Corrigendum to "Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities"
aa r X i v : . [ m a t h . A C ] M a r Corrigendum to “Reduced Tangent Conesand Conductor at MultiplanarIsolated Singularities”
Alessandro De Paris and Ferruccio Orecchia
Keywords:
Associated graded ring, Conductor, Singularities, Tangent cone.
MSC 2010: (cid:0) A (cid:1) is reduced. (1)It is also implicitly used in the middle of the proof:the hypothesis 3 in the statement implies that ν n is an isomorphismfor n >>
0. (2)In (1), the outcome Proj (G ( A )) ∼ = Proj (cid:0) G m (cid:0) A (cid:1)(cid:1) of the remark implies thatG (cid:0) A (cid:1) is reduced because of the hypothesis 1 in the statement and [1, Propo-sition 2.1]. In (2), the fact that ν n is an isomorphism for n >> m n = m n (and justifies the injectivity of S ֒ → G (cid:0) A (cid:1) ). Thoseoutcomes hold true in the hypotheses of [1, Theorem 3.2], simply because byadding the hypothesis that Proj (G ( A )) is reduced (hypothesis 1 in the theo-rem), [1, Remark 2.2] becomes true. The explanation is given by Proposition 1below, which therefore can be used as a replacement for the wrong remark(provided that the hypothesis 1 is also mentioned in (2)).In the statement of Proposition 1, the main thesis is that ν n is an isomor-phism for n >>
0. This also implies that m n = m n , by the Nakayama’s lemma,but this equality is not really needed (when we say ‘Then (2) is satisfied because m n + i = J n + i for i >> Proposition 1
Under the assumptions before [1, Remark 2.2], if
Proj (G ( A )) is reduced and √ b = m , then ν n is an isomorphism for n >> , and in turn thisimplies that Proj (G ( A )) ∼ = Proj (cid:0) G m (cid:0) A (cid:1)(cid:1) . Proof.
Since A is noetherian and √ b = m , we have m n ⊆ b for some n , hence m n = m n A ⊆ b A = b ⊆ m . 1f the class x of a f ∈ m n r m n +1 in G ( A ) n is in the kernel of ν n that is, f belongs to m n +1 , we have f n ∈ m nn + n = m nn + n A = m nn m n ⊆ m nn +1 , hence x n = 0 ∈ G ( A ) nn , that is, x is nilpotent. Since Proj (G ( A )) is reduced,there exists n such that for all n ≥ n there is no nilpotent x ∈ G ( A ) n r { } ,and therefore ν n is injective for all n ≥ n .Let us consider the powers m n and m n as A -modules, and denote by l ( M )the length of an arbitrary A -module M . Note also that the graded componentsG m (cid:0) A (cid:1) n = m n / m n +1 are vector spaces over the residue field k := A/ m , because m n +1 = mm n ; the same is obviously true for the graded components of G ( A ).For every n ≥ n we have m n ⊆ m n ⊆ m n ⊆ m , hence n − X i = n dim k G m (cid:0) A (cid:1) i = l (cid:18) m n m n (cid:19) ≤ l (cid:16) mm n (cid:17) = n − X i =1 dim k G ( A ) i . (3)Note that dim k G m (cid:0) A (cid:1) i ≥ dim k G ( A ) i for all i ≥ n , because ν i is injectivefor that values. Suppose now that it is not true that ν n is an isomorphismfor all n >>
0. Then we can find as many values of i as we want for whichdim k G m (cid:0) A (cid:1) i > dim k G ( A ) i . This contradicts (3) for a sufficiently large n .Finally, it is well known that if a graded ring homomorphism S → T preserv-ing degrees induces isomorphisms on all components of sufficiently large degrees,then it induces an isomorphism Proj ( T ) ∼ → Proj ( S ). (cid:3) Next, [1, Remark 2.2] is invoked in [1, Section 5, p. 2977, lines 20–21]:Since √ b = m and J = m A , we have Proj (G ( A )) ∼ = Proj (cid:0) G (cid:0) A (cid:1)(cid:1) by Remark 2.2. (4)At that point, we do not have that Proj (G ( A )) is reduced, and this fact is partof n. 3 of Claim 5.1 (the only point of that Claim that was still to be proved).In what follows we assume the notation of [1, Section 5]. To fix the mistake, thesentence (4) above must be dismissed, but we can keep the fact that J = m A (which is true, because it had been proved earlier that J = m B and A = B ).Let us look at the subsequent discussion in [1]. First of all, the isomorphismG (cid:0) A (cid:1) ∼ = Q ei =1 G a i ( k [ t, s ]) is pointed out. Here, let us also denote by π i :G (cid:0) A (cid:1) → G a i ( k [ t, s ]) = k (cid:2) t − a i , s i (cid:3) the projection on the i th factor. Let ussplit as follows the natural homomorphism displayed at [1, p. 2977, line 25]: k [ X , . . . , X r ] → G ( A ) ν −→ G (cid:0) A (cid:1) ∼ → e Y i =1 G a i ( k [ t, s ]) π i −→ k (cid:2) t − a i , s i (cid:3) . (5)(in the description X j x j given in the paper, x j denotes the class of x j ∈ R ⊂ k [ t, s ] in the degree one component of G a i ( k [ t, s ])). Assuming that X , . . . , X r act as the coordinate functions on k r +1 , and the linear forms in k [ X , . . . , X r ]act accordingly, for each choice of distinct i , i ∈ { , . . . , e } , since l i and l i are skew lines we can fix linear forms T i i , S i i such that2 T i i vanishes on a i , b i , b i , and takes value 1 /ρ i on a i , with ρ i beingas in [1, Footnote 4]; • S i i vanishes on a i , b i , a i , and takes value 1 on b i .Taking into account [1, Footnote 4], the images of T i i and S i i in k (cid:2) t − a i , s i (cid:3) through (5) vanish for i = i , and equal t − a i and s i , respectively, for i = i . Itfollows that the image of T i · · · T ( i − i T ( i +1) i · · · T ei in Q ei =1 G a i ( k [ t, s ]) through(5) is (cid:16) , . . . , t − a i e − , , . . . , (cid:17) , where the nonzero component occurs at the i -th place. In a similar way, every element of the form (cid:0) , . . . , t − a i a s b , , . . . , (cid:1) ,with a + b ≥ e −
1, is the image of a product of a + b linear forms, suitablychosen among the T i i s and the S i i s. It follows that the homomorphism k [ X , . . . , X r ] → Q ei =1 G a i ( k [ t, s ]) is surjective in degrees ≥ e −
1. Hence ν d issurjective for d ≥ e −
1, and is in fact an isomorphism by [1, Lemma 3.1]. Thus ν induces the required isomorphism Proj (G ( A )) ∼ = Proj (cid:0) G (cid:0) A (cid:1)(cid:1) .Finally, let us take this occasion to also fix a few typos and mild issues.1. In the summarized description of the main result [1, Theorem 3.2] givenin [1, Introduction] at the beginning of p. 2970, it is missed the hypothesis(duly reported in the statement of the theorem and in the abstract) thatProj (G ( A )) must be reduced.2. Typo at [p. 2971, lines 12–14]: “ P r +1 := Proj ( k [ X , . . . , X k ]) . . . sub-scheme of P r +1 ” should be replaced with “ P r := Proj ( k [ X , . . . , X r ]) . . . subscheme of P r ”.3. At the beginning of the proof of [1, Theorem 4.1] one findsThe hypothesis 1 states, in particular, that Y := Proj (G ( A )) isreduced. Then, taking also into account the hypothesis 2, we imme-diately deduce from the results of Section 2 that H (cid:0) A, n (cid:1) = ( n +1) e .Since Proj (G ( A )) is reduced, Proposition 1 can serve as a replacementfor [1, Remark 2.2] in this situation, too.4. In [1, Section 5] (main example), e must be assumed ≥ n has to bereplaced with r in some lists such as x , . . . , x n , f , . . . , f n , h , . . . , h n and X , . . . , X n , as well as in the exponent n + 1 in [1, Equation (6)].In the sentence below Equation (6), ‘such that g ( t ) takes one of the abovementioned special values’ should be completed with ‘or g ′ ( t ) = 0’. Later,‘vanishes in the ring T := B f ⊗ A f B f ’ should be ‘vanishes over the ring T := B f ⊗ A f B f ’ (that is, each component of the ( n + 1)-tuple underconsideration vanishes in T ). References [1] De Paris, A and Orecchia, F. Reduced Tangent Cones and Conductor at Mul-tiplanar Isolated Singularities Commun. Algebra36