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Abelian of rs mod e o bla aeoisre- categories Abelian for rem rwae supin,allow assumptions, weaker er incategory. lian Q C agae since language, ” nue nequivalence an induces : B → A oe o certain for eorem Q A ntr,Aeincate- Abelian unctors, and 3.3 / C tstates It . X egv,under give, we a the has or A This . / A Q ges, C of h Serre the : 3.2 we 2 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN an additional assumption, the following analogon of the fundamental homomorphism Theorem. If Q is a certain restriction of an ambient functor Q′ satisfying the assumptions of Gabriel’s characterization then Q still induces the desired equivalence of categories A/ ker Q ≃ B. Our original motivation is to establish a constructive setup for coherent sheaves on several classes of schemes X. This setup begins with describing quotient categories constructively by the 3-arrow formalism of so-called the Gabriel [BLH14b]. Then, the methods in this paper allow to recognize B = Coh X as a Serre quotient of the A = S-grmod of finitely presented graded modules over some graded coherent S. This gives an alternative and simplerproof than the one which can be derived from [Kra97, Theorem 2.6] using a result in [Len69]. The last step in this setup is modeling the category A = S-grmod over a computable ring S constructively through a presentation matrix [BLH11]. This setup turns out to be suitable and computationally efficient for a computer implementation. In Section 2 we collect some preliminaries about Serre quotients. In Section 3 we prove Propositions 3.2 and 3.3 which serve to identify certain categories as Serre quotient. In Sec- tion 4 we apply the two propositions to categories of coherent sheaves. We would like to thank Florian Eisele, Markus Perling, Sebastian Posur, and Sebastian Thomas for helpful discussions.

2. PRELIMINARIESON SERRE QUOTIENTS In this section we recall some results about Serre quotients. From now on A is an Abelian category. See [Gab62] for proofs and [BLH13] for detailed references. A non-empty full subcategory C of an Abelian category A is called thick if it is closed under passing to subobjects, factor objects, and extensions. In this case the (Serre) quotient category A/C is a category with the same objects as A and Hom-groups Hom (M, N) := lim Hom (M ′, N/N ′). A/C −→ A M ′֒→M,N ′֒→N M/M ′,N ′∈C The canonical functor Q : A→A/C is defined to be the identity on objects and maps a ϕ ∈ HomA(M, N) to its image in the direct HomA/C (M, N). The category A/C is Abelian and the canonical functor Q : A→A/C is exact and fulfills the following universal property. If G : A→D an of Abelian categories, and G (C) is zero then there exists a unique functor H : A/C→D with G = H ◦ Q. Let C ⊂A be thick. An object M ∈ A is called C-torsion-free if M has no nonzero subobjects in C. Denote by AC ⊂A the pre-Abelian full subcategory of C-torsion-free objects. If every object M ∈ A has a maximal subobject HC(M) ∈ C then (C, AC) is a hereditary torsion theory of A, i.e., C and AC are additive and full subcategories, C is closed under subobjects, M/HC(M) is C-torsion-free, i.e., lies in AC,

HomA(C, A)=0 for all C ∈ C and A ∈AC,

HomA(C, A)=0 for all C ∈ C implies A ∈AC for all A ∈A, and

HomA(C, A)=0 for all A ∈AC implies C ∈ C for all C ∈A. In this case we call C a thick torsion subcategory. CHARACTERIZINGSERREQUOTIENTSWITHNOSECTIONFUNCTOR 3

Remark 2.1. For C⊂A thick torsion the description of Hom-groups in A/C simplifies to Hom (M, N) = lim Hom (M ′,N/H (N)). A/C −→ A C M ′֒→M M/M ′∈C An object M ∈ A is called C-saturated if it is C-torsion-free and every extension of M by an object C ∈ C is trivial. Denote by SatC(A) ⊂A the full subcategory of C-saturated objects. We say that A has enough C-saturated objects if for each M ∈ A there exists a C-saturated object N and a morphism ηM : M → N such that ker ηM ∈ C. Any thick subcategory C⊂A is called a if the canonical functor Q : A→A/C admits a right adjoint S : A/C→A, called the section functor of Q. The category C ⊂A is localizing if and only if A has enough C-saturated objects. The section functor S : A/C→A is left exact and ∼ preserves products, the counit of the adjunction δ : Q ◦ S −→ IdA/C is a natural isomorphism, and an object M in A is C-saturated if and only if ηM : M → (S ◦ Q)(M) is an isomorphism, where η is the unit of the adjunction. Finally, a localizing C⊂A is a thick torsion subcategory with HC(M) = ker ηM .

3. RECOGNITION OF SERRE QUOTIENTS We say that an exact functor Q′ : A′ → B′ of Abelian categories admits a section functor ′ ′ ′ ′ ′ if Q admits a right adjoint S such that the counit of the adjunction δ : Q ◦ S → IdB′ is a natural isomorphism1. It follows that Q′ is essentially surjective. The next proposition characterizes Serre quotients A′/C′ for C′ a localizing subcategory of A′. Proposition 3.1 ([Gab62, Proposition III.2.5], [GZ67, Chap. 1.2.5.d]). Let Q′ : A′ → B′ be an exact functor of Abelian categories admitting a section functor S ′. Then C′ := ker Q′ is a localizing subcategory of A′ and the adjunction Q′ ⊣ (S ′ : B′ → A′) induces an adjoint equivalence of categories A′/C′ ≃ B′. The aim of this section is to formulate a characterization of certain Serre quotients A→A/C where the thick subcategory C is not necessarily localizing. The following two propositions are analogous to the second isomorphism Theorem and to the fundamental homomorphism Theorem. However, they need some additional assumptions. A e Proposition 3.2 (Second isomorphism theorem). Let A be an Abelian cate- A gory and C ⊂ A a thick torsion subcategory. Then for eache thick subcategory e e C A ⊂ A the intersection C := C∩A is a thick torsion subcategory of A. If fur- e thermoree the restricted canonicale functor is essentially surjective then A→ A/C C it induces and equivalence of categories e e A/C −→∼ A/C. e e Proof. From the thickness of A ⊂ A it follows that for each object N ∈ A the maximal C- e e subobject HCe(N) lies in C. Furthermore N/HCe(N) is C-free since it is C-free. Summing up, e e HC(N) := HC(N) is the maximal C-subobject of N ∈A establishing the first assertion. By the

1These adjunctions Q′ ⊣ (S ′ : B′ → A′) are the exact reflective localization of Abelian categories. 4 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN universal property of Q there exists a functor A/C → A/C which is essentially surjective by our assumption. We will now show that it is fully faithful.e e Let M, N ∈ A. We can replace N by its C-free factor N/HC(N) and without loss of generality assume that N is C-free. Because of the thickness of A ⊂ A and the definition of C, the A-subobjects M ′ of M with M/M ′ ∈ C are exactly the A-subobjectse with M/M ′ ∈ C and we obtaine e ′ HomA/C (M, N) = lim HomA(M , N) by Remark 2.1 and N ∈AC ,M ′֒→M in A M/M ′∈C ′ = lim HomA(M , N) ′ e ,M ֒→M in A ′ e M/M ∈C ′ =∼ lim Hom e(M , N) A ⊂ A full ′ e A M ֒→M in A, e ′ e M/M ∈C

e = HomAe/C(M, N) by Remark 2.1 and N ∈ ACe. e 

We say that an exact and essentially surjective functor Q : A → B of Abelian categories admits a sections functor up to extension if there exists an exact functor Q′ : A′ → B′ admitting a section functor with B ⊂ B′ a replete and full Abelian subcategory, A⊂A′ ′ thick, and Q = Q |A : A → B. Now we can formulate the analogon of the fundamental homomorphism Theorem. Q Q′ Proposition 3.3 (Fundamental homomorphism theorem). Let : B′ A′ A → B be an exact and essentially surjective functor of Abelian cat- egories which admits a section functor up to extension. Then Q induces Q an equivalence of categories A/C ≃ B, where C := ker Q is a thick B e A torsion subcategory of A. e Proof. By assumption there exists an exact functor Q′ : A′ → B′ of A Abelian categories admitting a section functor S ′ with B ⊂ B′ a replete 0 C ′ ′ e and full Abelian subcategory, A⊂A thick, and Q = Q |A : A → B. First note that C = ker Q = C∩A for C := ker Q′. Define A⊂A′ as C the preimage2 of B under Q′,e i.e., the fulle subcategory of A′ withe object class Obj A = {M ′ ∈ A′ | Q′(M ′) ∈ B}. A is a full replete Abelian subcategory of A′. The section functore S ′ maps e ′ ′ ′ objects in B to objects in A since the counit δ : Q ◦ S → IdB′ is an isomorphism. Hence, the e adjunction induces a restricted adjunction Q ⊣ (S : B→ A) and the counit δ : Q ◦ S → IdB is still an isomorphism. Proposition 3.1 impliese thatf A/C ≃e B since C = ker Qe =e kerfQ′ ⊂ A. The assertion B ≃ A/C ≃A/C now follows frome e Propositione3.2 oncee we have showne e e e that A → A/C is essentially surjective. As SatC(A) → A/C is essentially surjective (even an e e e e e e equivalence) we need to show that for every M ∈ SatC(A) there exists an M ∈A and M → M e 2Note that we didn’t need the preimage A in the statement of the proposition. e CHARACTERIZINGSERREQUOTIENTSWITHNOSECTIONFUNCTOR 5 with kernel and cokernel in C. Let M ∈ A be a preimage of Q(M) ∈ B under the essentially e′ e surjective restriction Q = Q = Q|A : A → B. Then |A e −1 ∼ ηe (S ◦ Q)(M) −−−−−−−−→= (S ◦ Q)(M) −−→M M. e e ∼ f e Q(M)=∼Q(M) f e = ηe Furthermore, M −−→M (S ◦ Q)(M) has kernel and cokernel in C.  f e e 4. APPLICATIONS TO COHERENT SHEAVES 4.1. Coherent sheaves on projective schemes. Let A be a commutative unitial ring and S = Z Pn A[x0,...,xn] the -graded polynomial ring over A with deg xi =1 for all i. Let A = Proj S Pn the n-dimensional projective space A. Denote by Coh A the category of coherent sheaves over Pn A. The category S-qfgrmod of quasi finitely generated graded S-modules is the full subcat- egory in the category of (not necessarily finitely generated) graded S-modules M where the truncated submodule M≥d is finitely generated for d ∈ Z high enough. Further, denote by 0 S-qfgrmod its thick subcategory of S-modules M with M≥d =0 for d ∈ Z high enough. 3 Pn Theorem 4.1 ([Ser55, GD61] ). The sheafification functor Sh : S-qfgrmod → Coh A is 0 Pn exact with kernel S-qfgrmod . The functor Γ• : Coh A → S-qfgrmod is right adjoint to Sh and the counit Pn is an isomorphism. The adjunction induces by δ : Sh ◦Γ• → IdCoh A Sh ⊣ Γ• Proposition 3.1ean adjoint equivalence of categories

Γ• - P S qfgrmod Coh A ∼ 0 . Sh S-qfgrmod Let S-grmod denote the category of finitely presented graded S-modules and S-grmod0 its thick subcategory of S-modules M with M≥d =0 for all d large enough. Corollary 4.2. Let A be a Noetherian ring4. The exact and essentially surjective sheafifica- Pn tion functor Sh : S-grmod → Coh A induces an equivalence of categories S-grmod ≃ Coh Pn , S-grmod0 A where S-grmod0 coincides with the kernel of the sheafification functor. Proof. Define A := S-grmod ⊂ S-qfgrmod =: A ⊃ C := S-qfgrmod0 and C := S-grmod0 = A ∩ C. To apply Proposition 3.2 to the thick subcategorye e A ⊂ A we need to show that the re- e e stricted canonical functor Q|A : A→ A/C is essentially surjective. By definition of S-qfgrmod e for each M ∈ S-qfgrmod there existse a de= d(M) ∈ Z high enough such that M≥d is finitely

3The case of A a field is treated in [Ser55, Prop. III.2.3, Prop. III.2.5, Theo. III.2.2, Prop. III.2.7, Prop. III.2.8, Prop. III.3.6] and the general case in [GD61, Prop. 3.2.4, Prop. 3.4.3.ii, Prop. 3.3.5, Theorem. 3.4.4]. The adjoint- ness is shown there by proving the zig-zag identities. 4This guarantees that S-grmod is an Abelan subcategory of S-qfgrmod. 6 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN

0 generated, i.e., lies in S-grmod. Further, M/M≥d lies in S-qfgrmod by definition of the latter. S-qfgrmod Hence and have isomorphic images in 0 and we obtain M≥d M S-qfgrmod S-qfgrmod S-grmod ≃ ≃ Coh Pn .  S-qfgrmod0 S-grmod0 A S-qfgrmod S-grmod Remark 4.3. Although the two Serre quotient categories S-qfgrmod0 ≃ S-grmod0 yield equi- Pn valent representations of the category Coh A we now show that the thick torsion subcategory S-grmod0 ⊂ S-grmod is not localizing, even though S-qfgrmod0 ⊂ S-qfgrmod is. S-grmod Therefore, let for a field and assume that there exists a functor S 0 S = k[x, y] k : S-grmod → S-grmod - right adjoint to the canonical functor Q - 0 . Define S grmod : S grmod → S-grmod M := S/hyi n −n Z P1 and N := x M for each n ∈ . The sheafification of M is a skyscraper on k. As Q(N n) =∼ Q(M) there exists by the Hom-adjunction a morphism ϕn : N n → (S ◦ Q)(M) with kernel and cokernel in S-grmod0. As N n and M are not in S-grmod0 the morphism ϕn is nonzero. Thus, ϕn must map the cyclic generator of N n of degree −n to a nonzero element of degree −n in (S ◦ Q)(M). In particular, dimk((S ◦ Q)(M))−n > 0 for all n ∈ Z and (S ◦ Q)(M) is not finitely generated. This is a contradiction. Pn Z In [BLH14a] we will prove that Coh A admits for each d ∈ yet another representation 0 as the Serre quotient S-grmod≥d/S-grmod≥d, where S-grmod≥d is the category of finitely 0 presented graded S-modules vanishing in degrees

Γ• : qCoh XΣ → S-grMod : F 7→ M Γ(XΣ, F(α)) α∈Cl XΣ is right inverse but not right adjoint to the exact sheafification functor Sh : S-grMod → qCoh XΣ. Recently Perling found the right adjoint of Sh, also valid in the singular case.

Theorem 4.4 ([Per14, Theorem 3.8 and Remark 3.9]). Let XΣ be a toric variety with no torus factor. There exists a so-called lifting functor Γ right adjoint to the exact sheafification b functor Sh : S-grMod → qCoh XΣ, where the counit of the adjunction δ : Sh ◦Γ → IdqCoh XΣ is a natural isomorphism. b This allows us to characterize toric sheaves as quotients of finitely generated modules. CHARACTERIZINGSERREQUOTIENTSWITHNOSECTIONFUNCTOR 7

Corollary 4.5. Let XΣ be a toric variety with no torus factor. The exact and essentially surjective sheafification functor Sh : S-grmod → Coh XΣ induces the equivalence S-grmod ≃ Coh XΣ, S-grmod0 of categories where S-grmod0 is defined as the kernel of the sheafification functor.

Proof. The essential surjectivity of Sh is the statement of [Mus02, Cor. 1.2] (cf. also [CLS11, Prop. 6.A.4]). Thus, the assumption of Proposition 3.3 is fulfilled with Q′ ⊣ (S ′ : B′ → A′) being the adjunction Sh ⊣ (Γ: qCoh XΣ → S-grMod) from Theorem 4.4.  b In fact Perling proved Theorem 4.4 in a more general setup described in [Per14] following [Hau08, ADHL]. This setup covers toric varieties over arbitrary fields, Mori dream spaces, and categories of equivariant coherent sheaves on them. Furthermore, Trautmann and Perling proved in [PT10, Proposition 5.6.(2),(4)] that the sheafification functor restricted to the subcate- gory of finitely generated graded modules is essentially surjective onto the category of coherent sheaves.

APPENDIX A. THENONEXISTENCEOFAFUNDAMENTALHOMOMORPHISMTHEOREM In this appendix we show that a naive fundamental homomorphism theorem of exact functors between Abelian categories cannot exist. Such a theorem would imply that the corestriction of an exact faithful functor to its image5 is full. To justify our counterexample we need the following argument. Lemma A.1. An exact faithful functor of Abelian categories is conservative6.

Proof. Let F : A → B be such a functor and ϕ : M → N be a morphism in A. Consider ϕ the exact A-sequence 0 → ker ϕ −→ι M −→ N −→π coker ϕ → 0. As B is Abelian and F exact, F (ϕ) is an isomorphism iff the morphisms F (ι) and F (π) are zero in B. The faithfulness of F implies that then ι and π are zero. Finally, since A is Abelian it follows that ϕ is an isomorphism. 

Example A.2. Let B be the category of k-vector spaces, G a nontrivial group, and A the category of G-representations with object (V, ρ : G → GL(V )) and G-equivariant morphisms in B. The F : A → B, (V, ρ) 7→ V is exact and faithful with im F = B. First note that 1V ∈ HomB(V,V ) is in the image of 1(V,ρ) under F . However, 1V ∈ HomB(V,V ) ′ cannot be in the image under F of HomA((V, ρ), (V, ρ )) for two inequivalent representa- tions (V, ρ) 6∼= (V, ρ′); a preimage would be an A-isomorphism since F is conservative by Lemma A.1. A contradiction.

5The image of a functor G : A→B, denoted by im G , is the smallest subcategory of B which contains all image morphisms G (ϕ). 6A functor F is called conservative if reflects isomorphisms, i.e., if F (ϕ) iso =⇒ ϕ iso. 8 MOHAMEDBARAKATANDMARKUSLANGE-HEGERMANN

This shows that the corestriction of an exact faithful functor to its image is not necessarily full, in particular, it is not an equivalence7 of categories. Any exact functor G : A → B of Abelian categories with kernel C := ker G induces a unique faithful exact functor H : A/C → B such that G = H ◦ Q. This is the universal property of the canonical functor Q : A→A/C. Note that im Q is in general strictly contained in A/C and that im G is strictly contained in im H . In the rest of the appendix we want to define an image-notion for which equality holds in both cases. We define the conservative image of a functor G : A → B, denoted by conim G , to be the smallest subcategory of B which contains im G and inverses of B-isomorphisms in im G . We call G conservatively surjective if conim G = B. The name is motivated by conim G = im G for any G . Lemma A.3. Let Q : A→D, G : A → B, and H : D → B be functors with H ◦ Q = G and Q conservatively surjective. Then conim G = conim H .

Proof. It is clear that conim H ⊃ conim G . Every morphism α in conim H is a finite com- ǫi position of morphisms of the form H (ϕi) , with ǫi ∈ {±1}. Since conim Q = D each ϕi is σij in turn a finite composition of morphisms of the form Q(ψij ) , with σij ∈ {±1}. Finally α is ǫiσij then a finite composition of morphisms of the form G (ψij) .  Proposition A.4. Let G : A → B be an exact functor of Abelian categories, C := ker G , H : A/C → B the unique functor such that G = H ◦ Q. Then H is an exact, faithful, and conservative functor. Furthermore im H = conim H = conim G .

Proof. H is exact and faithful by definition and hence conservative by Lemma A.1. Hence im H = conim H . The canonical functor Q is conservatively surjective, as every morphism ι ϕ π ϕ : M → N in A/C is of the form Q(M ′ ֒→ M)−1Q(M ′ −→ N/N ′)Q(N ։ N/N ′)−1, with M/M ′, N ′ ∈ C. The equality conim H = conim G follows from Lemma A.3. 

REFERENCES [ADHL] Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cox rings (project) (http://www.mathematik.uni-tuebingen.de/~hausen/CoxRings/download.php?name=coxrings.pdf). 7 [BLH11] Mohamed Barakat and Markus Lange-Hegermann, An axiomatic setup for algorithmic and an alternative approach to localization, J. Algebra Appl. 10 (2011), no. 2, 269–293, (arXiv:1003.1943). MR 2795737 (2012f:18022) 2

7 Enlarging the image only deepens the problem. In the following example, due to Eisele [Eis], an inclusion functor of an Abelian subcategory has the inequivalent target category as its essential image: Let k be a field and (R, m) be a finite dimensional local k-algebra with R/m =∼ k. Let A be the category of R- modules and B be the category with the same objects as A but HomB(M,N) := Homk(M,N), i.e., all k- homomorphisms. The forgetful (identity on objects) functor F : A→B is clearly exact and surjective (on objects). B is equivalent to the category of all k-vector spaces as every k-vector space can be seen as an R- via k =∼ R/m. The kernel of F is the subcategory 0A of zero objects in A. But A/0A ≃ A is not equivalent to B (if R 6= k). CHARACTERIZINGSERREQUOTIENTSWITHNOSECTIONFUNCTOR 9

[BLH13] Mohamed Barakat and Markus Lange-Hegermann, On monads of exact reflective localizations of Abelian categories, Homotopy Appl. 15 (2013), no. 2, 145–151, (arXiv:1202.3337). MR 3138372 2 [BLH14a] Mohamed Barakat and Markus Lange-Hegermann, A constructive approach to coherent sheaves via Gabriel monads,(arXiv:1409.6100), 2014. 6 [BLH14b] Mohamed Barakat and Markus Lange-Hegermann, Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves,(arXiv:1409.2028), 2014. 2 [CLS11] DavidA. Cox, JohnB. Little, andHenryK. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322 (2012g:14094) 6, 7 [Eis] Florian Eisele, Does there exist a wide but not full abelian subcategory of an abelian category?, Math- Overflow, (accessed 2012-08-21): (http://mathoverflow.net/questions/103868). 7 [Gab62] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. MR 0232821 (38 #1144) 2, 3 [GD61] Alexandre Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique II, Publications Mathématiques, vol. 8, Institute des Hautes Études Scientifiques., 1961. 5 [Gro57] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221, Translated by Marcia L. Barr and Michael Barr: Some aspects of homological algebra, (ftp://ftp.math.mcgill.ca/barr/pdffiles/gk.pdf). MR MR0102537 (21 #1328) 1 [GZ67] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125 (35 #1019) 3 [Hau08] Jürgen Hausen, Cox rings and combinatorics. II, Mosc. Math. J. 8 (2008), no. 4, 711–757, 847. MR 2499353 (2010b:14011) 7 [Kra97] HenningKrause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), no. 3, 259–271. MR 1426488 (98e:18006) 2 [Len69] Helmut Lenzing, Endlich präsentierbare Moduln, Arch. Math. (Basel) 20 (1969), 262–266. MR 0244322 (39 #5637) 2 [Mus02] Mircea Musta¸ta,˘ Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no. 3, 451–470. MR 1916637 (2003e:14013) 6, 7 [Per14] Markus Perling, A lifting functor for toric sheaves, Tohoku Math. J. (2) 66 (2014), no. 1, 77–92, (arXiv:1110.0323). MR 3189480 6, 7 [PT10] M. Perling and G. Trautmann, Equivariant primary decomposition and toric sheaves, Manuscripta Math. 132 (2010), no. 1-2, 103–143, (arXiv:0802.0257). MR 2609290 (2011c:14051) 7 [Ser55] Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278. MR MR0068874 (16,953c) 5

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KAISERSLAUTERN, 67653 KAISERSLAUTERN, GER- MANY E-mail address: [email protected]

LEHRSTUHL B FÜR MATHEMATIK, RWTH AACHEN UNIVERSITY, 52062 AACHEN,GERMANY E-mail address: [email protected]