Elements of Computer-Algebraic Analysis 2 G

Operator algebras, partial classiﬁcation Operator algebras, partial classiﬁcation More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks)

1 J. C. McConnell, J. C. Robson, ”Noncommutative Noetherian Rings”, Graduate Studies in Mathematics, 30, AMS (2001) Elements of Computer-Algebraic Analysis 2 G. R. Krause and T. H. Lenagan, ”Growth of Algebras and Gelfand-Kirillov Dimension”, Graduate Studies in Mathematics, 22, AMS (2000) Viktor Levandovskyy 3 S. Saito, B. Sturmfels and N. Takayama, “Gröbner Lehrstuhl D fürMathematik, RWTH Aachen, Germany Deformations of Hypergeometric Differential Equations”, Springer, 2000 4 J. Bueso, J. Gómez–Torrecillas and A. Verschoren, 22.07. Tutorial at ISSAC 2012, Grenoble, France ”Algorithmic methods in non-commutative algebra. Applications to quantum groups”, Kluwer , 2003 5 H. Kredel, ”Solvable polynomial rings”, Shaker Verlag, 1993 6 H. Li, ”Noncommutative Gröbnerbases and filtered-graded transfer”, Springer, 2002

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classiﬁcation Operator algebras, partial classiﬁcation More general framework: G-algebras More general framework: G-algebras Recommended literature (textbooks and PhD theses) Software

D-modules and algebraic analysis: 1 V. Ufnarovski, ”Combinatorial and Asymptotic Methods of Algebra”, Springer, Encyclopedia of Mathematical Sciences 57 kan/sm1 by N. Takayama et al. (1995) D-modules package in Macaulay2 by A. Leykin and H. Tsai 2 F. Chyzak, ”Fonctions holonomes en calcul formel”, Risa/Asir by M. Noro et al. PhD. Thesis, INRIA, 1998 OreModules package suite for Maple by D. Robertz, 3 V. Levandovskyy, ”Non-commutative Computer Algebra for A. Quadrat et al. polynomial algebras: Gr¨obner bases, applications and Singular:Plural with a D-module suite; by V. L. et al. implementation” PhD. Thesis, TU Kaiserslautern, 2005 holonomic and D-ﬁnite functions: 4 C. Koutschan, ”Advanced Applications of the Holonomic Groebner, Ore algebra, Mgfun, ... by F. Chyzak Systems Approach”, PhD. Thesis, RISC Linz, 2009 HolonomicFunctions by C. Koutschan 5 K. Schindelar, ”Algorithmic aspects of algebraic system theory”, PhD. Thesis, RWTH Aachen, 2010 Singular:Locapal (partly under development) by V. L. et al.

Algebraic Analysis Operator algebras and their partial classification 1 As a notion, it arose in 1958 in the group of Mikio Sato More general: G-algebras and Gröbnerbases in G-algebras (Japan) Module theory; Dimension theory; Gel’fand-Kirillov dimension 2 Main objects: systems of linear partial DEs with variable Linear modeling with variable coefficients coefficients, generalized functions Elimination of variables and Gel’fand-Kirillov dimension 3 Main idea: study systems and generalized functions in a Ore localization; smallest Ore localizations coordinate-free way (i. e. by using modules, sheaves, Solutions via homological algebra categories, localizations, homological algebra, . . . ) The complete annihilator program 4 Keywords include D-Modules, (sub-)holonomic D-Modules, Some computational D-module theory, Weyl closure regular resp. irregular holonomic D-Modules Purity; pure modules, pure functions, preservation of purity 5 Interplay: singularity theory, special functions, . . . . Purity filtration of a module; connection to solutions Other ingredients: symbolic algorithmic methods for discrete resp. continuous problems like symbolic summation, symbolic integration Jacobson normal form etc.

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classiﬁcation Operator algebras, partial classiﬁcation More general framework: G-algebras More general framework: G-algebras What is computer-algebraic Analysis? Some important names in computer-algebraic analysis

Algebraization as a trend W. Gr¨obnerand B. Buchberger: Gr¨obnerbases and Algebra: Ideas, Concepts, Methods, Abstractions constructive ideal/module theory O. Ore: Ore Extension and Ore Localization Computer algebra works with algebraic concepts in a I. M. Gel’fand and A. Kirillov: GK-Dimension (semi-)algorithmic way at three levels: B. Malgrange: M. isomorphism, M. ideal, . . . 1 Theory: Methods of Algebra in a constructive way J. Bernstein, M. Sato, M. Kashiwara, C. Sabbah, 2 Algorithmics: Algorithms (or procedures) and their Z. Mebkhout, B. Malgrange et al.: D-module theory Correctness, Termination and Complexity results (if possible) N. Takayama, T. Oaku, B. Sturmfels, M. Saito, M. Granger, 3 Realization: Implementation, Testing, Benchmarking, U. Walther, F. Castro, H. Tsai, A. Leykin et al.: (not only) Challenges; Distribution, Lifecycle, Support and computational D-module theory software-technical aspects ...

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Operator algebras Operator algebras, partial classification Operator algebras More general framework: G-algebras Partial classification of operator algebras More general framework: G-algebras Partial classification of operator algebras Operator algebras: partial Classification

Let K be an effective field, that is (+, , , :) can be performed − · algorithmically. Moreover, let be a K-vector space (”function space”). F Let x be a local coordinate in . It induces a K-linear map Part I. Operator algebras and their partial classification. F X : , i. e. X (f ) = x f for f . Moreover, let F → F · ∈ F o : be a K-linear map. x F → F Then, in general, o X = X o , that is x ◦ 6 ◦ x o (x f ) = x o (f ) for f . x · 6 · x ∈ F

The non-commutative relation between ox and X can be often read oﬀ by analyzing the properties of ox like, for instance, the product rule.

Let g be a sequence in discrete argument k and s is the shift operator s(g(k)) = g(k + 1). Note, that s is multiplicative. Let f : C C be a diﬀerentiable function and ∂(f (x)) := ∂f . → ∂x Thus s(kg(k)) = (k + 1)g(k + 1) = (k + 1)s(g(k)) holds. Product rule tells us that ∂(x f (x)) = x ∂(f (x)) + f (x), what is translated into the following relation between operators The operator algebra, corr. to s is the 1st shift algebra

(∂ x x ∂ 1) (f (x)) = 0. S1 = K k, s sk = (k + 1)s = ks + s . ◦ − ◦ − h | i Intermezzo The corresponding operator algebra is the 1st Weyl algebra For a function in diﬀerentiable argument x and in discrete argument k the natural operator algebra is D1 = K x, ∂ ∂x = x∂ + 1 . h | i A = D S = K x, k, ∂ , s ∂ x = x∂ + 1, s k = ks + s , 1 ⊗K 1 h x k | x x k k k xk = kx, xs = s x, ∂ k = k∂ , ∂ s = s ∂ . k k x x x k k x i

Let k K be fields and q K . ⊂ ∈ ∗ In q-calculus and in quantum algebra three situations are common Algebra with linear (affine) relation for a fixed k: (a) q k, (b) q is a root of unity over k, and Let q K and α, β, γ K. Define ∈ ∗ (c) q is transcendental over k and k(q) K. ∈ ∈ ⊆ (1)(q, α, β, γ) := K x, y yx q xy = αx + βy + γ f (qx) f (x) A h | − · i Let ∂q(f (x)) = (q −1)x be a q-differential operator. x − Because of integration operator (f (x)) := f (t)dt for a R, The corr. operator algebra is the 1st q-Weyl algebra I a ∈ obeying the relation x x = 2 we need yet more general I − I −I R D(q) = K x, ∂ ∂ x = q x∂ + 1 . framework. 1 h q | q · q i Algebra with nonlinear relation (2) The 1st q-shift algebra corresponds to the q-shift operator Let N N and c0,..., cN , α K. Then (q, c0,..., cN , α) is s (f (x)) = f (qx): ∈ n ∈ i A q K x, y yx q xy = i=1 ci y + αx + c0 or h | − · n i i K x, y yx q xy = i=1 ci x + αy + c0 . Kq[x, sq] = K x, sq sqx = q xsq . h | − · P i h | · i P

Theorem (L.–Koutschan–Motsak, 2011) Theorem (L.–Makedonsky–Petravchuk, new) (1)(q, α, β, γ) = K x, y yx q xy = αx + βy + γ , A h | − · i (2) where q K ∗ and α, β, γ K For N 2 and c ,..., c , α K, (q, c ,..., c , α) ∈ ∈ ≥ 0 N ∈ A 0 N is isomorphic to the 5 following model algebras: = K x, y yx q xy = N c y i + αx + c is isomorphic to ... h | − · i=1 i 0i (q) 1 K[x, y], 1 K [x, s] or D , if q = 1, q 1 P6 2 the 1st Weyl algebra D1 = K x, ∂ ∂x = x∂ + 1 , 2 S = K x, s sx = xs + s , if q = 1 and α = 0, h | i 1 h | i 6 3 the 1st shift algebra S1 = K x, s sx = xs + s , 3 K x, y yx = xy + f (y) , where f K[y] with deg(f ) = N, if h | i h | i ∈ 4 the 1st q-commutative algebra K [x, s] = K x, s sx = q xs , q = 1 and α = 0. q h | · i (q) 5 the 1st q-Weyl algebra D = K x, ∂ ∂x = q x∂ + 1 . 1 h | · i

Lemma (L.–Makedonsky–Petravchuk, new) Given a system of equations S in terms of other operators, K x, y yx = xy + f (y) = K z, w wz = zw + g(w) one can look up a concrete isomorphism of K-algebras (e. g. from h | i ∼ h | i the mentioned papers) if and only if λ, ν K and µ K, such that g(t) = νf (λt + µ) (in and rewrite S as S0 in terms of the operators above. ∃ ∈ ∗ ∃ ∈ particular deg(f ) = deg(g)). Further results on S0 after performing computations can be transferred back to original operators. Lemma (L.–Makedonsky–Petravchuk, new)

Example: diﬀerence and divided diﬀerence operators ∆n = Sn 1, For any algebra of the type B = K a, b ba = ab + f (a) for f = 0 (q) (q) − h | i 6 ∆n = Sn 1 etc. there exists an injective homomorphism into the 1st Weyl algebra. −

Let N = deg f (y) = 2 and K be algebraicaly closed ﬁeld of char K > 2. Then there are precisely two classes of non-isomorphic algebras of the type K x, y yx = xy + f (y) : h | i K x, y yx = xy + y 2 type Let A be a bivariate algebra as before. h | i integration algebra K x, x = x 2 , If S is a multiplicatively closed Ore set (see next parts), then h I | I I − I i 1 1 the algebra K x 1, ∂ = d ∂x 1 = x 1∂ (x 1)2 , there exists localization S− A, such that A S− A holds. h − dx | − − − − i ⊂ 1 1 1 1 2 Problem: establish isomorphy classes for the localized algebras the algebra K x, ∂− ∂− x = x∂− (∂− ) etc. 1 h | − i S− A, depending on the type of S. K x, y yx = xy + y 2 + 1 type Example: in the part on localization. h | i tangent algebra K tan, ∂ ∂ tan = tan ∂ + tan2 +1 (take h | · · i y = tan, x = ∂) − the subalgebra of the 1st Weyl algebra, generated by Y = x − and X = (x2 + 1)∂; then YX = XY + Y 2 + 1 etc.

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Construction of G-algebras Operator algebras, partial classification Construction of G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras More general framework: G-algebras Towards G-algebras

α1 α2 αn Let R = K[x1,..., xn]. The standard monomials x1 x2 ... xn , Suppose we are given the following data αi N, form a K-basis of R. ∈ 1 a field K and a commutative ring R = K[x1,..., xn], α α1 α2 αn n Mon(R) x = x x ... x (α1, α2, . . . , αn) = α N . 2 a set C = cij K ∗, 1 i < j n 3 1 2 n 7→ ∈ { } ⊂ ≤ ≤ 3 a set D = d R, 1 i < j n { ij } ⊂ ≤ ≤ 1 a total ordering on Nn is called a well–ordering, if ≺ Assume, that there is a monomial well–ordering on R such that F Nn there exists a minimal element of F , in particular ≺ ∀ ⊆ n a N , 0 a 1 i < j n, lm(dij ) xi xj . ∀ ∈ ≺ ∀ ≤ ≤ ≺ 2 an ordering is called a monomial ordering on R, if ≺ α, β Nn α β x α x β To the data (R, C, D, ) we associate an algebra ∀ ∈ ≺ ⇒ ≺ α, β, γ Nn such that x α x β we have x α+γ x β+γ . ≺ ∀ ∈ ≺ ≺ α A = K x ,..., x x x = c x x + d 1 i < j n . 3 Any f R 0 can be written uniquely as f = cx + f 0, h 1 n | { j i ij · i j ij } ∀ ≤ ≤ i ∈ \{ } α α α with c K ∗ and x 0 x for any non–zero term c0x 0 of f 0. lm(f )∈ = xα, the≺leading monomial of f A is called a G–algebra in n variables, if lc(f ) = c, the leading coefficient of f . c c d x x d + c x d c d x + d x c c x d = 0. ik jk · ij k − k ij jk · j ik − ij · ik j jk i − ij ik · i jk VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Construction of G-algebras Operator algebras, partial classification Construction of G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras G-algebras Classical examples: full shift algebra

Adjoining the backwards shift s 1 : f (x) f (x 1) to the shift − 7→ − algebra, we incorporate several more relations, which deﬁne a Theorem (Properties of G-algebras) so-called full shift algebra: Let A be a G-algebra in n variables. Then 1 1 1 1 1 K x, s, s− sx = (x + 1)s, s− x = (x 1)s− , s− s = s s− = 1 A is left and right Noetherian, h | − · i A is an integral domain, Note: full shift algebra is not a G-algebra, due to the relation the Gel’fand-Kirillov dimension over K is GKdim(A) = n, 1 s s− = 1. But it can be realized as a factor algebra of a the global homological dimension gl. dim(A) n, · ≤ G-algebra the generalized Krull dimension Kr. dim(A) n. A = K x, s, s 1 sx = (x + 1)s, s 1x = (x 1)s 1, s 1s = ss 1 ≤ h − | − − − − − i A is Auslander-regular and a Cohen-Macaulay algebra. modulo the two-sided ideal s 1s 1 . h − − i We can also realize this algebra as an Ore localization of the shift algebra, see next parts.

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Construction of G-algebras Operator algebras, partial classification Construction of G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras More general framework: G-algebras Properties and Gröbnerbases in G-algebras GröbnerBases in G-algebras GröbnerBases in G-algebras

Let A be a G-algebra in x1,..., xn. From now on, we assume that a given ordering is a well-ordering. There exists a generalized Buchberger’s algorithm (as well as Definition other generalized algorithms for Gröbner bases), which works along the lines of the classical commutative algorithm. We say that xα xβ, i. e. monomial xα divides monomial xβ, if | αi βi i = 1 ... n. There exist algorithms for computing a two-sided Gröbner ≤ ∀ basis, which has no analogon in the commutative case. β α n It means that x is reducible by x , that is there exists γ N , G-algebras are fully implemented in the actual system α γ β ∈ such that β = α + γ. Then lm(x x ) = x , hence Singular:Plural, as well as in older systems MAS, α γ β x x = cαγx + lower order terms. Felix. Definition In Singular:Plural there are many thorougly implemented Let be a monomial ordering on A, I A be a left ideal and functions, including Gröbnerbases, Gröbnerbasics (module ≺ ⊂ G I be a finite subset. G is called a (left) Gröbnerbasis of I , arithmetics) and numerous useful tools. ⊂ if f I r 0 there exists a g G satisfying lm(g) lm(f ). ∀ ∈ { } ∈ |

VL Elements of CAAN VL Elements of CAAN Operator algebras, partial classification Construction of G-algebras Systems, modules, solutions From system of equations to modules More general framework: G-algebras Properties and Gröbnerbases in G-algebras Modeling From modules to solutions of systems Dimensions From functions to modules GröbnerTechnology = Gröbnertrinity + Gröbner basics Gröbnertrinity: left Gröbnerbasis of a submodule of a free module left syzygy module of a given set of generators left transformation matrix, expressing elements of Gröbner basis in terms of original generators Gröbnerbasics (Buchberger, Sturmfels, ...) Part II. Dimension theory. Ideal (resp. module) membership problem (NF, reduce) Intersection with subrings (eliminate) Intersection and quotient of ideals (intersect, quot) Kernel of a module homomorphism (modulo) Kernel of a ring homomorphism (preimage) Algebraic dependencies of commuting polynomials Hilbert polynomial of graded ideals and modules ...

VL Elements of CAAN VL Elements of CAAN Systems, modules, solutions From system of equations to modules Systems, modules, solutions From system of equations to modules Modeling From modules to solutions of systems Modeling From modules to solutions of systems Dimensions From functions to modules Dimensions From functions to modules From system of equations to modules

O := K n, s s n = ns + s K x, ∂ ∂ x = x∂ + 1 . h n | n n ni ⊗K h x | x x i Consider Legendre’s differential equation (order 2 in ∂ ) x From the system of equations 2 2 (x 1)P00n(x) + 2xP0n(x) n(1 + n)Pn(x) = 0 2 2 − − (x 1)P00n(x) + 2xP0n(x) n(1 + n)Pn(x) = 0, − − (n + 1)Pn+1(x) (2n + 1)xPn(x) + nPn 1(x) = 0. − − x is differentiable with ∂x as corr. operator one obtains the matrix P O2 1; thus M = O/O1 2P and ∈ × × if n Z, n is discretely shiftable with sn as corr. op. ∈ 2 2 (x 1)∂x + 2x∂x n(1 + n) 0 − 2 − Pn(x) = . (n + 2)sn (2n + 3)xsn + n + 1 • 0 then there is a recursive formula of Bonnet (order 2 in shift sn) − With the help of Gröbnerbases over O: a minimal generating set (n + 1)Pn+1(x) (2n + 1)xPn(x) + nPn 1(x) = 0. of the left ideal P contains a compatibility condition − − (n + 1)s ∂ (n + 1)x∂ (n + 1)2 (n + 1)(s ∂ x∂ + n + 1). n x − x − ≡ n x − x

Let f1(x1,..., xn),..., fm(x1,..., xn) be unknown generalized n Different matrices P can represent the same module M. functions, for instance from C ∞(R ). i Then a homogeneous system of linear functional (operator) ` ` equations with coefficients from K[x1,..., xn] can be presented via For instance, for any unimodular T O × one has 1 ∈m 1 ` 1 m 1 ` the matrix equation in the corresponding operator algebra O: Pf = 0 (TP)f = 0 and also O × /O × TP = O × /O × P. ⇔ ∼ For various purposes we might utilize different presentations of M. f1 0 . . ` m The invariants of a module M, like dimensions, do not depend on P . = . , P O × ·  .   .  ∈ the presentation. fm 0     Algebraic manipulations from the left on P often need algorithms     1 m 1 ` for left Gröbnerbases for a submodule of a free module, generated One associates to the system a left O-module M = O × /O × P, saying M is finitely presented by a matrix P. by rows or columns of P (thus not only GBs of ideals).

Let be a left O-module (not necessarily finitely presented), and F f . Consider Of = o f o O , which is an O-submodule Let be a left O-module (not necessarily finitely presented), and ∈ F { • | ∈ } F of . P a system of equations as before, then F Consider a homomorphism of left O-modules m 1 SolO(P, ) := f × : P f = 0 . φ : O , o o f , in other words φ (1) = f . Then F { ∈ F • } f → F 7→ • f ∈ F Imφ = Of , Ker φ = o O : o f = 0 =: Ann f f f { ∈ • } O Noether-Malgrange Isomorphism as left O-modules, one has Of ∼= O/ AnnO f There exists an isomorphism of K-vector spaces hence Of is finitely presented left O-module.

1 m 1 ` Hom (M, ) = Hom (O × /O × P, ) = Sol (P, ), An element m is called a torsion element, if AnnO m = 0. O F O F ∼ O F ∈ F 6 m 1 Many classical functions in common functional spaces are torsion. (φ : M ) (φ([e ]), . . . , φ([e ])) × . → F 7→ 1 m ∈ F Hence, algorithms for the computation of the left ideal AnnO m (which is ﬁnitely generated when O is Noetherian) are very important. VL Elements of CAAN VL Elements of CAAN Systems, modules, solutions From system of equations to modules Systems, modules, solutions From system of equations to modules Modeling From modules to solutions of systems Modeling From modules to solutions of systems Dimensions From functions to modules Dimensions From functions to modules From functions to modules From functions to modules

Let be a left O-module, and f ,..., f be torsion elements. F 1 m ∈ F Consider M = Of1 + ... + Ofm. As we know, every Ofi is finitely presented O-submodule of . Many classical functions in common functional spaces are torsion. F But not all. Consider a homomorphism of left O-modules Example: f = tan(x) is not a torsion element in a module over m φ : Om = Oe , o e o f , Weyl algebra, since there exists no system of linear ODEs with i → F i i 7→ i • i variable coefficients, having tan(x) as solution. However, there is a Mi=1 X X 2 nonlinear ODE f 0 = 1 + f . in other words φ(e ) = f . Then Im φ = M = Of , i i ∈ F i m Recall: we are able to treat polynomials in the operator tan(x) as Ker φ = [o1,..., om] O : oi fi = 0 =: MannO M · { ∈ • } P coefficients in an algebra with differentiation w.r.t x. m as left O-modules, one has MP= Ofi ∼= O / MannO M hence M = i Ofi is finitely presentedP left O-module. Clearly Ker φ eP Mann M. ⊕ fi i ⊆ O VL Elements of CAAN In general there is no left idealVLI ElementsO, such of CAAN that m ⊂ O / MannO M ∼= O/I . Systems, modules, solutions Systems, modules, solutions Gel’fand-Kirillov dimension Modeling VMPUM Modeling GK-dimension and elimination Dimensions Dimensions Dimensions Idea: Model polynomial-exponential signals by linear systems. Question: What is more precise in such a modeling: operator Generalized Krull dimension (for an algebra or a module, algebras with constant or with polynomial coefficients? Kr. dim M) is called Krull-Rentschler-Gabriel dimension; not algorithmic Answer: algebras with polynomial coefficients. Theorem (Zerz–L.–Schindelar, 2011) projective dimension of a module, p. dim M; algorithmic ` (relatively expensive), implemented Let K = R, pi K[x1,..., xn] and V = Kp1 + + Kpm. Let O ∈ ··· global homological dimension of an algebra, gl. dim A = be the n-th Weyl algebra and O AnnO(V ) := AnnO pi be the ⊃ ∩ sup p. dim M : M A mod , in general not algorithmic left ideal of operators, simultaneously annihinalting p1,..., pm. { ∈ − } Then ` SolO( O/ AnnO(V ), C ∞(R )) = V . homological grade of a module, j(M); algorithmic (a little less expensive than p. dim M), implemented Keywords: Variant Most Powerful Unfalsified Model, cf. two recent papers by Zerz, L. and Schindelar. Gel’fand-Kirillov Dimension; algorithmic (relatively cheap), implemented; intuition: similar to usual Krull dimension

Let M M be a finite K-vector space, spanned by the generators 0 ⊂ Let A be a K-algebra, generated by x1,..., xm. of M. That is dim M < and AM = M. K 0 ∞ 0 Degree filtration Hd := Vd M0, d N is an induced ascending filtration on M. { ∈ } Let V = Kx ... Kx be a vector space. 1 ⊕ ⊕ m The Gel’fand-Kirillov dimension of M is defined as follows Set V = K, V = K V and V = V V k+1. If 0 1 ⊕ k+1 k ⊕ GKdim(M) = lim sup (logd (dimK Hd )) d ∞ →∞ Vi Vi+k , Vi Vj Vi+j , A = Vk , ⊆ · ⊆ In the standard construction one puts deg xi := 1 and defines k[=0 V := f deg f = d and V d := f deg f d . d { | } { | ≤ } then Vk k N is the standard (ascending) filtration of A. { | ∈ } Conventions: GKdim(0) = . GKdim (Q) = 0. −∞ Q

VL Elements of CAAN VL Elements of CAAN Systems, modules, solutions Systems, modules, solutions Gel’fand-Kirillov dimension Gel’fand-Kirillov dimension Modeling Modeling GK-dimension and elimination GK-dimension and elimination Dimensions Dimensions Gel’fand-Kirillov dimension: examples and properties Lemma Let A be a K-algebra and a domain. If the standard ﬁltration on A is compatible with the PBW Basis xα α Nm , then { | ∈ } Free associative algebra T = K x1,..., xn , n 2 GKdimK (A) = m. h i ≥ d d nd+1 1 nd+1 1 d dim Vd = n , dim V = n −1 . Note, that n −1 > n . d − d − Since logd n = d logd n = , d , it follows that d + m 1 d + m logn d → ∞ → ∞ dim V = − , dim V d = . GKdim(T ) = . d m 1 m ∞ − d+m (d+m)...(d+1) dm Thus m = m! = m! + ... and Properties GKdim M = sup GKdim(N): N A mod, N M , d + m { ∈ − ⊆ } GKdim(A) = lim sup log = m. GKdim A = sup GKdim(S): S A, S ﬁn. gen. subalgebra d d m →∞ { ⊆ } Hence, if K = , then GKdim(K[[x ,..., x ]]) = for n 1. | | ∞ 1 n ∞ ≥ Hence for any G-algebra A in n variables has GKdimK (A) = n.

VL Elements of CAAN VL Elements of CAAN Systems, modules, solutions Systems, modules, solutions Gel’fand-Kirillov dimension Gel’fand-Kirillov dimension Modeling Modeling GK-dimension and elimination GK-dimension and elimination Dimensions Dimensions Gel’fand-Kirillov dimension for modules Lemma (R is commutative) (i) Let R be a commutative affine K-algebra. Then (by Noether There is an algorithm by Gomez-Torrecillas et. al., which computes normalization) S = K[x ,..., x ] R and R is finitely Gel’fand-Kirillov dimension for finitely presented modules over ∃ 1 t ⊆ generated S-module. Then GKdimK R = Kr. dim S = t. G-algebras over ground field K. It is implemented e. g. in Singular:Plural. (ii) If R is an integral domain, GKdimK R = tr. degK Quot(R). GKdimK (F ) For any K-algebra R: GKdim R[x ,..., x ] = GKdim R + m. 1 m Let A be a G-algebra in variables x ,..., x . Curiosity: GKdim(R) 0, 1 [2, + ). 1 n ∈ { } ∪ ∞ Input: Left generating set F = f ,..., f Ar ◦ { 1 m} ⊂ Exactness r r Output: k N, k = GKdim(A /M), where M = A F A . Let R be an affine algebra with finite standard fin.-dim. filtration, ◦ ∈ h i ⊆ G =LeftGrobnerBasis¨ (F ) = g ,..., g ; such that Gr R is left Noetherian. Then GKdim is exact on short { 1 t } L = lm(g ) = xαi e 1 i t ; exact sequences of fin. gen. left R-modules. That is, { i s | ≤ ≤ } N = L ; K[x1,...,xn]h i 0 L M N 0 GKdim M = sup GKdim L, GKdim N r → → → → ⇒ { } return Kr. dim(K[x1,..., xn] /N);

Recall Legendre’s example:

Lemma (MR, KL) O := K n, sn snn = nsn + sn K K x, ∂x ∂x x = x∂x + 1 . h | i ⊗ h | i Let I A be a left ideal and S A be a subalgebra. Then ⊂ ⊂ Then GKdimK O = 4. I S = 0 implies GKdim A/I GKdim S, ∩ ≥ The Gr¨obnerbasis of the ideal P is GKdim A/I < GKdim S implies I S = 0. ∩ 6 (x2 1)∂2 + 2x∂ n(1 + n), (n + 2)s2 (2n + 3)xs + n + 1, − x x − n − n Recall: Bernstein’s inequality 2 (n + 1)sn∂x (n + 1)x∂x (n + 1) . Let A be the n-th Weyl algebra over K with − − char K = 0 = GKdim K, then GKdim(A) = 2n.

2 2 2 Let 0 = M be an A-module, then GKdimK M n. The leading monomials are x ∂x , nsn , nsn∂x . Hence 6 ≥

GKdim O/P = Kr. dim K[n, s , x, ∂ ]/ x2∂2, ns2, ns ∂ = 2. K n x h x n n x i

VL Elements of CAAN VL Elements of CAAN Systems, modules, solutions Ore Localization and its recognition Localization: commutative case Gel’fand-Kirillov dimension Modeling Ore localization on modules Localization: non-commutative case, Ore GK-dimension and elimination Dimensions The complete annihilator program Smallest Ore localizations Elimination and GK-dimension

Let f , such that Ann f K[x ,..., x ] = 0. Then ∈ F O ∩ 1 n GKdim O/ Ann f n. K O ≥ Proposition (Existence of elimination via dimension) Let O = n O , O = K x , o ... . Moreover, let I O and i=1 i i h i i | i ⊂ GKdim O/I = m. Then for any subalgebra S O, such that Part III. Ore localization. N ⊂ GKdim S m + 1 one has I S = 0. ≥ ∩ 6 Application: For I such that GKdim O/I = m we guarantee that 2n (m + 1) = 2n m 1 variables can be eliminated from I , − − − for instance, if m = n, we can eliminate all but one operators, all but one coordinate variables. More applications will follow ... in the parts, which follow.

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition Localization: commutative case Ore Localization and its recognition Localization: commutative case Ore localization on modules Localization: non-commutative case, Ore Ore localization on modules Localization: non-commutative case, Ore The complete annihilator program Smallest Ore localizations The complete annihilator program Smallest Ore localizations Localization in commutative case Example

Let A = K[x1,..., xn]. for f A K, consider S = f i : i N . Then Let A be a commutative Noetherian domain and ∈ \ { ∈ } 1 1 S a multiplicatively closed set in A, where 0 S. S− A ∼= K[x1,..., xn, f ]. 6∈ this type is called a monoidal localization. 1 The localization of A w.r.t S is a ring AS := S− A together with Another instance: for f1,..., fm A K, deﬁning an injective homomorphism φ : A AS , such that i1 im ∈ \ → S = f ... fm : ij N results in { 1 · ∈ } (i) for all s S φ(s) is a unit in AS , ∈ 1 1 1 1 K[x1,..., xn, ,..., ] = K[x1,..., xn, ] (ii) for all f AS , a A, s S s. t. f = φ(s)− φ(a). f1 fm ∼ f1 fm ∈ ∃ ∈ ∈ ··· i 1 = ( (f1 fm) : i N )− K[x1,..., xn]. ∼ { ··· ∈ }

Example Let A = K[x ,..., x ]. 1 n Example 1 If S = A∗ := A 0 , then S− A = Quot(A) = K(x1,..., xn). \{ } ∼ Let A = K[[x1,..., xn]]. this type is called a rational localization. If S = A∗ := A 0 , then n For p K , consider mp := x1 p1,..., xn pn , a maximal 1 \{ } ∈ h − − i S− A ∼= Quot(A) = K((x1,..., xn)). ideal in K[x1,..., xn]. Deﬁne S = K[x1,..., xn] mp. Then \ Notably, K[[x1,..., xn]] is a local ring (i. e. there is exactly 1 g one maximal ideal). Thus for f := x1 xn and S− A = K[x1,..., xn]p = g K[x1,..., xn], h / mp i 1 ··· { h | ∈ ∈ } S = f : i N one has S− A = Quot(A) = K((x1,..., xn)). { ∈ } ∼ this type is called a geometric localization, it is widely used in algebraic geometry.

Example Let A be a non-commutative Noetherian domain and Let S = A := A 0 . Then S 1A = Quot(A) (quotient S a multiplicatively closed set in A, where 0 S. ∗ \{ } − ∼ 6∈ division ring of a domain). If S is additionally an Ore set in A, then S 1A. ∃ − If K ( S ( A∗, then A AS Quot(A), → → 1 Ore condition For any S, S− A is an A-module (not ﬁnitely generated), For all s S, r A there exist s S, r A, such that in general A is not an S 1A-module. 1 ∈ 1 ∈ 2 ∈ 2 ∈ − 1 1 1 1 r s = s r , that is s r = r s . S− gives rise to a functor A-mod S− A-mod. 1 2 1 2 1− 1 2 2− → Ore condition holds S is an Ore set in A. ⇒

We take A to be one of model algebras and f A K. We will ∈ \ analyze, whether S = f i : i N is an Ore set in A. { ∈ } S is not Ore in A = K x, s sx = (x + 1)s Weyl algebra: S is an Ore set h | i Take s and f k (x) S and suppose, that f `(x) and t A, such Suppose we are given g = d b (x)∂j A with b = 0 and f k ` k ∈ k ∃ ` ∃ ∈ j=0 j ∈ 1 d 6 that sf (x) = f (x)t. Thus f (x)t = f (x + 1) s. But for a ﬁxed k N. ∈ P f (x) - f (x + 1) for f / K, thus such t A does not exist. j i+1 i j+1 j j ∈ ∈ For j N and i + 1 j one has ∂ f = f − (f ∂ + vij ), ∈ ≥ · · where the terms of v A have degree at most j 1 and contain ij ∈ 1 − derivatives up to f (j). Then Let us introduce the notion of Ore closure of a multiplicatively closed set S: (S) is the smallest (w.r.t inclusion) two-sided M d d multiplicative superset of S, which has an Ore property in A. d+k j d+k k j j d j g f = b (x)∂ f = f b (x)(f ∂ + v )f − . · j · · j j+k,j Xj=0 Xj=0

Consider S = si : i N . Then S is an Ore set in A. Lemma (L.–Schindelar, 2011) { ∈ } n This follows from For the shift algebra, (S) = f (x z) n, z N0 . M { ± | ∈ } i k k i ai(x)s s = s ai (x k)s d j k · · − Given g = b (x)s A with b = 0 and h(x) = f (x + z ) i i j=0 j d 0 X X ∈ 6 d with k N, z0 Z. Let us deﬁne gf (x) := h(x i) S. ∈ P ∈ i=0 − ∈ Then Q The resulting algebra is already mentioned full shift algebra:

d d i 1 ( s : i )− K x, s sx = (x + 1)s = d j N ∼ g gf (x) = h(x) bj (x) h(x + j i)s . { ∈ } h | i · · − 1 1 1 1 j=0 i=0,i=j K x, s, s− sx = (x + 1)s, s− x = (x 1)s− , s− s = 1 X Y6 h | − i

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition Localization: commutative case Ore Localization and its recognition Ore localization on modules Localization: non-commutative case, Ore Ore localization on modules The complete annihilator program Smallest Ore localizations The complete annihilator program Smallest localizations: quantum plane With Ore localization we can recognize, that S = f i : i N is not Ore in A = K(q) x, y yx = qxy . { ∈ } h | i 1 K(X )[∂ ; σ , δ ] [∂ ; σ , δ ] = (K[X ] 0 )− K X , ∂ , . . . , ∂ ... 1 1 1 ··· m m m ∼ \{ } h 1 m | i Lemma (L.–Schindelar, 2011) and the functor S 1 connects categories of modules. n z − (S) = f (q± x) n, z N0 is an Ore set in A. M { | ∈ } Algorithmic aspects m m For any g(x) K[x] one has y g(x) = g(q x)y. 1 ∈ Algorithmic computations over S− A can be replaced completely Suppose we are given g = d b (x)y j A and j=0 j ∈ with computations over A. h(x) = f k (q`x) S . Let us define g (x) := d h(q i x) S. ∈ 1 P f i=0 − ∈ Keywords: integer strategy, fraction-free strategy. Then Q For instance, a Gröbnerbasis theory over A induces a Gröbner 1 basis theory over S− A. d d d d j i j i j There are implementations for the rational localization g gf (x) = bj (x)y h(q− x) = h(x) bj (x) h(q − x)y . · · · K(X ) ∂1,... . j=0 i=0 j=0 i=0,i=j X Y X Y6 h i

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition Ore Localization and its recognition Ore localization on modules Ore localization on modules The complete annihilator program The complete annihilator program Induced Gr¨obnerbasis theory in the localization Properties of localized modules

Let A be a G-algebra K[x1,..., xn] ∂1, . . . , ∂n ... . Let A be a K-algebra and S A a mult. closed Ore set in A. h | i ⊂ Suppose that S = K[x1,..., xn] 0 is an Ore set in A. Moreover, let \{ } Moreover, let be an admissible monomial ordering on A, n m ≺X M = A /A P, a finitely presented left A-module, having the elimination property for x, that is ∼ a left A-module, F α n 1 1 X x X ∂i , α N , 1 i n. a left S− A-module. ≺ ≺ ∀ ∈ ∀ ≤ ≤ F S 1M = (S 1A)n/(S 1A)mP. Lemma e− ∼ − − 1 r SolA(M, ) = Sol 1 (S− M, ), A Gröbnerbasis of a submodule N A w.r.t ∼ S− A ⊂ ≺X F F is a non-reduced Gröbnerbasis of a submodule Assume, that as left A-modules. Then e F ⊂ F e 1 1 r r S− N (S− A) = K(x ,..., x ) ∂ , . . . , ∂ ... . e SolA(M, ) SolA(M, ), ⊂ 1 n h 1 n | i F ⊆ F e

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition Ore Localization and its recognition Ore localization on modules Ore localization on modules The complete annihilator program The complete annihilator program Properties of localized modules Properties of localized modules

Here is a typical situation of behaviour of modules under localization. Let Mi be A-modules, satisfying Message: In order to compute generalized solutions, work over unlocalized ring and thus employ target spaces, having torsion under localization. 0 ( M1 ( ... ( Mi ( ... ( Mj ( ... ( Mk ( ... ( Mr ( A 1 Technology: the information, obtained for the localized module After applying S− to this sequence, we obtain (and homomorphism of such etc.), can be and should be used for studying the original module (and homomorphism of such etc.). 1 1 1 1 0 = ... = S− Mi 1 ( S− Mi = ... = S− Mj 1 ( S− Mj ... − − ⊂ 1 1 1 ... S− Mk ( S− Mk+1 = ... = S− Mr = A. ⊂

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition Ore Localization and its recognition Ore localization on modules Ore localization on modules The complete annihilator program The complete annihilator program Elimination, dimension and localization GK-dimension and localization

Drawback of Gel’fand-Kirillov dimension of localized algebras: Lower bound for nontrivially localizable modules it is mathematically hard to determine. It is known, that Suppose that I , S O are such that GKdim S 1A GKdim A. ⊂ − ≥ S is an Ore set in O (so S 1O exists) − Lemma (Very exceptional result) 1 1 (S− O)I = S− O (i. e. I is proper in the localized algebra). 6 Let A be the n-th Weyl algebra, S = K[x1,..., xn] 0 . Then Then I S = 0, what implies GKdim O/I GKdim KS, where KS 1 \{ } ∩ ≥ GKdim S− A = GKdim A = 2n. is the monoid algebra. In the analogous situation for A being n-th shift, q-Weyl algebra or 1 1 Note, that for every J S− O there exists I O such that a quantum space, we have GKdim S− A 3n. 1 1 ∈ ∈ ≥ S− OI = S− OJ (idea: clear denominators). Lemma (Corollary from Makar-Limanov) 1 1 In general, if S− OL = S− O, one has 6 Let K = C, A be the n-th shift algebra and 1 1 GKdim S− O/(S− O)L GKdim O/L. S = K[x1,..., xn] 0 . Then 1 \{ } ≥ GKdim S− A = 3n > 2n = GKdim A.

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition An interesting D-module example Ore Localization and its recognition An interesting D-module example Ore localization on modules Weyl closure Ore localization on modules Weyl closure The complete annihilator program Bernstein-Sato polynomial for varieties The complete annihilator program Bernstein-Sato polynomial for varieties The complete annihilator program The complete annihilator program

There exists no general algorithm, which can compute the Let be function spaces, i. e. K-vector spaces and left complete annihilator program of f over O (where O is an algebra G ⊂ F O-modules over a ﬁxed operator algebra O. with polynomial coeﬃcients).

Let f , then AnnF f := p O : pf = 0 is the ∈ F O { ∈ ∈ F} annihilator of f , which is a left ideal in O. Therefore one investigates some classes of f and develops special methods for the classes.

Let I ( O be an ideal and suppose, that dimK ( ) < . One of successes is computational D-module theory, where G ∞ I is called the complete annihilator of over O, if the following among other one can compute the complete annihilators of G properties hold: s1 sm ”most powerful”: if g rg = 0 for r O, then r I f (x, s) = f1(x1,..., xn) ... fm(x1,..., xm) , fi (x) K[x1,..., xn] ∀ ∈ G ∈ ∈ · · ∈ ”unfalsiﬁed”: Sol (O/I , ) = . n O F G over O = K x , ∂ ∂ x = x ∂ + 1 K[s ,..., s ] K h i i | i i i i i ⊗K 1 m Oi=1 in an algorithmic way. There are implementations. VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition An interesting D-module example Ore Localization and its recognition An interesting D-module example Ore localization on modules Weyl closure Ore localization on modules Weyl closure The complete annihilator program Bernstein-Sato polynomial for varieties The complete annihilator program Bernstein-Sato polynomial for varieties Some computational D-module theory Some computational D-module theory

Let char K = 0, Some very easy examples: Dn(K) = K x1,..., xn, ∂1, . . . , ∂n ∂j xi = xi ∂j + δij be the n-th s+1 s h | i ∂x (x) = (s + 1) (x) , Weyl algebra and D [s] = D K[s]. • · n n ⊗K (1/4)∂2 (x2)s+1 = (s + 1)(s + 1/2) (x2)s , x • · Theorem (J. Bernstein, 1971/72) (2x∂ + ∂ 4s 4) (x2 + x)s+1 = (s + 1) (x2 + x)s . x x − − • · Let f (x) C[x1,..., xn]. Then there exist ∈ Some facts an operator P(s) Dn C C[s], ∈ ⊗ M. Kashiwara: all roots of bf (s) are negative rationals a monic polynomial 0 = bf (s) C[s] of the smallest degree 6 ∈ 1 is always a root; in general the roots lie in ( n, 0) (called the global Bernstein-Sato polynomial), − − such that for arbitrary s the following functional equation holds bf (s) = s + 1 if and only if V (f ) is smooth B. Malgrange: if bf ,p(ξ) = 0 (local Bernstein-Sato polynomial s+1 s 2iπξ P(s) f = bf (s) f . at p V (f )), then e is an eigenvalue of the action on • · ∈ monodromy Let Ann (f s ) = Q(s) D[s] Q(s) f s = 0 D[s] be the D[s] { ∈ | • } ⊂ Complicated Bernstein-Sato polynomials appear for such f , annihilator, then P(s)f b (s) Ann (f s ) holds. − f ∈ D[s] that V (f ) possess complicated singularities VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition An interesting D-module example Ore Localization and its recognition An interesting D-module example Ore localization on modules Weyl closure Ore localization on modules Weyl closure The complete annihilator program Bernstein-Sato polynomial for varieties The complete annihilator program Bernstein-Sato polynomial for varieties More interesting D-module example Numerology of Bernstein data

2 9 2 2 3 2 3 9 2 3 Consider f = (x + 4 y + z 1) x z 80 y z K[x, y, z]. − − − ∈ s Then AnnD[s] f has 13 generators with leading terms

4617 y 3∂ , 513 x2y∂ ,..., 102400 x2z5∂ , 37428480 y 4z5∂ ; · x · x · y · y

Bernstein-Sato poly:b (s) = (s + 1)2 (s + 2 ) (s + 4 ) (s + 5 ) f · 3 · 3 · 3

A reduced operator Pf (S) has 1261 terms, here some leading part of them 1 1 7084781 1 ( xy 2z3 xy 2z2)∂3∂2 + ( yz3 yz2)∂2∂ ∂2 24 − 5760 x z 177292800 − 4104 x y z

Sing(f ) = V1 V2, where 2 ∪ 2 Another success of computational D-module theory is the V1 = V ( x + 9/4y 1, z) an ellipse at z = 0 plane; h 2 − possibility to compute the Weyl closure of certain ideals. V2 = V ( x, y, z 1 ) consists of 2 different points; h 2 − i 2 V3 = V ( 19x + 1, 171y 80, z ) consists of 4 different Let A be a K-algebra, S A a m. c. Ore set in A. Moreover, let h − i 1 ⊂ points; moreover, V3 V1, V2 V3 = . 0 = J ( S− A a left ideal. The restriction of J to A is the ideal ⊂ ∩ ∅ 6 1 (S− A)J A. L. and Mart´ın-Morales, 2012: algorithm for constructing a ∩ 3 stratification of C into constructible sets such that bf ,p(s) is Let A be the n-th Weyl algebra, S = K(x ,..., x ) 0 and a left constant on each stratum. 1 n \{ } ideal J satisfies dim S 1A/(S 1A)J < . Then the Weyl K(x1,...,xn) − − ∞ 1 p C3 V (f ), closure of J is defined to be the restriction of J to A and there is ∈ \ s + 1 p V (f ) (V V ), an algorithm to compute it in finitely many steps.  ∈ \ 1 ∪ 2  2 bf ,p(s) = (s + 1) (s + 4/3)(s + 2/3) p V1 V3, There are implementations of algorithms, computing Weyl closure  ∈ \ (s + 1)2(s + 4/3)(s + 5/3)(s + 2/3) p V , after H. Tsai (Macaulay2, recently in Singular:Plural). ∈ 3 (s + 1)(s + 4/3)(s + 5/3) p V2.  ∈  VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition An interesting D-module example Ore Localization and its recognition An interesting D-module example Ore localization on modules Weyl closure Ore localization on modules Weyl closure The complete annihilator program Bernstein-Sato polynomial for varieties The complete annihilator program Bernstein-Sato polynomial for varieties Weyl closure Bernstein-Sato polynomial for varieties

s s1 sr Example (1st Weyl algebra) In the following, f := f fr . 1 ··· 3 2 Let I = (x + 2)∂x 3x D1. Gr¨obnerbasis of J is then Theorem (Budur, Mustat¸ˇaand Saito, 2006) h − i ⊂ r 3 2 2 2 Let char K = 0. For every r-tuple f = (f1,..., fr ) K[x] there ∂x + x∂x 3, x∂x 2∂x , x ∂x + ∂x 3x ∈ { − − − } exists a non-zero univariate polynomial b(ξ) K[ξ] and r ∈ diﬀerential operators P (S),..., P (S) D S such that w.r.t degree reverse lexicographical ordering and 1 r ∈ nh i 3 2 2 2 r (x + 2)∂x 3x , ∂x + x ∂x 3x s s { − − } Pk (S)fk f = b(s11 + + srr ) f . • ··· · w.r.t the ordering, eliminating x (compatible with localization). Xk=1 Here Dn is the n-th Weyl algebra in xi , ∂i and Open problem Can one develop algorithms for computing analogous closure for Dn S = Dn K K s11,..., snn 1 i, j, k, l n h i ⊗ h | ∀ ≤ ≤ other model algebras? Possible bottleneck: localizations of other s s s s = δ s δ s algebras are more involved, as we know from before. ij kl − kl ij jk il − il kj i

VL Elements of CAAN VL Elements of CAAN Ore Localization and its recognition An interesting D-module example Purity of modules Dimension function Ore localization on modules Weyl closure Pure functions Purity w.r.t dimension function The complete annihilator program Bernstein-Sato polynomial for varieties Jacobson normal form Bernstein-Sato polynomial for varieties

Theorem (Andres–L.–Mart´ın-Morales, ISSAC 2009) Let f = (f ,..., f ) be an r-tuple in K[x]r and let D ∂ , S be the 1 r nh t i K-algebra generated by ∂t and S over Dn subject to relations on S and s ∂t ∂t s = δ ∂t . Consider the left ideal in { ij · k − k · ij jk i } Part IV. Purity. D ∂ , S nh t i r ∂fk 1 i, j r F := sij + ∂ti fj , ∂xm + ∂tk ≤ ≤ . ∂xm 1 m n k=1 ≤ ≤ D X E s Then AnnD S (f ) = Dn ∂t , S F Dn S . h i h i ∩ h i Thus, this is another type of objects, for which complete annihilator program is successful.

VL Elements of CAAN VL Elements of CAAN Purity of modules Purity of modules Dimension function Dimension function Pure functions Pure functions Purity w.r.t dimension function Purity w.r.t dimension function Jacobson normal form Jacobson normal form Dimension function Purity w.r.t dimension function

Let A be a Noetherian algebra. A dimension function δ assigns a Let A be a K-algebra and δ a dimension function on A-mod. value δ(M) to each finitely generated A-module M and satisfies A module M = 0 is δ-pure (or δ-homogeneous), if 6 the following properties: (i) δ(0) = . 0 = N M, δ(N) = δ(M). −∞ ∀ 6 ⊆ (ii) If 0 M0 M M00 0 is exact sequence, then → → → → A simple module is pure. Thus, purity is a useful weakening of δ(M) sup δ(M0), δ(M00) with equality if the sequence is ≥ { } the concept of simplicity of a module. split. Unlike simplicity, the purity (w.r.t a dimension function) is (iii) If P is a (two-sided) prime ideal with P Ann (M) and M is ⊆ A algorithmically decidable over many common algebras. a torsion module over A/P, then δ(M) δ(A/P) 1. ≤ − generalized Krull dimension is an exact dimension function M. Barakat, A. Quadrat: Algorithms for the computation of the Gel’fand-Kirillov dimension is a dimension function, not purity filtration of a module with δ = homological grade; there are always exact several implementations: in homalg, OreModules(Maple) and Singular:Plural.

Consider purity with respect to Gel’fand-Kirillov dimension. Lemma (L.) Lemma (L.) Let A be a G-algebra, S A a m. c. Ore set in A. Let be a set ⊂ M Let A be a K-algebra and δ a dimension function on A-mod. 1 of left A-modules M, satisfying S− M = 0 and having dimension Moreover, let 0 = M1, M2 N be two δ-pure modules with 6 6 ⊂ GKdim KS, where KS is the monoid algebra. Then consists of δ(M1) = δ(M2). Then M pure modules. the set of δ-pure submodules (of the same dimension) of a module is a lattice, i. e. Example (Pure modules)

1 M M is either 0 or it is δ-pure with δ(M M ) = δ(M ), modules of Krull dimension 0 over K[x1,..., xn], i. e. modules 1 ∩ 2 1 ∩ 2 1 M, such that dimK M < 2 M1 + M2 is δ-pure with δ(M1 + M2) = δ(M1). ∞ any set of modules of smallest possible dimension in A, for instance holonomic modules over the n-th Weyl algebra over a ﬁeld with char K = 0; it is known that they have GK dimension n over K. VL Elements of CAAN VL Elements of CAAN Purity of modules Purity of modules Dimension function Operations with pure functions Pure functions Pure functions Purity w.r.t dimension function Purity ﬁltration Jacobson normal form Jacobson normal form Ubiquity of pure modules Pure functions and operations with them

Let O be an operator algebra and an O-module. A torsion F element f (that is a ”function” having nonzero annihilator) is Recall ∈ F called pure, is the corresponding left O-module Of ∼= O/ AnnO f Let A be an operator algebra over K[x1,..., xn] and is pure. S = K[x ,..., x ] 0 A be a m. c. Ore set in A. 1 n \{ } ⊂ This definition generalizes both the notion of Zeilberger-holonomic A left A-module M is called D-finite, if dim S 1M < . K(x1,...,xn) − ∞ or D-finite function as well as some other. Thus D-finite modules are pure. Lemma (L.) Let f be a pure function. Then for any o O 0 h = of Note: we can do much more with the concept of purity ∈ F ∈ \{ } is pure as well. We can consider pure modules of any reasonable dimension, without restricting ourselves to the modules of smallest possible Proof: Og = Oof Of is a natural submodule, hence it is pure. ⊂ dimension! Moreover, AnnO of = r O : r(of ) = (ro)f = 0 = s Ann f : r O, s = ro = { ∈ } { ∈ O ∃ ∈ } AnnO f : o = KerO(O O/ AnnO f , 1 o) is computable. VL Elements of CAAN → VL Elements of CAAN7→ Purity of modules Purity of modules Operations with pure functions Operations with pure functions Pure functions Pure functions Purity filtration Purity filtration Jacobson normal form Jacobson normal form Operations with pure functions Identities, Elimination, Purity Filtration

Let 0 M M M /M 0 be an exact sequence of → 1 → 2 → 2 1 → Lemma (L.) fin. pres. O-modules. Moreover, let be an arbitrary O-module. F Let f , g be pure functions. Then for any p, q O 0 Then we have that SolO(M2/M1, ) SolO(M2, ). ∈ F ∈ \{ } F ⊆ F h = pf + qg is pure as well. If is injective O-module, the natural map F SolO(M2, ) SolO(M1, ) is surjective (not true for general ). Proof: by the previous lemma Mf = Opf and Mg = Oqg are pure F → F F Purity filtration with δ = GKdim modules. By another lemma before Mf + Mg is pure. Hence Oh M + M is pure as well. Let O be a Noetherian domain, being Auslander-regular and ⊆ f g Moreover, (AnnO f : p) (AnnO g : q) AnnO h. Cohen-Macaulay algebra with GKdim O = n. ∩ ⊆ Given a fin. pres. O-module M of dimension n > d 0, then the ≥ More operations, preserving the purity, are under investigation. purity filtration of M is the sequence

M = Mn d Mn d+1 ... Mn 1 Mn = 0. Observation : many (but not all) special functions give rise to pure − ⊃ − ⊃ − ⊃ modules. where GKdim Mn (d i) = d i. Moreover, Mn d+k /Mn d+k+1 is − − − − − either 0 or pure of dimension d k. VL Elements of CAAN VL− Elements of CAAN Purity of modules Purity of modules Operations with pure functions Operations with pure functions Pure functions Pure functions Purity ﬁltration Purity ﬁltration Jacobson normal form Jacobson normal form

Identities, Elimination, Purity Filtration The purity ﬁltration of M = t(M) is 0 ( M3 ( M2 = M,

M = O/ n + 1, s , ∂ with GKdim M = 1. 3 ∼ h n x i 3 Consider the mixed system, annihilating Legendre polynomials What are the most general solutions g(n, x) of this system? Since ∂ (g) = 0, one has g(n, x) = g(n). O = K n, s s n = ns + s K x, ∂ ∂ x = x∂ + 1 . x h n | n n ni ⊗K h x | x x i however, g(n) should not be identically zero: in case n 1, 0, 1,... , one can select g( 1) K arbitrary M = O/P, ∈ {− } − ∈ (step of the jump function). P = (x2 1)∂2 + 2x∂ n(1 + n), (n + 2)s2 (2n + 3)xs + n + 1, h − x x − n − n (n + 1)(s ∂ x∂ + n + 1) . Localization n x − x i The ideal n + 1, s is two-sided and maximal. Hence the GKdim O = 4, GKdim M = 2, t(M) = M = O/P. h ni submodule M3 vanishes under any nontrivial Ore localization w. r. t S K n, sn ... , for instance when n S or sn S (then 1 ⊂ h i ∈ ∈ 1 s− is present and therefore n Z should hold). And S− M is n ∈ then a pure module.

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The purity ﬁltration of M = t(M) is 0 ( M3 ( M2 = M. The pure part of GK dimension 2 is t(M)/M3 ∼=

2 2 2 The hypergeometric series is deﬁned for z < 1 and c / N0 as O/ (x 1)∂x + 2x∂x n(1 + n), (n + 2)Sn (2n + 3)xSn + n + 1, | | − ∈ h − − − follows: 2 ∞ (a) (b) zn (1 x )∂x + (n + 1)Sn (n + 1)x . F (a, b, c; z) = n n − − i 2 1 (c) n! n=0 n For further investigations of M over localizations w.r.t. n or Sn one X should then take the simpliﬁed equations from the ideal P above. 0 We derive two annihilating ideals from the anihilator of Elimination leads to new identities 2F1(a, b, c; z):

The elimination property guarantees, that 1 arbitrary variable of O Ja which does not contain a,

can be eliminated from P and from P0; so one gets for instance Jc which does not contain c, x free :(n + 1)(n + 2) (S2 1)∂ (2n + 3)S P (x) = 0, and analyze corresponding modules for purity. − · n − x − n • n n free : (1 x2) (S2 2xS + 1)∂ S ) P (x) = 0. − − · n − n x − n • n VL Elements of CAAN VL Elements of CAAN Purity of modules Purity of modules Operations with pure functions Operations with pure functions Pure functions Pure functions Purity ﬁltration Purity ﬁltration Jacobson normal form Jacobson normal form

Case Ja

The ideal in O = K[b, c, z] Sb, Sc, Dz ... is generated by: h | i bcSb czDz bc The purity filtration of M = t(M) − − bSbSc bSc + cSc c ... and − − bSb2 zSbDz bSb + Sb2 Sb − − − M5 ∼= O/ c, Sb, b + 1, zDz Dz 1 , GKdim M5 = 2. b2Sb bzDz b2 + bSb zDz b h − − i − − − − bzSbDz z2Dz2 bzDz bSbDz + zDz2 bSb + bDz + b + Dz The solutions can be read off: − − − − Let M = M = O/J . Then GKdim O = 6, GKdim M = 4. a a δc,0 δb, 1 (ln(z 1) + k0), k0 K · − · − ∈ The purity filtration of M = t(M)

0 ( M5 = M4 ( M3 = M2 = M, where

M/M = O/ bSb zDz b, zDzSc + cSc c , GKdim M/M = 4 5 ∼ h − − − i 5

Case Jc

The ideal in O = K[b, c, z] Sb, Sc, Dz ... is generated by: h | i aSa bSb a + b The purity ﬁltration of M = t(M) − − bSb2 SbzDz bSb + Sb2 Sb ... and − − − b2Sb bzDz b2 + bSb zDz b − − − − M6 ∼= O/ Sb, b + 1, Sa, a + 1, zDz Dz 1 , GKdim M6 = 2. abSb azDz ab + bSb zDz b h − − i − − − − bSbzDz z2Dz2 bSbDz bzDz + zDz2 bSb + bDz + b + Dz The solutions: − − − − Let M = M = O/J . Then GKdim O = 6, GKdim M = 4. c c δa, 1 δb, 1 (ln(z 1) + k0), k0 K − · − · − ∈ The purity ﬁltration of M = t(M)

0 ( M6 = M5 = M4 ( M3 = M2 = M, where

M/M = O/ bSb zDz b, aSa zDz a , GKdim M/M = 4. 6 ∼ h − − − − i 6

One of the most important questions in algebra is undecidable in general:

Let A be a (Noetherian) K-algebra and M, N are two ﬁnitely presented A-modules. Can we decide, whether M ∼= N as A-modules? Part V. Jacobson normal form. Yet another application of localization as a functor: Let S A be a m. c. Ore set, then S 1A exists. ⊂ − Given an A-module homomorphism ϕ : M N (M, N are ﬁnitely → presented). Then there is an induced homomorphism of S 1A-modules S 1ϕ : S 1M S 1N. − − − → − Application to the isomorphism problem If there exists such m. c. Ore set S˜ A, that S˜ 1ϕ is not an ⊂ − isomorphism, then ϕ is not an isomorphism.

Let R be a non-commutative Euclidean domain and M Rm n. ∈ × Then there exist Above we have seen several dimensions of modules, some of them unimodular matrices U Rm m, V Rn n; are computable. What can one achieve with the help of ∈ × ∈ × localization? a matrix D Rm n with elements d ,..., d on the main ∈ × 1 r diagonal and 0 outside of the main diagonal . . . Let S = A 0 . Then the rank of f. g. A-module M is \{ } 1 such that d d (total divisibility), meaning deﬁned to be dim 1 S M. i i+1 S− A − || d d d Oh i+1iO ⊆ Oh i i ∩ h i iO Let R = A[∂; σ, δ] for an integral domain A and S = A 0 . such that U M V = D. 1 1 \{ } · · Then S− M is a vector space over Quot(A) = S− A and 1 dim 1 S M is an invariant of the module. S− R − In particular there is an isomorphism of R-modules

1 n 1 m 1 n 1 m R × /R × M ∼= R × /R × D.

Recognizing the localization Example L.–Schindelar (2011, 2012) presented two algorithms, computing Let A be the polynomial and B = (K[x] 0 ) 1A the rational 1 1 \{ } − 1 matrices U, V , D by using Gr¨obnerbases. Weyl algebra. Consider the matrix

A fraction-free algorithm performs only operations over polynomial ∂2 1 ∂ + 1 M = − . (i.e. unlocalized) algebra. A minor modiﬁcation allows to produce ∂2 + 1 ∂ x matrices U, V , D with polynomial entries. − The algorithm returns Theorem (L.–Schindelar) x2∂2 + 2x∂2 + ∂2 2x∂ 2∂ x2 1 0 Let A be a G-algebra in variables x1,..., xn, ∂ and assume that D = − − − − , 0 1 x1,..., xn generate a G-algebra B ( A. Suppose, there exists an { } admissible monomial ordering on A, satisfying x ∂ for all ≺ k ≺ x∂ ∂ + x2 + x + 1 x∂ + ∂ + x 1 k n. Then the following holds U = − − , ≤ ≤ ∂ x ∂ 1 B∗ is multiplicatively closed Ore set in A. − − − 1 1 0 (B∗)− A can be presented as an Ore extension of Quot(B) by V = . x∂2 + ∂2 + 2∂ x + 1 1 the variable ∂. −

Let us analyze, under which localizations U, V will be invertible. Let f = x + 1. Then U from above will be unimodular over any Indeed, V is unimodular over A , since it admits an inverse: 1 localization, where f is invertible. In particular, the smallest one, 1 i 1 0 as we know, is C1 := Sf− A1, where Sf = f : i N . V 1 = { ∈ } − (x + 1)∂2 + x 2∂ 1 1 − − − Thus the isomorphism of B1-modules, provided by the Jacobson form, holds not only over B = (K[x] 0 ) 1A , but also over C . 1 \{ } − 1 1 On the contrary, U is NOT unimodular over A , since U Z = W 1 · and W is ﬁrst invertible in the localization: General strategy: depending on the concrete questions, analyze U resp. V for unimodularity over localizations, less greedy than the 2∂ + 2 (x + 1)∂ + x 2 0 4x2 8x 4 Z = − , W = − − − rational one. 2(∂ x)(x + 1)∂ x2 x 3 2 5x + 5 − − − − Note: the steps of such an analysis are algorithmic. For the invertibility of W we need only to divide by x + 1 =: f .

VL Elements of CAAN VL Elements of CAAN Purity of modules Purity of modules Isomorphism problem Isomorphism problem Pure functions Pure functions Unimodularity and localization Unimodularity and localization Jacobson normal form Jacobson normal form Recognize and lift localized problems Merci beaucoup Strategical remarks for conclusion. pour votre attention! use the information from the localized situation - for instance, implementations of numerous good algorithms - for the analysis of the unlocalized, ”global” situation; in algorithms: perform fraction-free computations, if possible or keep track of operations, requiring localized computations use this tracking information and determine a smaller localization, where desired properties still hold. Lift the obtained results to that smaller localization. study obstructions to the lifting: this provides several cases, which again hints at the treatment of the problem at a global level by using local ones. obtain new powerful and useful results! http://www.singular.uni-kl.de/

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