same description with any small pre-additive category A in place of R. The result is a topological space (with points the \primes" of A) with associated presheaf (of \localisations" of A). We may, in particular, apply this de¯nition with mod-R (the category of ¯nitely presented right R-modules) in place of R to obtain a non-commutative geometry associated with R. Admittedly this has moved us one `representation- level' up since it is rather the spectrum of mod-R than of R. Nevertheless the \usual spectrum" of R, if it has one, sits inside this richer structure (more accurately, inside that with (R-mod)op replacing R). For example if R is com- mutative noetherian then the (Zariski) spectrum of R is a subspace of the larger space, 2.7, and the associated sheaf of rings is a part of the larger presheaf, see 7.11. If discovery could peer at itself through hindsight this, no doubt, would have been how I had come to the de¯nition. In fact I ¯rst de¯ned this space as the dual of another, the Ziegler spectrum, and only later realised that it could be presented as a natural generalisation of the Zariski spectrum [36]. Here is the exact de¯nition. Let A be a small preadditive category and let Mod-A denote the category of right A-modules (that is, the category, (Aop; Ab), of contravariant additive functors from A to the category, Ab, of abelian groups). Let inj-A denote the set of isomorphism classes of indecomposable injective A- modules: this is the set underlying our topological space. The topology is deter- mined by declaring the following sets to be open: [F ] = fE 2 inj-A : (F; E) = 0g where F ranges over (isomorphism classes of) ¯nitely presented A-modules. Since F © G is ¯nitely presented if F and G are, these basic open sets are closed under ¯nite intersection, so an arbitrary open set will just be a union of sets of the form [F ]. We will call this the Gabriel-Zariski spectrum of A because the idea of representing prime ideals by the corresponding indecomposable injective repre- sentations goes back to Gabriel's thesis [10]. We write GZspec(A). If A itself 2 has the form (R-mod)op for some ring (or small preadditive category) R then we will refer to this as the rep-Zariski spectrum of R and we write just ZarR. This means that if R is commutative noetherian then we write Spec(R) for the usual spectrum and ZarR for this larger space in which, as we will see, the former embeds. We have yet to describe the ring(oid)ed structure on this space: the descrip- tion requires a little setting up. For GZspec(A) this is the presheaf of localisations de¯ned as follows. Given M 2 mod-A, the indecomposable objects in the basic open set [M] = fE 2 inj-A : (M; E) = 0g together cogenerate a hereditary torsion theory on Mod-A: the torsionfree objects are those which embed in some direct product of copies of members of [M] and the torsion objects are those with no non-zero morphism to any member of [M]. Recall, [55], that each of the classes, T , of torsion and, F, of torsionfree objects determines the other and the pair ¿ = (T ; F) is referred to as a hereditary torsion theory. Throughout this paper when we say \torsion theory" we always mean a hereditary torsion theory. In fact we want a torsion theory which is determined by the ¯nitely pre- sented torsion objects, that is, one of ¯nite type, so we use instead the torsion theory whose torsion class, which we denote T[M], is generated as such by the ¯nitely presented objects with no non-zero morphism to any member of [M] (so T[M] ⊆ T ). Denote by F[M] (¶ F) the corresponding torsionfree class. The localisation Mod-A ¡! (Mod-A)¿[M] (the latter category we also write just as (Mod-A)[M]) at this ¯nite type torsion theory, ¿[M] = (T[M]; F[M]), correspond- ing to [M], is the Grothendieck category which is obtained from Mod-A by forcing all objects of T[M] to become zero. The image of A in (Mod-A)¿[M] (via the Yoneda embedding of A in Mod-A), we denote it A¿[M] , is the localisation of A which is assigned to the basic open set [M]. With the natural restriction maps, this gives a presheaf de¯ned on the given basis (which is enough to de¯ne the corresponding sheaf). We would also like to say explicitly how the above de¯nition reads for the rep-Zariski spectrum, ZarR, of a ring R or, more generally, for ZarA where A is a small preadditive category. So let mod-A denote (a small version of) the category of ¯nitely presented right A-modules. Recall that an A-module M is ¯nitely presented if there is an exact sequence P ¡! Q ¡! M ¡! 0 where P; Q are ¯nitely generated pro- jective A-modules (that is, are ¯nite direct sums of representable functors from Aop to Ab). It is equivalent to require that the hom-functor (M; ¡) commute with direct limits. Similarly A-mod denotes the category of ¯nitely presented left A-modules. Now consider the category (A-mod; Ab) (of left (A-mod)- modules). There is a full embedding [16] of Mod-A into (A-mod; Ab) given by sending the right A-module M to the functor M ­A ¡ : A-mod ¡! Ab (this functor is determined by its being right exact and having the action, (M ­ ¡) : (A; ¡) 7! MA (A 2 A) on this generating set of projectives in 3 A-Mod) and having the obvious e®ect on morphisms. In particular, since A embeds as a full subcategory of mod-A (via the Yoneda embedding A 7! (¡; A) for A an object of A), there is also a copy of A sitting as a full subcategory of the functor category (A-mod; Ab). This may be identi¯ed with the Yoneda-image of the Yoneda-image of A in A-Mod. That is, map A 2 A to (A; ¡) 2 A-mod and then this to the representable functor ((A; ¡); ¡) in (A-mod; Ab). This latter doesn't look quite so bad in the case where A is a ring R since the usual practice is to denote the projective left R-module (R; ¡) also by R (or RR) and then the image of this in the functor category is denoted (RR; ¡) (and is just the forgetful functor). Now let F 2 (A-mod; Ab)fp (the full subcategory of ¯nitely presented func- tors, which we could, though won't, write as (A-mod)-mod). As described above, the indecomposable injective functors in the basic open set [F ] together cogenerate a torsion theory and we take the largest torsion theory of ¯nite type smaller (in the sense of inclusion of torsion classes) than this, denoting it ¿[F ]. Let (A-mod; Ab)[F ] denote the quotient category of (A-mod; Ab) with respect to this torsion theory. Denote by A[F ] the image of A (regarded as a full subcat- egory of the functor category) in this localisation. It is a category with the same objects as A but, in general, modi¯ed morphism groups and we can think of it as a kind of localisation of A. Assign to the basic open set [F ] this localisation of A. Note that in the case where A is a ring R (identi¯ed with the forgetful functor (RR; ¡) sitting in (R-mod; Ab)) the localisation at F is again a one- point category, that is, a ring which, qua ring, is the endomorphism ring R[F ] = End((RR; ¡)[F ]).
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