Locality for qc-sheaves associated with tilting - ICART 2018, Faculty of Sciences, Mohammed V University in Rabat Jan Trlifaj Univerzita Karlova, Praha Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 1 / 28
Overview I. Tilting modules and classes over commutative rings. II. Quasi-coherent sheaves and Zariski locality. III. Mittag-Leffler conditions and tilting. IV. Zariski locality of qc-sheaves associated with tilting. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 2 / 28
I. Tilting modules and classes over commutative rings. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 3 / 28
Tilting modules and classes Definition A (right R-) module T is tilting provided that (T1) T has finite projective dimension, (T2) Ext i R (T, T (κ) ) = 0 for each cardinal κ and each i > 0, and (T3) there exists an exact sequence 0 R T 0 T r 0 such that r < ω, and for each i r, T i Add(T ), i.e., T i is a direct summand in a direct sum of copies of T. If pd R (T ) n, then T is called n-tilting. 0-tilting modules = projective generators (possibly infinitely generated). B = i>0 KerExti R (T, ) is the (right) tilting class induced by T. A = KerExt 1 R (, B) is the left tilting class induced by T. A tilting module T is equivalent to T in case Add(T ) = Add( T ). Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 4 / 28
Basic restriction for the commutative setting Let R be a commutative ring. (i) If T is a tilting module of projective dimension > 0, then T is not finitely generated. (ii) If T mod R and 1 pd R (T ) = n <, then Ext n R (T, T ) 0. Hence T fails condition (T2). Proof of (ii) All syzygies of T are finitely presented, so pd R (T ) = max m pd Rm (T m ). Take m mspec(r) such that pd Rm (T m ) = n. Then T m mod R m and (Ext n R (T, T )) m = Ext n R m (T m, T m ), so w.l.o.g., we can assume that R is local. Then T has a minimal free resolution, F, given by an iteration of projective covers. So d k (F k ) mf k 1 for all k > 0 where d k is the differential. As Ext n R (T, ) is right exact, the epimorphism T T /mt induces a surjection Ext n R (T, T ) Extn R (T, T /mt ). However, Ext n R (T, T /mt ) = Ext 1 R (Ωn 1 (T ), T /mt ) = Hom R (F n, T /mt ) 0. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 5 / 28
Example: tilting abelian groups Let P Spec(Z) \ {0}, and Z A P Q and A P /Z = p P Z p. T P = A P A P /Z is a tilting abelian group, B P = {A Mod Z Ap = A for all p P}. Each tilting abelian group is equivalent to T P for some P Spec(Z) \ {0}. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 6 / 28
Finite type Theorem Let T be a tilting module with the induced left and right tilting classes A and B, respectively. Then there is a set S consisting of strongly finitely presented modules in A, such that B = KerExt 1 R (S, ). One can always take S = A mod R; in this case A lim S. Terminology: The set S witnesses the finite type of T. Abelian groups, cont. For P Spec(Z) \ {0}, we can take S P = {Z p p P}. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 7 / 28
Characteristic sequences over commutative rings A subset P Spec(R) is Thomason, if P = I I V (I ) for a set I consisting of finitely generated ideals of R. Here, V (I ) = {p Spec(R) I p}. If M Mod R and p Spec(R), then p is weakly associated to M, if R/p is contained in the smallest subclass of Mod R containing M and closed under submodules and direct limits. Example: If R is noetherian, then Thomason subsets = upper subsets, and weakly associated primes = associated primes. Let R be a commutative ring. A sequence P = (P 0,..., P n 1 ) of Thomason subsets of Spec(R) is called characteristic, if P 0 P 1 P n 1, and Vass(Ω i (R)) P i = for all i < n. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 8 / 28
Structure of tilting classes over commutative rings For R commutative and n 1, there is a bijection between: characteristic sequences of length n, and right n tilting classes in Mod R. The right n-tilting class T P corresponding to a characteristic sequence P = (P 0,..., P n 1 ), where P i = I I i V (I ) for each i < n, equals T P = {M Mod R Tor R i (R/I, M) = 0 i < n I I i }. For an n-tilting module T inducing the n-tilting class T, the corresponding characteristic sequence P T = (P 0,..., P n 1 ) satisfies for each i < n P i = {p Spec(R) i < k n : Tor R k (E(R/p), T ) 0}. Open problem: the structure of the corresponding tilting modules. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 9 / 28
II. Quasi-coherent sheaves and Zariski locality. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 10 / 28
Quasi-coherent sheaves as representations Let X be a scheme and R = (R(U) U X, U open affine ) be its structure sheaf. A quasi-coherent sheaf M on X can be represented by an assignment to every affine open subset U X, an R(U)-module M(U) of sections, and to each pair of open affine subsets V U X, an R(U)-homomorphism f UV : M(U) M(V ) such that id R(V ) f UV : R(V ) R(U) M(U) R(V ) R(U) M(V ) = M(V ) is an R(V )-isomorphism. + compatibility conditions: if W V U, then f UV f VW = f UW. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 11 / 28
Properties of the representations Exactness The functors R(V ) R(U) are exact, i.e., the R(U)-modules R(V ) are flat (in fact, they are very flat ). Non-uniqueness of the representations Not all affine open subsets are needed: a set of them, S, covering both X, and all U V where U, V S, will do. The affine case (Grothendieck) If X = Spec(R) for a commutative ring R, then S = {X } is enough, so quasi-coherent sheaves on X = R-modules. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 12 / 28
Extending properties from modules to qc-sheaves Definition Let P be a property of modules. A qc-sheaf M on a scheme X is a locally P qc-sheaf provided that for each open affine set U of X, M(U) satisfies P as an R(U)-module. Requirement: Properties studied for qc-sheaves should be independent of coordinates, i.e., for each scheme X, it should be possible to test for the property using an (arbitrary) open affine covering of X. Definition The notion of a locally P qc-sheaf is Zariski local if for each scheme X, each open affine covering X = V S V of X, and each qc-sheaf M on X, if M(V ) satisfies P as R(V )-module for each V S, then M is a locally P-qc-sheaf. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 13 / 28
Ascent and descent along flat base changes Definition A property of modules P is called an ad-property provided that if ϕ : R S is any flat ring homomorphism, then P ascends along ϕ (i.e., if M satisfies P as R module, then so does M R S as S module), and if ϕ : R S is any faithfully flat ring monomorphism, then P descends along ϕ (i.e., if M R S satisfies P as S-module, then so does M as R module). Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 14 / 28
Sufficient conditions for Zariski locality Lemma Let P be an ad-property of modules over commutative rings. Then the notion of a locally P qc-sheaf is Zariski local. Affine Communication Lemma (ACL) A weaker property is sufficient: (1) P ascends along all localizations ϕ : R R f where f R, and (2) P descends along all faithfully flat ring monomorphisms of the form ϕ f0,...,f m 1 : R j<m R f j where R = j<m f jr. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 15 / 28
The basic example: vector bundles Definition A qc-sheaf M on a scheme X is an (infinite dimensional) vector bundle, if M(U) is a projective R(U)-module for each open affine set U of X. (So vector bundles are exactly the locally projective qc-sheaves.) Theorem The notion of a vector bundle is Zariski local. Conjectured by Grothendieck in the 1960 s, proved in 1971 by Raynaud and Gruson, by showing that projectivity is an ad-property. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 16 / 28
Locally tilting quasi-coherent sheaves Definition Let n < ω. A qc-sheaf M on a scheme X is locally n-tilting, if M(U) is an n-tilting R(U)-module for each open affine set U of X. Let 0 n. A qc-sheaf M on a scheme X is locally left n-tilting (locally Add-n-tilting), if for each open affine set U of X, there exists an n-tilting R(U)-module T (U) such that M(U) A T (U) (M(U) Add(T (U))). Problem: Are these three notions Zariski local for each n? (The latter two are Zariski local for n = 0 by the Theorem of Raynaud and Gruson above, as they coincide with the notion of a vector bundle.) Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 17 / 28
III. Mittag-Leffler conditions and tilting. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 18 / 28
Tilting and flat base change Ascent-Descent Lemma Let ϕ : R S be a flat ring homomorphism of commutative rings, and T be an n-tilting R-module with the left and right tilting classes A and B, respectively. T = T R S is an n-tilting S-module. So the property of being an n-tilting module ascends along ϕ. Let B be the right n-tilting class induced by T. Then B = B Mod S, and for each module N Mod R, N B, iff N R S B. If ϕ is a faithfully flat ring monomorphism, then for each M Mod R, M A, iff M R S A, where A is the left n-tilting class induced by T. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 19 / 28
Tools: Mittag-Leffler conditions Definition An inverse system of modules H = (H i, h ij i j I ) is Mittag-Leffler, if for each k I there exists k j I, such that Im(h kj ) = Im(h ki ) for each j i I, that is, the terms of the decreasing chain (Im(h ki ) k i I ) of submodules of H k stabilize. Let B be a module and M = (M i, f ji i j I ) a direct system of finitely presented modules. An application of Hom R (, B) yields the induced inverse system H = (H i, h ij i j I ), where H i = Hom R (M i, B) and h ij = Hom R (f ji, B) for all i j I. Let B be a class of modules. A module M is B-stationary, if M is a direct limit of a direct system M of finitely presented modules so that for each B B, the induced inverse system H is Mittag-Leffler. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 20 / 28
Tools: Mittag-Leffler conditions and generalized dévisage Lemma In the setting of the Ascent-Descent Lemma, let S = A mod R. Let C Mod R and C = C R S. If C lim S is countably presented, then C A, iff C is B-stationary. Assume that ϕ is faithfully flat, and C lim(s R S) is countably presented. Then C lim S and C is countably presented. Moreover, the equivalent conditions above are further equivalent to C being B -stationary, and hence to C A. If ϕ is faithfully flat, then for each M Mod R, M A, iff M R S A. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 21 / 28
IV. Zariski locality of qc-sheaves associated with tilting. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 22 / 28
Characteristic sequences and flat base change Let ϕ : R S be the faithfully flat ring monomorphism from condition (2) of the ACL. If P is a characteristic sequence in Spec(R) corresponding to an n-tilting module T, then the characteristic sequence in Spec(S) corresponding to T = T R S is Q P, where for each i < n, Q i is defined by Q i = {q Spec(S) p P i : ps q}. Let T Mod R be such that T = T R S is an n-tilting S-module. Then the characteristic sequence Q in Spec(S) corresponding to T is of the form Q P for some characteristic sequence P in Spec(R). Hence T satisfies conditions (T1) and (T2). Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 23 / 28
Zariski locality for qc-sheaves associated with tilting Theorem The notions of a locally left n-tilting, locally Add-n-tilting, and locally n-tilting quasi-coherent sheaves are Zariski local for each n 0. Sketch of proof: The first two claims follow by the Lemma above. The third claim then follows using the fact that condition (T3) can be reformulated as D(R) is the smallest localizing subcategory of itself containing T and employing the fact that localizing subcategories are tensor ideals. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 24 / 28
Ascent-descent for tilting Theorem (i) If R is noetherian, then for each n 0, the property of being an n-tilting module is an ad-property. (ii) The property of being a 1-tilting module is an ad-property. Sketch of proof: (i) The descent for conditions (T1) and (T2) follows by the Lemma above. For the proof of (T3), we use its refomulation above, and employ Neeman s classification of localizing subcategories of D(R), for R commutative noetherian, by the subsets of Spec(R). (ii) Again, the descent for conditions (T1) and (T2) follows by the above. (T3) is proved in its equivalent form of KerHom R (T, ) KerExt 1 R (T, ) = 0. Open problem: Is the property of being an n-tilting module an ad-property for n > 1 and R non-noetherian? Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 25 / 28
References I M. Raynaud, L. Gruson, Critères de platitude et de projectivité, Invent. Math. 13(1971), 1-89. L. Angeleri Hügel, D. Herbera, Mittag-Leffler conditions on modules, Indiana Math. J. 57(2008), 2459-2517. J.Šaroch, Tilting approximations and Mittag-Leffler conditions - the tools, Israel J. Math. 226(2018), 737-756. L.Angeleri Hügel, J.Šaroch, J.Trlifaj, Tilting approximations and Mittag-Leffler conditions - the applications, Israel J. Math. 226(2018), 757-780. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 26 / 28
References II E.E. Enochs and S. Estrada, Relative homological algebra in the category of quasi-coherent sheaves, Adv. Math. 194(2005), 284-295. L. Angeleri Hügel, D. Pospíšil, J. Šťovíček, J. Trlifaj, Tilting, cotilting, and spectra of commutative noetherian rings, TAMS 366(2014), 3487-3517. M. Hrbek, J. Šťovíček, Tilting classes over commutative rings, to appear in Forum Math., arxiv:1701.05534v1. M. Hrbek, J. Šťovíček, J. Trlifaj, Zariski locality of quasi-coherent sheaves associated with tilting, arxiv:1712.08899v1. Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 27 / 28
An invitation... Workshop and 18th International Conference on Representations of Algebras ICRA 2018 Workshop: August 8-11 Conference: August 13-17 Venue: Prague, Czech Republic www.icra2018.cz Charles University and Czech Technical University Jan Trlifaj (Univerzita Karlova, Praha) Locality for tilting qc-sheaves HAMRC 28 / 28