Mora's holy grail: Algorithms for computing in localizations at prime ideals
aa r X i v : . [ m a t h . A C ] A p r MORA’S HOLY GRAIL : ALGORITHMS FOR COMPUTINGIN LOCALIZATIONS AT PRIME IDEALS MAGDALEEN S. MARAIS AND YUE REN
Abstract.
This article discusses a computational treatment of the lo-calization A L of an affine coordinate ring A at a prime ideal L and itsassociated graded ring Gr a ( A L ) with the means of standard basis tech-niques. Building on Mora’s work [6], we present alternative proofs ontwo of the central statements and expand on the applications mentionedby Mora: resolutions of ideals, systems of parameters and Hilbert poly-nomials, as well as dimension and regularity of A L . All algorithms areimplemented in the library graal.lib for the computer algebra system Singular . Introduction
Computer algebra systems like
Macaulay2 and
Singular have becomea staple in studying affine or projective varieties globally, by examining theircoordinate rings or homogeneous coordinate rings.The behaviour of an affine variety X ⊆ k n around a single point p ∈ X is described by the localization A ( X ) m p of its coordinate ring A ( X ) at thecorresponding maximal ideal m p . This local ring is commonly realized byapplying an affine coordinate transformation ϕ : k n ∼ −→ k n , shifting thepoint p into the origin . The respective localization A ( ϕ ( X )) m can then besimulated by working with a local monomial ordering. This technique hasbeen applied in the study of isolated singularities to great success.In this article, we discuss a related approach for the realization of the lo-calization A L of an affine coordinate ring A at a prime ideal L , see Lemma 1.Introduced by Mora in [6], it uses a local ordering on a specific ring to mimicthe local structure of A L . From it, one can also deduce a correspondingrepresentation of its associated graded ring Gr a ( A L ) , a E A L denoting itsmaximal ideal, see Lemma 3. In Section 1, we discuss the aforementionedLemmata, giving alternative proofs than in [6].Because a graded ring Gr a ( A L ) is computationally easier to handle thana local ring A L , we show in Section 2 how to exploit the a -adic topology on A L , allowing us to work over Gr a ( A L ) instead. Albeit not viable for every Mathematics Subject Classification.
Primary 13H99; Secondary 13P10, 14Q99.
Key words and phrases. local ring, associated graded ring, localization, resolution.This research was supported by the African Institute for Mathematical Sciences, theDepartment of Mathematics and Applied Mathematics of the University of Pretoria anda grant awarded by Wolfram Decker. We are thankful to all of them. see [6]. problem, a lot of information about A L can be found preserved in Gr a ( A L ) .The problems covered are the Hilbert polynomials of ideals in A L , a systemof parameters for A L as well as the regularity and dimension of A L . Inparticular, as stated in [6], we present an algorithm for lifting a resolutionof an initial ideal in a ( I ) E Gr a ( A L ) to a resolution of the ideal I E A L .All algorithms and examples in this article have been implemented inthe Singular library graal.lib [5], which also showcases some of the newobject-oriented features of
Singular . The library will be available in thenext
Singular distribution.1.
An Algorithm for the Associated Graded Ring Gr a ( A L ) In this article, let Q = k [ X ] be a multivariate polynomial ring over aground field k , and H ⊆ J E Q two prime ideals. Let A := Q/H be theaffine coordinate ring of the variety defined by H , and let L := J · A E A bea prime ideal describing an irreducible subvariety.Let U ⊆ X be a maximal independent set of variables with respect to J ,and let V := X \ U denote the remaining variables, so that J · k ( U )[ V ] E k ( U )[ V ] becomes a maximal ideal. Set Q := k ( U )[ V ] , H = H · Q and J = J · Q .Next, fix a system of generators J = h f , . . . , f s i , and consider the polyno-mial ring Q [ Y ] := Q [ Y , . . . , Y s ] . Set I := h f − Y , . . . , f s − Y s i + H E Q [ Y ] .Let a := L · A L denote the maximal ideal of A L , K := A L / a its residue field. Q := k [ X ] Q := k ( U )[ V ] a E A L A := Q/H E J h f , . . . , f s i = ⊆ H E L := J · A X \ U = maximal independent setwith respect to J K maximal ideal Throughout the entire article, we will be abbreviating ( Y , . . . , Y s ) with Y , ( f , . . . , f s ) with f and also, for α = ( α , . . . , α s ) ∈ N s , Y α · · · Y α s s with Y α , f α · · · f α s s with f α .Set w ∈ R | V | × R | Y | to be the weight vector which weights the variables V with and the variables Y with − . In particular, for a polynomial f ∈ Q [ Y ] , − deg w ( f ) will then be the lowest degree in Y occurring in it.Let > be a weighted ordering on Mon(
V, Y ) with weight vector w and anyglobal ordering > ′ on Mon( V ) as tiebreaker, i.e. V β · Y α > V δ · Y γ : ⇐⇒ (cid:0) | α | < | γ | (cid:1) or (cid:0) | α | = | γ | and V β > ′ V δ (cid:1) with V β > ′ for all β ∈ N | V | .Let Q [ Y ] > denote the localization of Q [ Y ] at the monomial ordering > , Q [ Y ] > := (cid:26) fu (cid:12)(cid:12)(cid:12)(cid:12) f, u ∈ Q [ Y ] and LM > ( u ) = 1 (cid:27) . LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 3
The notions of leading monomials then extend naturally from Q [ Y ] to Q [ Y ] > . Lemma 1 (corresponding to Lemma 6.3 in [6]) . The map Q [ Y ] → A , Y α f α extends naturally to a map φ : Q [ Y ] > → A L , inducing an exact sequenceof filtered rings −→ I · Q [ Y ] > −→ Q [ Y ] > φ −→ A L −→ . h Y i d · Q [ Y ] > a d · A L Y α f α ⊆ ⊆ h f − Y , . . . , f s − Y s i + H = Proof.
Before we show exactness, note that the map φ is well-defined becausethe residues of the algebraically independent variables U in A are naturallynot contained in L , making them invertible in A L . Moreover, any polynomial g ∈ Q [ Y ] with LM > ( g ) = 1 has to be of the form g = c + p for some c ∈ k ( U ) , p ∈ h Y i , and hence it is also mapped to something invertible in A L .It is clear that I · Q [ Y ] > is the kernel of φ . In order to show that φ issurjective, it suffices to find elements mapping to p − for p / ∈ L .Let p / ∈ L and let g = P α c α · Y α ∈ Q [ Y ] > , with c α ∈ Q , be a preimageof p . Clearly, c = 0 since Y is mapped into L . We will now show that c is invertible modulo I , which implies that g is invertible modulo I andhence completes the proof. Because J E Q is maximal, there exist a / ∈ J and b ∈ J such that a · c = 1 + b . And since J = h f , . . . , f s i and I ⊇ h f − Y , . . . , f s − Y s i , a · c = 1 + p b mod I , for some p b ∈ h Y , . . . , Y s i .Due to the ordering > , p b is invertible. (cid:3) Definition 2.
We define the associated graded ring of A L with respect to a to be Gr a ( A L ) := ∞ M n =0 a n / a n +1 = A L / a ⊕ a / a ⊕ . . . . Given c ∈ A L \ { } , the valuation ν a ( c ) of c with respect to a is the unique n ∈ N such that c ∈ ( a n · A L ) \ ( a n +1 · A L ) , and the initial form in a ( c ) of c with respect to a is its residue class c ∈ ( a n · A L ) / ( a n +1 · A L ) ⊆ Gr a ( A L ) .For an ideal I E A L , the initial form of I with respect to a is in a ( I ) = h in a ( c ) | c ∈ I i . For our weight vector w ∈ R | V | + | Y | and any g = P α,β c α,β · V β Y α ∈ Q [ Y ] , we define the initial form of g with respect to w to be in w ( g ) = P w · ( β,α ) max c α,β · V β Y α ∈ Q [ Y ] . Naturally, for an ideal J E Q [ Y ] , the initial form of J with respect to w is in w ( J ) = h in w ( g ) | g ∈ J i . The map in w : Q [ Y ] → Q [ Y ] naturally induces a map Q [ Y ] > → ( Q /J )[ Y ] = K [ Y ] , as LM > ( p ) = 1 implies in w ( p ) ∈ Q whose image thenlies in K and therefore is invertible. We will denote this map with () in : MAGDALEEN S. MARAIS AND YUE REN Q [ Y ] > K [ Y ] Q [ Y ] Q [ Y ]() in in w ⊆ Abusing the notation by abbreviating the ideal I · Q [ Y ] with I , we thenget: Lemma 3 (Lemma 6.6 in [6]) . We have an exact sequence of graded rings −→ I in · K [ Y ] −→ K [ Y ] λ −→ Gr a ( A L ) −→ .K [ Y ] d −→ a d / a d +1 Y α f α ⊆ ⊆ Proof.
It is clear that λ is surjective, so remains to show that ker( λ ) = I in .Suppose P | α | = d a α Y α ∈ ker( λ ) for some a α ∈ A L . Then P | α | = d a α f α ∈ a d +1 , say P | α | = d a α f α = P | β | = d +1 b β f β for suitable b β ∈ A L . Recall theexact sequence in Lemma 1. Pick a α, , b β, ∈ Q such that φ ( a α, ) = a α and φ ( b β, ) = b β for all α, β . Then P | α | = d a α, Y α − P | β | = d +1 b β, Y β ∈ ker( φ ) = I . Thus (cid:16) X | α | = d a α, Y α − X | β | = d +1 b β, Y β (cid:17) in = (cid:16) X | α | = d a α, Y α (cid:17) in = X | α | = d a α Y α ∈ I in . Conversely, suppose g in ∈ I in for some g ∈ I . Writing g = h + r , where h is the sum over the terms of lowest degree in Y , we have φ ( g ) = φ ( h ) |{z} ∈ a d + φ ( r ) |{z} ∈ a d +1 = 0 , where d = − deg w ( g ) is the degree in Y of g . This implies that λ ( g in ) = λ ( h in ) = 0 ∈ a d / a d +1 . (cid:3) Proposition 4.
The previous two Lemmas yield a diagram −→ I · Q [ Y ] > −→ Q [ Y ] > φ −→ A L −→ −→ I in · K [ Y ] −→ K [ Y ] λ −→ Gr a ( A L ) −→ . () in () in in a () For any f ∈ Q [ Y ] > we can calculate the initial form of its image in A L by in a ( φ ( f )) = λ (NF( f, G I ) in ) with ν a ( φ ( f )) = − deg w (NF( f, G I )) . In particular, if LM > ( f ) / ∈ LM > ( G I ) , then in a ( φ ( f )) = λ ( f in ) and ν a ( φ ( f )) = − deg w ( f ) . Proof.
The diagram is merely a concatenation of the sequences in Lemma 1and 3.
LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 5
The formula for the initial form and valuation follow immediately from a = h φ ( f ) ,. . . , φ ( f k ) i , I = h f − Y , . . . , f k − Y k i and our chosen ordering > . (cid:3) Remark 5.
Note that if G is a standard basis for I E Q [ Y , . . . , Y s ] > withrespect to our chosen weighted ordering > , then so is { in w ( g ) | g ∈ G } for in w ( I ) . This implies that a standard basis of I yields a Gröbner basis of I in . Remark 6.
For practical reasons, it is recommended to compute the min-imal polynomial of K over k ( U ) , as we have a natural isomorphism K ∼ = Q /J = k ( U )[ V ] /J . This can be done by applying a generic coordinatetransformation ϕ : Q = k ( U )[ V , . . . , V n ] → k ( U )[ T , . . . , T n − , T ] to push it into general position with respect to a lexicographical ordering(see [3], Proposition 4.2.2). Depending on the lexicographical ordering, ϕ ( J ) has a Gröbner basis of the form h T − g , . . . , T n − − g n − , g i , where g , . . . , g n − , g ∈ k ( U )[ T ] . We then obtain K ∼ = Q /J ∼ = k ( U )[ T , . . . , T n − , T ] /ϕ ( J ) ∼ = k ( U )[ T ] / h g i , where the last isomorphism is defined by V i g i , for i = 1 , . . . , n − .2. Applications
In this section we demonstrate how to exploit of the a -adic topology inseveral calculations on A L . We give some exemplary questions which allowus to compute in Gr a ( A L ) and lift the result back to A L .2.1. Dimension.
It is well known that the dimension of a local Noetherianlocal ring coincides with that of its associated graded ring (see [3] Theorem5.6.2), dim( A L ) = dim(Gr a ( A L )) . Therefore, we can determine the dimension of A L by computing the dimen-sion of Gr a ( A L ) .2.2. Regularity.
As we have mentioned in the introduction, localizationof affine coordinate rings at the origin is an established tool in the studyof singularities. In the nature of these studies, the object of interest arelocalizations which are not regular.
Definition 7. A L is a regular local ring if dim K ( a / a ) = dim A L . Geometrically, this means that the irreducible subvariety defined by L isnot contained in the singular locus of the affine variety given by A . Therepresentation of Gr a ( A L ) in Lemma 3 implies following criterion for theregularity of A L : MAGDALEEN S. MARAIS AND YUE REN
Proposition 8.
We have A L regular ⇐⇒ Gr a ( A L ) isomorphic to a polynomial ring ⇐⇒ I in generated by linear elements.Proof. Let H := { h , . . . , h n } be a reduced Gröbner basis of I in , therefore H contains only homogeneous elements. Let dim K ( a / a ) = d and without lossof generality suppose { Y , . . . , Y d } is an independent set of variables over I in .Then Gr a ( A L ) ∼ = K[ Y , . . . , Y d ] / I ′ in , where I ′ in is generated by appropriatelinear transformations of all the nonlinear elements of H . If I ′ in = h i , it isclear that dim(Gr a ( A L )) = d and thus A L is regular.Conversely, suppose I ′ in = h i , i.e. Gr a ( A L ) is not a polynomial ring, andsuppose without loss of generality that Y d is a divisor of one of the terms ofone of the elements of I ′ in . Then, since K[ Y , . . . , Y d ] / I ′ in is a finitely gener-ated K algebra, dim(Gr a ( A L )) = dim (K[ Y , . . . , Y d ] / I ′ in ) / h Y , . . . , Y d − i ) + height h Y , . . . Y d − i ≤ d − . Hence A L is not regular. (cid:3) Example 9.
Consider the affine surface X defined by y + x − x z ∈ Q [ x, y, z ] . Intersecting X with affine planes z − t yield nodal curves withsingular point (0 , , t ) for t = 0 , which degenerate into a cuspidal curve for t = 0 . h x, y i h x − z , y i z < z = 0 z > Figure 1. V ( y + x − x z ) Consequently, the singular locus of X is the line given by L := h x, y i E A := Q [ x, y, z ] / h y + x − x z i . Indeed, setting f := x and f := y we see that z · f + f = f ∈ A ⊆ A L , which means in the isomorphism induced by Lemma 3 that z · Y + Y =0 ∈ Gr a ( A L ) . By Proposition 8, A L is then not regular.On the other hand for L ′ := h x − z , y i E A := Q [ x, y, z ] / h y + x − x z i . we can compute that Gr a ( A L ) = K [ Y , Y ] , where K = Q [ T ] /T and Y , Y represent the generators x − z , y . Therefore, A L ′ is regular. LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 7
System of parameters.
A system of parameters can be thought of asa local coordinate system, which determines the points around the subvarietyup to finite ambiguity.
Definition 10.
Let d := dim( A L ) . Then { a , . . . , a d } is called a system ofparameters of A L , if h a , . . . , a d i is a -primary. If h a , . . . , a d i = a then it iscalled a regular system of parameters.Note that if a , . . . , a d ∈ A L is such that the radical of h in a ( a ) , . . . , in a ( a d ) i E Gr a ( A L ) is h Y , . . . , Y s i / I in , then for I := h a , . . . , a d i , we have ( √ I ∩ a i ) / a i +1 = a i / a i +1 for all i . But, since ∩ i a i = 0 , this implies that √ I = a .Hence, knowing that an ideal in A L or Gr a ( A L ) is primary if its radical ismaximal, finding a system of parameters for A L boils down to finding a setof homogeneous elements { λ , . . . , λ d } ⊂ K[ Y , . . . , Y s ] such that the radicalof H = h λ , . . . , λ d i + I in is h Y , . . . , Y s i .Now, it follows from linear algebra that for a generic choice of c ij ∈ Q , λ i = P c ij Y j satisfies the above condition (which can be tested by computingthe dimension of H ). Choosing a i as P c ij f i , where h f , . . . , f s i = L ⊆ A ⊆ A L , we have that in a ( a i ) = λ i and we are done.A regular system of parameters only exists when A L is regular, which canbe checked using the method described in Section 2.2. In such a case, a regu-lar system of parameters can be determined in the same way, except checkingwhether H equals h Y , . . . , Y s i instead of checking whether the radical of H equals h Y , . . . , Y s i . Example 11.
For A L as in Example 9, a simple possible system of param-eters is { y } . It means that if we look alongside the line given by L E A , thefunction y ∈ A distinguishes the points of A in the near vicinity of L up tofinite ambiguity as in Figure 2. V ( L ) X ( A ) y = ε for some ε < Figure 2. V ( y + x − x z ) around V ( x, y ) Hilbert-Samuel function and Hilbert-Samuel polynomial.
For ahomogeneous ideal I E Gr a ( A L ) , i.e. an ideal with an homogeneous preimage,we denote the Hilbert function of I by H Gr a ( A L ) ( I, n ) = dim K (Gr a ( A L ) /I ) n . MAGDALEEN S. MARAIS AND YUE REN
Since the the Hilbert-Samuel function of an ideal J E A L , denoted by HS A L ( J ) ,evaluated at n equals dim( J/ ( a n · J )) by definition, it follows that HS A L ( J, n ) = n X i =1 H Gr a A L (in a ( J ) , i − . Suppose the Hilbert polynomial of in a ( J ) E Gr a ( A L ) is of the form(1) P Gr a ( A L ) (in a ( J ) , x ) = s − X ν =0 a ν (cid:18) xν (cid:19) , for a ν ∈ Z , so that P Gr a ( A L ) (in a ( J ) , n ) = H Gr a ( A L ) (in a ( J ) , n ) for n suffi-ciently large. Then the Hilbert Samuel polynomial is given by(2) HSP A L ( J, x ) := s X ν =1 a ν − (cid:18) xν (cid:19) + c, where c = dim( J/ ( a l · J )) − P sν =1 a ν − (cid:0) lν (cid:1) , for any sufficiently large l , so that HSP A L ( J, n ) = HS A L ( J, n ) . And, by Corollaries 5.1.5 and 5.5.5 in [3], any l ≥ d is sufficiently large, where d is the degree of the second Hilbert seriesof in a ( J ) .Taking the relationship between (1) and (2) into account it suffices tocompute P Gr a ( A L ) (in a ( J )) to determine HSP A L ( J ) . Since Gr a ( A L ) / in a ( J ) ∼ = K [ Y ] / ( I in + in a ( J )) it follows that H Gr a ( A L ) ( J in ) = H K [ Y ] ( I in + in a J ) =H K [ Y ] ( L ( I in + in a J )) , where L ( I ) , denotes the leading ideal of I . Remark 12.
This algorithm is still subject to ongoing work, because alge-braic field extensions over transcendental field extensions are not availablein Singualr yet.2.5.
Syzygies and Resolutions.
For this section, fix an ideal I E A L andconsider an A L -free resolution ←− A L /I ←− A L ←− M ←− M ←− · · · , where, for sake of simplicity, all M i are free A L modules of finite rank, andall maps are compatible with the filtrations. That means for example thatthe first two maps are of the form g i, ←− [ e i, A L ←− M M ←− M g i, ←− [ e i, = L k i =1 A L ( − ν a ( g i, )) , = L k i =1 A L ( − ord( g i, )) , where e i, , e i, refer to the canonical basis elements of M and M , and g i, , g i, refer to their images respectively. Moreover, ord denotes the orderfunction of the filtration on M , which has been twisted so that ord(1 · e i, ) =0 + ν a ( g i, ) , hence the map takes the n -th filtration module of M to the n -thfiltration module of A L for any n ∈ N . LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 9
Definition 13.
Let M be an A L -module as above, say M = ⊕ ki =1 A L ( − d i ) .Then its associated graded module is the free graded Gr a ( A L ) -module givenby Gr a ( M ) := k M i =1 Gr a ( A L )( − d i ) , so that for any d ∈ N its degree d component is given by Gr a ( M ) d := k M i =1 Gr a ( A L ) d + d i = k M i =1 a d + d i / a d + d i +1 . For any g = P ki =1 g i · e i ∈ M , say ord( g ) = n , the initial form of g isdefined to be in a ( g ) := k X i =1 g i · e i ∈ Gr a ( M ) n ⊆ Gr a ( M ) , so that g i · e i = 0 ∈ Gr a ( M ) n for all ord( g i · e i ) > ord( g ) . The initial module ofa submodule M ′ ⊆ M is naturally defined to be in( M ′ ) := h in( g ) | g ∈ M ′ i .It is straightforward to see how our A L -free resolution of I yields a graded, Gr a ( A L ) -free resolution of in a ( I ) due to the functorial nature of taking initialforms: ←− A L /I ←− A L ←− M ←− M ←− · · · ←− Gr a ( A L ) / in a ( I ) ←− Gr a ( A L ) ←− N ←− N ←− · · · . in a in a in a in a Given the notation from the very beginning, the first two maps of ourresolution would be of the form in a ( g i, ) ←− [ e i, Gr a ( A L ) ←− N N ←− N in a ( g i, ) ←− [ e i, = L k i =1 Gr a ( A L )( − ν a ( g i, )) , = L k i =1 Gr a ( A L )( − ord( g i, )) , where, for sake of simplicity, we abuse notation and use e i, respectively e i, to refer to the canonical basis elements of both M and N respectively M and N .The goal of this section is to establish an inverse process, in which wewill lift a graded Gr a ( A L ) -free resolution of in a ( I ) to an A L -free resolutionof I . For that, fix a graded Gr a ( A L ) -free resolution for the remainder of thechapter ←− Gr a ( I ) / in a ( I ) ←− Gr a ( A L ) ←− N ←− N ←− · · · . Roughly speaking, the lift of the whole resolution consists of repeated liftsof syzygies over Gr a ( A L ) to syzygies over A L , i.e. for any free Gr a ( A L ) -module N = N j in our resolution and two sets Θ = { θ , . . . , θ k } ⊆ N and ∆ = { δ , . . . , δ k } ⊆ M := M j with in a ( δ i ) = θ i , compute for a given η ∈ syz(Θ) a γ ∈ syz(∆) with in a ( γ ) = η . In our setting, both Θ and ∆ willtypically be systems of generators of previous syzygy modules, while η is theimage of a canonical basis element in the resolution over Gr a ( A L ) and γ willthen be the image of a canonical basis element in the resolution over A L . . . . ←− M ←− L ki =1 A L ( − ord( δ i )) . . . ←− N ←− L ki =1 Gr a ( A L )( − deg( θ i )) ←− . . . ∆Θ ⊆ ⊆ in a in a γη ∈ ∈ e i ←− [ find such that in a ( γ ) = η (cid:18) so that here γ ← [ e i (cid:19) Suppose N = ⊕ i Gr a ( A L )( − d i ) and M := ⊕ i A L ( − d i ) . Then Lemma3 immediately induces a surjective morphism λ from the free K [ Y ] -module b N = ⊕ i K [ Y ]( − d i ) onto N with kernel ker( λ ) = I in · b N and Lemma 1 inducesa surjective morphism φ from a free Q [ Y ] > -module c M = ⊕ i Q [ Y ] > ( − d i ) onto M with kernel ker( φ ) = I · c M . Note that b N and c M are shifted similarlyto their images so that λ remains a graded map of degree and φ remainscompatible with the filtrations.Moreover, if we consider a weight on the module variables and an orderingon the module monomials of c M that are compatible with the filtration,Proposition 4 immediately carries over to our module setting. For that, weextend our existing weight vector w to weight the canonical basis elements e i with weight d i . Note that our lifted resolution will induce a sequence on thefree Q [ Y ] > -modules c M . Extend > to the Schreyer ordering on the modulemonomials on c M .Note that a Gröbner basis of ker( λ ) = I in · b N and a standard basis ker( φ ) = I · c M are not hard to obtain, as they can be easily derived from the Gröbnerbasis of I in and the standard basis of I respectively. −→ I · c M −→ c M φ −→ M −→ −→ I in · b N −→ b N λ −→ N −→ . () in () in in a () L i A L ( − d i ) L i Gr a ( A L )( − d i ) = = L i K [ Y ]( − d i ) = L i Q [ Y ] > ( − d i ) = Now let { δ , . . . , δ k } be a generating set of a submodule M ′ ≤ M suchthat { θ , . . . , θ k } , where θ i := in a ( δ i ) , generates N ′ := in a ( M ′ ) ≤ N . Tolift syzygies of ( θ , . . . , θ k ) to syzygies of ( δ , . . . , δ k ) will require a standard LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 11 basis of φ − ( M ′ ) ≤ c M as well as a representation of the basis elements viasuitable preimages of δ i .Since our utmost goal is to avoid complicated computations over A L ,it stands out of question to compute this standard basis of φ − ( M ′ ) fromscratch. However, this is not needed since Gröbner bases of φ − N ′ can benaturally lifted to standard bases of φ − ( M ′ ) . −→ I · c M −→ c M φ −→ M −→ −→ I in · b N −→ b N λ −→ N −→ . () in () in in a () M ′ ⊇ { δ , . . . , δ k } N ′ ⊇ { θ , . . . , θ k } ≤ ≤ λ − N ′ { h , . . . , h k } ⊆ φ − M ′ { g , . . . , g k } ⊆ ≤≤ { h ′ , . . . , h ′ l , h ′ l +1 , . . . } ⊆ standard basis with h ′ j = P ki =1 q ji · h i + s j For that, consider preimages g , . . . , g k of δ , . . . , δ k with LM > ( g i ) / ∈ LM > ( I · c M ) , so that h , . . . , h k , where h i := g i, in , are homogeneous preimages of θ , . . . , θ k with LM > ( h i ) / ∈ LM > ( I in · b N ) . Consider a homogeneous Gröb-ner basis { h ′ , . . . , h ′ l , h ′ l +1 , . . . } of λ − N ′ with LM > ( h ′ i ) / ∈ LM > ( I in · b N ) for ≤ i ≤ l and LM > ( h ′ i ) ∈ LM > ( I in · b N ) otherwise. Then there exists q ji ∈ Q [ Y ] such that h ′ j = k X i =1 q ji · h i + s j for some s i ∈ I in for all i = 1 , . . . , l. Since all h ′ j and h i are homogeneous (with variables Y ), we may assumethat all q ji are weighted homogeneous with − deg w ( q ji ) = deg( h ′ i ) − deg( h j ) ,unless q ji = 0 . Moreover, we may assume that LM > ( q ji ) / ∈ I unless q ji = 0 ,as this can always be achieved by modifying s i , which in particular impliesthat ν a ( φ ( q ji )) = − deg w ( q ji ) .With this, we have everything necessary to formulate our algorithm forlifting Gröbner bases. Algorithm 1 lifting Gröbner bases
Input:
Given two submodules M ′ ≤ M and N ′ ≤ N with in a ( M ′ ) = N ′ ,(1) a standard basis G of I · c M ,(2) G = { g , . . . , g k } ⊆ c M with LM > ( g i ) / ∈ LM > ( I · c M ) ,(3) { h ′ , . . . , h ′ l , h ′ l +1 , . . . } a Gröbner basis of λ − N ′ with LM > ( h ′ i ) / ∈ LM > ( I in · b N ) for ≤ i ≤ l and LM > ( h ′ i ) ∈ LM > ( I in · b N ) other-wise, Input: (continued)(4) q ji ∈ Q [ Y ] homogeneous with respect to weight vector w and LM > ( q ji ) / ∈ I unless q ji = 0 such that h ′ j = k X i =1 q ji · h i + r j for all j = 1 , . . . , l, where h i := g i, in for i = 1 , . . . , k . Output: G ′ = { g ′ , . . . , g ′ l } ⊆ c M and q ′ ji ∈ Q [ Y ] such that(1) G ′ ∪ G is a standard basis of φ − M ′ ,(2) g ′ j = P ki =1 q ′ ji · g i + r j for some r j ∈ I ,(3) ord( φ ( g ′ j )) = ord( φ ( q ′ ji · g i )) unless q ′ ji · g i = 0 . for j = 1 , . . . , l do Compute g ′ j := NF (cid:16) k X i =1 q ji · g i , G (cid:17) ∈ Q [ Y ] . if LM > ( g ′ j ) = s j · Y α for some s j ∈ Q , s j = 1 then Find a j ∈ Q such that a j · s j = 1 ∈ K = Q /J . Redefine g ′ j := NF( a j · g ′ j , G ) . else Set a j := 1 ∈ Q . end if end for return { g ′ , . . . , g ′ l } and ( a i · q ji ) . Proof.
In order to show that G ′ ∪ G is a standard basis of φ − M ′ , we needshow that LM > ( g ′ j, in ) = LM > ( h ′ j ) for all j = 1 , . . . , l first. For sake of clarity,rename g ′ j as it appears in Steps 2 to 5 to g ′′ j , and set r j, := k X i =1 q ji · g i − NF (cid:16) k X i =1 q ji · g i , G (cid:17) as in Step 2 ,r j, := ( a j · g ′′ j − NF( a j · g ′′ j , G ) as in Step 5 if s j = 1 , otherwise,so that g ′′ j = P ki =1 q ji · g i − r j, and g ′ j = a j · g ′′ j − r j, , or rather g ′ j = a j · k X i =1 q ji · g i − a j · r j, − r j, . LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 13
By definition of > and unless r j, = 0 or r j, = 0 , we have deg w ( r j, ) ≤ deg w (cid:16) k X i =1 q ji · g i (cid:17) , deg w ( r j, ) ≤ deg w ( a j · g ′′ j ) ≤ deg w (cid:16) a j · k X i =1 q ji · g i (cid:17) . Setting c and c to be either or depending on the disparity of weighteddegree, together, this yields(*) g ′ j, in = (cid:16) a j · k X i =1 q ji · g i (cid:17) in | {z } = a j · h ′ j by (4) − c · ( a j · r j, ) in | {z } ∈ I in − c · ( r j, ) in | {z } ∈ I in . Because LM > ( g ′ j ) / ∈ LM > ( I · c M ) due to the normal form computations, wehence must have LM > ( g ′ j, in ) / ∈ LM > ( I in · b N ) , which leaves LM > ( g ′ j, in ) =LM > ( a j · h ′ j ) = LM > ( h ′ j ) as the only possibility.To show that G ′ ∪ G is a standard basis of φ − M ′ , consider an element f ∈ φ − M ′ . If LM > ( f ) ∈ LM > ( I · c M ) , then by assumption (1) there existsan element of G with leading monomial dividing LM > ( f ) .If LM > ( f ) / ∈ LM > ( I · c M ) , then in particular LM > ( f in ) / ∈ LM > ( I in · b N ) , andby assumption (3) there exists an j = 1 , . . . , l such that LM > ( h ′ j ) | LM > ( f in ) .Suppose s i = 1 during Step in the j -th iteration of the for loop. In Step we then find an a j ∈ Q such that a j · s j = r + 1 for some r ∈ J . Byconstruction of I and the choice of our monomial ordering > , we thereforehave LM > ( g ′ i ) = 1 · Y α after the normal form computation in Step . Eitherway, before the end of our for loop iteration in Step , we have LM > ( g ′ i ) = Y α = LM > ( g ′ i, in ) = LM > ( h ′ i ) | LM > ( f in ) . And since LM > ( f in ) = LM > ( f ) | V =1 it follows that LM > ( g ′ i ) | LM > ( f ) .Moreover, our previous considerations imply for all j = 1 , . . . , lg ′ j = a j · k X i =1 q ji · g i − a j · r j, − r j, = k X i =1 q ′ ji · g i − a j · r j, − r j, | {z } ∈ I · c M , showing the second condition of our output.For the last condition note that we have ord( φ ( g ′ j )) = deg( λ ( a j · h ′ j )) − deg w ( h ′ i ) = = − deg w ( q ji ) − deg w ( h i ) = − deg w ( q ji ) − deg w ( g i )ord( φ ( q ji )) + ord( φ ( g i )) = LM > ( g ′ j ) / ∈ LM > ( I · c M ) and (*) deg w ( q ji ) = deg w ( h ′ j ) − deg w ( h i ) g i, in = h i in which we also use that LM > ( h ′ i ) / ∈ I in · b N , LM > ( q ji ) / ∈ I and LM > ( g i ) / ∈ I · c M for the correlation between degree resp. order and weighted degree. (cid:3) Now that we have this algorithm for lifting standard bases of preimagesin b N to the corresponding preimage in c M , we can write down our algorithmfor lifting syzygies over Gr a ( A L ) to syzygies over A L . · · · N L ki =1 Gr a ( A L )( − deg θ i ) · · · M L ki =1 A L ( − ν a ( δ i )) · · · b N L ki =1 K [ Y ]( − deg θ i ) · · · c M L ki =1 Q [ Y ] > ( − ν a ( δ i )) λ λφ φ ∆ = { δ i } ki =1 ⊂ Θ = { θ i } ki =1 ⊂ syz(∆) ∋ γ ⊂ syz(Θ) ∋ η ⊂ { g i } ki =1 ⊆ G ′ ⊂ g ∈ { h i } ki =1 ⊆ H ′ ⊂ h ∈ Algorithm 2 lifting syzygies
Input: η ∈ syz( θ , . . . , θ k ) ⊆ L ki =1 Gr a ( A L )( − deg θ i ) homogeneous and(1) g i ∈ c M with LM > ( g i ) / ∈ LM > ( I · c M ) such that in a ( φ ( g i )) = θ i .(2) a standard basis G of I · c M ,(3) a set G ′ = { g ′ , . . . , g ′ l } ⊆ c M such that G ′ ∪ G is a standard ba-sis of φ − M ′ , where M ′ = h θ , . . . , θ k i and q ji ∈ Q [ Y ] weightedhomogeneous with LM > ( q ji ) / ∈ LM > ( I · c M ) , g ′ j = k X i =1 q ji · g i + r j ∈ c M for some r j ∈ I · c M , and ord( φ ( g ′ j )) = ord( φ ( q ′ ji · g i )) unless q ′ ji · g i = 0 , as in the outputof Algorithm 1. Output: γ ∈ syz(∆) ⊆ L ki =1 A L ( − ord( δ i )) such that in a ( γ ) = η . LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 15 Pick a homogeneous representative h of η , say h = k X i =1 c i, in · e i ∈ k M i =1 K [ Y ]( − deg θ i ) with c i ∈ Q [ Y ] and LM > ( c i ) / ∈ LM > ( I ) . Compute a normal form r := NF (cid:16) k X i =1 c i · g i , G (cid:17) ∈ c M . Compute a standard representation r = l X j =1 d j · g ′ j ∈ c M . Set g := k X i =1 (cid:16) c i − l X j =1 q j,i · d j (cid:17) · e i ∈ k M i =1 Q [ Y ] > ( − ord( δ i )) . return γ := φ ( g ) ∈ L ki =1 A L ( − ord( δ i )) Proof.
To show that γ lies in syz(∆) , it suffices to show that the image of g in c M lies in ker( φ ) = I · c M . That image is given by P ki =1 ( c i − P lj =1 q j,i · d j ) · g i = P ki =1 c i · g i − P ki =1 P lj =1 q j,i · d j · g i = P ki =1 c i · g i − P lj =1 d j · P ki =1 q j,i · g i = P ki =1 c i · g i − P lj =1 d j · ( g ′ j − r j )= P ki =1 c i · g i − r | {z } ∈ I · c M − P lj =1 d j · r j | {z } ∈ I · c M , as r = NF( P ki =1 c i · g i , G ) and r j ∈ I · c M for j = 1 , . . . , l .It remains to show that η = in a ( γ ) . Note that we have for any j = 1 , . . . , l ord( φ ( r )) = − deg w ( r ) ≤ − deg w ( d j · g ′ j ) ≤ ord( φ ( d j · g ′ j )) . Step 2 andProp. 4 Step 3 anddefinition of > always as φ : Y f Furthermore, observe that η ∈ syz(Θ) implies that P ki =1 λ ( c i, in · h i ) = 0 .Using Proposition 4, our choice of g i ∈ c M and c i ∈ Q [ Y ] therefore yields P ki =1 in a ( φ ( c i · g i )) = P ki =1 in a ( φ ( c i )) · in a ( φ ( g i )) P ki =1 λ ( c i, in ) · λ ( g i, in ) = P ki =1 λ ( c i, in · h i ) = 0 , = = LM > ( g i ) / ∈ LM > ( I · c M )LM > ( c i ) / ∈ LM > ( I · c M ) g i, in = h i which implies that ord( P ki =1 φ ( c i · g i )) > ord( φ ( c s · g s )) for any s = 1 , . . . , k .Together, we get for all i = 1 , . . . , k and all j = 1 , . . . , l ord( φ ( d j · g ′ j )) ≥ ord( φ ( r )) > ord( φ ( c i · g i )) . Recall that by assumption (4) ord( φ ( g ′ j )) = ord( φ ( q ji · g i )) = ν a ( φ ( q ji )) +ord( g i ) , which implies ord( e i ) = ord( φ ( g ′ j )) − ν a ( φ ( q ji )) . Therefore ord( φ ( q ji · d j · e i )) = ν a ( φ ( d j )) + ord( φ ( q ji · e i ))= ν a ( φ ( d j )) + ord( φ ( q ji · g i ))= ν a ( φ ( d j )) + ord( φ ( g ′ j ))= ord( φ ( d j · g ′ j )) > ord( φ ( c i · g i )) = ord( φ ( c i · e i )) so that in a ( γ ) = in a φ (cid:16) k X i =1 (cid:16) c i − l X j =1 q j,i · d j (cid:17) · e i (cid:17) = in a φ (cid:16) k X i =1 c i · e i (cid:17)! = η. (cid:3) Now that we are able to lift syzygies with Algorithm 2, we obtain as animmediate consequence:
Corollary 14.
Let I E A L be an ideal. For any graded, Gr a ( A L ) -free reso-lution C • there exist a graded, A L -free resolution D • such that the followingdiagram commutes: C • : 0 ←− A L /I ←− A L ←− M ←− M ←− · · · D • : 0 ←− Gr a ( A L ) / in a ( I ) ←− Gr a ( A L ) ←− N ←− N ←− · · · , in a in a in a in a where if N k = L j ∈ Z Gr a ( A L )( − j ) b j − k,k we have M k = L j ∈ Z A L ( − j ) b j − k,k .In particular, if A L is regular, there always exists a finite A L -free resolu-tion of I . LGORITHMS FOR COMPUTING IN LOCALIZATIONS AT PRIME IDEALS 17
Proof.
Suppose η , . . . , η k ∈ N i are the images of the canonical basis ele-ments of N i +1 , so that they generate the syzygy module of the images of thecanonical basis elements of N i .We can then apply Algorithm 1 to lift them to θ , . . . , θ k ∈ M i , whichwe will set as the images of the canonical basis elements of M i +1 . By ouralgorithm, they are mapped to so that our resulting sequence C • of A L modules is a complex.However, we also know that the initial forms of η , . . . , η k generate theinitial of the syzygy module, hence the η i must generate the syzygy moduleand our complex is exact, giving us a A L -free resolution D • of A L /I .Clearly finite graded Gr a ( A L ) -free resolutions of in a ( I ) lifts to finite A L -free resolutions. If Gr a ( A L ) is isomorphic to a polynomial ring, i.e. Gr a ( A L ) is regular by Proposition 8, it follows from Hilbert’s Syzygy Theorem that afinite free-resolution of in a ( I ) exists. (cid:3) Example 15.
Consider the union of a circle on a hyperplane and the inter-section of a twisted cubic with that hyperplane I := h ( x − + y − , z i ∩ h xz − y , yw − z , xw − yz, z i E A := Q [ x, y, z, w ] . The twisted cubic makes the resolution of I more complicated, a minimalresolution would be of the form ←− A/I ←− A ←− A ←− A ←− A ←− . However if we localize at a subvariety outside the twisted cubic, say on twoconjugate points on the circle, L := I + h x − i = h z, x − y, y w − w, y − y i E A, the local ring will be of the form A L = Q ( w )[ Y , . . . , Y s , x, y, z ] . h f − Y , . . . , f − Y , wY − w Y + Y i , where f , . . . , f are the four generators of L stated above.The associated graded ring of A L is then isomorphic to Gr a ( A L ) = K [ Y , Y , Y , Y ] / h · Y − w · Y i , where K = Quot( A/L ) ∼ = Q ( w )[ t ] / ( t − t − ,x, y, z = 0 , − t , , and the initial ideal of I has the simple form of in a ( I ) = h w Y , Y i E Gr a ( A L ) . It is easy to see that in a ( I ) allows for a Koszul resolution ←− Gr a ( A L ) / in a ( I ) ←− Gr a ( A L ) (cid:16) w Y Y (cid:17) ←− Gr a ( A L ) − Y w Y ) ←− Gr a ( A L ) ←− , which then lifts to an equally simple resolution ←− A L /I ←− A L M ←− A L M ←− A L ←− , but with more complicated matrices M = (cid:16) w Y + 3 w xY + 3 wY Y − Y Y (cid:17) and M = − Y w Y + 3 w Y + w Y Y − w Y + 3 w Y − Y Y + w Y + Y Y . acknowledgements We would like to thank Janko Boehm for insightful converstations andTheo Mora for clarifications of the current state of research.
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E-mail address : [email protected] Yue Ren, Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany
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