Satyagopal Mandal |
Department of Mathematics |
Office: 502 Snow Hall Phone: 785-864-5180 |
We first state the following theorem of Bloch-Murthy-Szpiro ([BMS]).
Theorem 1.1 Let R=k[X_{1},X_{2},X_{3},X_{4}] be a polynomial ring over an algebraically closed field k and I be an ideal of height 2 in R and A=R/I, such that X = Spec(R/I) is smooth over k. Let $\omega $_{I}=$\Lambda $^{2}(I/I^{2})^{-1} and K_{X} = C_{1}($\omega $_{I}) be the canonical divisor of X. If rK_{X}^{2} =0 in the Chow group CH^{2}(X), for some integer r>0, then X is set-theoretically complete intersection.
Following theorem is helpful to understand the statement of this theorem K-theoretically.
Theorem 3.2 Let X =Spec(A) be a smooth affine algebra over an algebraically closed field k. Assume that projective A-modules of rank 2 have cancellation property. Let L be a projective A-module of rank one. Then the following conditions are equivalent:
We will try to give a 5-dimentional version of the theorem 1.1 assuming that projective moduled of rank n-1 have cancellation property, where n is the dimention of the ring.
For (an increasing sequence of) positive integers n, Mohan Kumar ([MK]) constructed smooth affine algebras A with dim A = n and stably free projective A-modules of rank n-2 that are not free. He also posed the following question.
LRC-problem: Let A be a (smooth) affine algebra of dimension n over an algebraically closed field k. Does the projective A-modules of rank n-1 have the cancellation property? That is, for a projective A-module P with rank(P)=n-1, does P $\oplus $ A $\equiv $ Q $\oplus $ A $\Rightarrow $ P $\equiv $ Q?
If LRC-problem has an affirmative answer for 3-folds, then we prove a three dimensional analogue of the above theorem of Bloch-Murthy-Szpiro.
Lemma 2.1 Let R= A[X] be a polynomial ring over a noetherian commutative ring A and I be a locally complete intersection ideal of height 2 in R. Assume that I contains a monic polynomial and there is a surjective map I/I^{2} $\to \; \omega $ _{I} = Ext^{2}(R/I,R). Then I is set theoretically generated by 2 elements.
Proof.
Let J be defined by the exact sequence
0 $\to $ J/I^{2} $\to $ I/I^{2} $\to $ $\omega $_{I} $\to $ 0 |
The generator e of Ext^{1}(J,R)
correcpond to an exact sequence
0 $\to $ R $\to $ P $\to $ J $\to $ 0 |
Exercise. Why P is projective?
So, the main Idea is to get a map like in this lemma.
Lemma 2.2 Let R=R'[X] be polynomial ring over a noetherian commutative ring R' with dim R' =4 (so dim R=5). Let I be a locally complete intersection ideal of height 2 in R that contains a monic polynomial and A= R/I. Assume that all rank 2 projective A=R/I modules have cancellation property. Then
Proof.
0 $\to $ P $\to $ R^{n} $\to $ I $\to $ 0 |
Hence, from theorem of Quillen and Suslin it follows P $\equiv $ Rn-1.$$
Tensoring the above sequence with A we get the following exact sequence:
0 $\to $ L $\to $ A^{n-1} $\to $ A^{n} $\to $ I/I^{2} $\to $ 0 |
Now we prove the second statement:
a) $\Rightarrow $ b): By 1) we have
I/I^{2} $\equiv $ $\omega $^{-1}
$\oplus $ A and also A^{2} $\equiv $ $\omega $
$\oplus \; \omega $^{-1}. So,
I/I^{2} $\oplus $
A $\equiv $ (A $\oplus \; \omega $^{-1})
$\oplus $ A
$\equiv $ $\omega $^{-1} $\oplus $
A^{2}
$\equiv $ $\omega $^{-1}
$\oplus $
($\omega $ $\oplus \; \omega $^{-1})
$\equiv $ $\omega $ $\oplus $($\omega $ ^{-1} $\otimes $ A^{2}) $\equiv $ $\omega $ $\oplus \; \omega $^{-1} $\otimes $ ($\omega $ $\oplus \; \omega $^{-1}) $\equiv $ $\omega $ $\oplus $ A $\oplus \; \omega $^{-2}. |
Since all rank 2 projective A-modules have cancellation property, we have I/I^{2}$\equiv $ $\omega $ $\oplus \; \omega $^{-2}.
b)$\Rightarrow $ c): By 1) and b) it follows that
A $\oplus \; \omega $^{-1} $\equiv $ $\omega $ $\oplus \; \omega $^{-2}. |
A $\oplus \; \omega $ $\equiv $ $\omega $^{-1} $\oplus \; \omega $^{2}. |
A^{2} $\oplus \; \omega $ $\equiv $ (A $\oplus \; \omega $^{-1}) $\oplus \; \omega $^{2} $\equiv $I/I^{2} $\oplus \; \omega $^{2} $\equiv $($\omega $ $\oplus \; \omega $^{-2}) $\oplus \; \omega $^{2} |
Following theorem relates the cancellation property of rank 2 projective modules with set theoretic complete intersection property of height 2 locally complete intersection ideals.
Theorem 2.1 Let R=R'[X] be a polynomial ring over a noetherian commutative ring R' with dim R' =4 (so dim R=5). Let I $\subseteq $ R be a locally complete intersection ideal of height 2 that contains a monic polynomial and A = R/I. Assume that projective A-modules of rank 2 have the cancellation property. Let $\omega $ = Ext^{2}(A,R). Assume that $\omega $^{r} is generated by 2 elements, for some integer r ≥1. Then I is a set-theoretic complete intersection ideal.
Proof.
0 $\to $ A $\to $ I/I^{2} $\to $ $\omega $^{-1} $\to $ 0, |
Let L be the cokernel of the map given by f as follows
f: R/J $\to $ J/J^{2} $\to $ L $\to $ 0. |
Since rad(I) = rad(J), all rank 2 projective R/J-modules also have cancellation property. By the above lemma, we have J/J^{2}$\equiv $ R/J $\oplus \; \omega $_{J}^{-1} and L $\equiv $ $\omega $_{J}^{-1}. It follows that
Consider the exact sequence
0 $\to $ R/J$\to $ J/J^{2} $\to $ $\omega $_{J}^{-1}$\to $. |
0 $\to $ R/I $\to $ J/IJ $\to $ $\omega $_{J}^{-1} $\otimes $ R/I $\to $ 0 |
$\omega $_{J}^{-1} $\otimes $ R/I $\equiv $ J/(IJ,f) $\equiv $ (I^{r},f)/(I^{r+1},f) $\equiv $ $\omega $_{I}^{r}, |
J/J^{2} $\to $$\omega $_{J}. |
Now it follows from lemma 2.1 that J is set theoretically generated by 2 elements. This completes the proof of this theorem.
Now we give a criterion for locally complete intersection ideal of height 2, in the polynomial ring R, to be ideal theoretically generated by dim R -2 elements.
Theorem 2.2 Let R=R'[X] be the polynomial ring over a noetherian commutative R'. Assume dim R =n ≥ 3. Suppose I is a locally complete intersection ideal of height 2 that contains a monic polynomial and A=R/I. Assume that all projective A-modules of rank n-3. have cancellation property. Write $\omega $=Ext^{2}(A,R). Then the following conditions are equivalent:
Proof.
2) $\Rightarrow $ 3): We have the following exact sequence:
0 $\to $ P $\to $ R^{m} $\to $ I $\to $ 0 |
So, by the theorem of Quillen and Suslin, P $\equiv $ R^{m-1}.
By tensoring the above sequence with A=R/I, we get the following
exact sequence:
0 $\to $ L $\to $ A^{m-1} $\to $ A^{m} $\to $ I/I^{2} $\to $ 0 |
L $\equiv \; \Lambda $^{2} I/I^{2} $\equiv $ $\omega $^{-1}. |
Since I/I^{2} is generated by n-2 elements, we have a surjective map A^{n-2+k} $\to \; \omega $^{-1} $\oplus $ A^{k+1}.
Therefore,
Q $\oplus \; \omega $^{-1} $\oplus $ A^{k+1} $\equiv $ A^{n-2+k}. |
3) $\Rightarrow $ 1):
Since $\omega $ is n-3 generated,
by the argument of Serre (see [Mu]),
there is an exact sequence
0 $\to $ R^{n-3} $\to $ P $\to $ I $\to $ 0 |
Theorem 2.3 Let R' be a smooth affine algebra, with dim R'=n-1, over an algebraically closed field k, with trivial differential module $\Omega $ _{R'/k} $\equiv $ R'^{n-1}. Let R=R'[X] be the polynomial ring. Assume n=dim R ≥ 5. Let I be an ideal of R so that A = R/I is smooth and let I contain a monic polynomial. Assume that n-2 ≥ dim A +2 or height(I)=1,2. In case height (I) =1, assume that all projective A-modules of rank n -2 have cancellation property. Then the following conditions are equivalent:
Proof.
2) $\Rightarrow $ 3):
Consider the sequence
0 $\to $ I/I^{2} $\to \; \Omega $_{R/k} / I$\Omega $_{R/k}→ Ω_{A/k} $\to $ 0 |
The sequence is exact. Also note that
$\Omega $_{R/k} $\equiv \; \Omega $_{R'/k} ⊗ R $\oplus $ RdX $\equiv $ R^{n}. |
By 2), we have I/I^{2} $\oplus $ Q $\equiv $ A^{n-2}. So, in K_{0}(A), we have [A^{2}] + [Q] = [$\Omega $_{ A/k }]. By cancellation theorem of Suslin, we have $\Omega $_{ A/k } $\equiv $ A^{2} $\oplus $ Q.
3) $\Rightarrow $ 2):
We have $\Omega $_{ A/k
} $\equiv $ A^{2} $\oplus $
P
and
I/I^{2} $\oplus \; \Omega $_{ A/k
}$\equiv $ A^{n}
So,
I/I^{2} $\oplus $ A^{2} $\oplus $ P $\equiv $ A^{n}. |
I/I^{2} $\oplus $ P $\equiv $ A^{n-2}. |
2) $\Rightarrow $ 1): If n-2 ≥ dim (A) +2, then it follows from [Ma1] that I is n-2 generated. If height(I) =1 then I is one generated. If height(I)=2 it follows from the above theorem.
Notation 3.1 Let A a smooth affine algebra over an algebraically closed field k and X=Spec(A).
Following is a restatement of the theorem of Murthy and Mohan Kumar ([MKM]) in our context of LRC-question.
Theorem 3.1 Let X =Spec(A) be a smooth affine 3-fold over an algebraically closed field k. Assume that projective A-modules of all rank have cancellation property. Then, for any two projective modules P and Q of same rank, P $\equiv $ Q if and only if C_{i}(P) = C_{i}(Q) for i=1,2,3.
Proof.
($\Leftarrow $) The theorem of Mohan Kumar and Murthy ([MKM]) implies that P and Q are stably isomorphic. Now P $\equiv $ Q by the cancellation property.
Theorem 3.2 Let X =Spec(A) be a smooth affine algebra over an algebraically closed field k. Assume that projective A-modules of rank 2 have cancellation property. Let L be a projective A-module of rank one. Then the following conditions are equivalent:
Proof.
2) $\Rightarrow $ 3): It follows that the total Chern class C(L $\oplus $ L^{-1}) =1. Hence [L] $\oplus $ [L^{-1}] = [A^{2}] in K_{0}(A) and by cancellation L $\oplus $ L^{-1} $\equiv $ A^{2}.
3) $\Rightarrow $ 1): This implication is obvious.
Theorem 3.3 Suppose k is an algebraically closed field and R=R'[X] a polynomial ring over an affine algebra R' with dim R' =4. Let I be a locally complete intersection ideal in R of height 2 that contains a monic polynomial. Assume A =R/I is smooth and rank 2 projective A-modules have cancellation property. Then the following conditions are equivalent:
In all these cases, we have I/I^{2} $\equiv $ $\omega $_{I} $\oplus \; \omega $_{I}^{-2} and that I is a set-theoretic complete intersection ideal.
Proof.
By lemma 2.2, we have I/I^{2} $\equiv $ A $\oplus \; \omega $^{-1}. Since K^{2}=0, the total Chern class C(I/I^{2}) = C(A $\oplus \; \omega $_{I}^{-1}) =C($\omega $_{I} $\oplus \; \omega $_{I}^{-2}). By cancellation I/I^{2} $\equiv $ $\omega $_{I} $\oplus \; \omega $_{I}^{-2}. Also I is set-theoretic complete intersection by Theorem 2.1. So, the proof of this theorem is complete.
Theorem 3.4 Suppose k is an algebraically closed field and R=R'[X] is a polynomial ring over an affine algebra R' with dim R' =4. Let I be a locally complete intersection ideal of R with height(I) = 2 and A=R/I is smooth over k. Assume that all rank 2 projective A-modules have cancellation property. Suppose rK^{2} = 0 for some r > 0 where K=C_{1}( $\omega $_{I}). Then I is set-theoretic complete intersection.
Proof.
C(L $\oplus $ L^{-1}) = (1+rK)(1 - rK) = 1. |
So, L $\oplus $ L^{-1} $\equiv $ A^{2}. So, $\omega $_{I} ^{r} is two generated. By Theorem 2.1, I is a set-theoretic complete intersection.
Following corollary follows from the above thoerem.
Corollary 3.1 Let k be an algebraically closed field and R=k[X_{1}, ... ,X_{5}] be polynomial ring and $\phi $ : R $\to $ A =A'[X] be a surjective map onto a polynomial algebra over a smooth algebra A' of dimension 2. Let I be a the kernel of $\phi $ and K=C_{1} ($\omega $_{I}). If rK^{2} =0 for some positive integer r then I is set theoretic complete intersection.
Proof.
Theorem 3.5 Let R=k[X_{1}, .... , X_{5}] be a polynomoal ring over an algebraically closed field k and I be a locally complete intersection ideal of height 2. Assume A=R/I is smooth and rank 2 projective A-modules have cancellation property. If X=spec (A) has a projective smooth completion Y with rK_{Y}^{2} =0 for some positive integer r, then X is set-theoretic complete intersection.
Proof.