Bockstein Cohomology of Associated Graded Rings
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Bockstein Cohomology of Associated Graded Rings Tony J. Puthenpurakal1 Received: 27 March 2018 / Revised: 25 December 2018 / Accepted: 27 December 2018 / Published online: 13 February 2019 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract Let (A, m) be a Cohen-Macaulay local ring of dimension d and let I be an m-primary ideal. Let G be the associated graded ring of A with respect to I and let R = A[I t, t −1 ] be the extended Rees ring of A with respect to I . Notice t −1 is a nonzero divisor on R and R/t −1 R = G. So, we have Bockstein operators β i : HGi + (G)(−1) → HGi+1 (G) for + i ≥ 0. Since β i+1 (+1) ◦ β i = 0, we have Bockstein cohomology modules BH i (G) for i = 0, . . . , d. In this paper, we show that certain natural conditions on I implies vanishing of some Bockstein cohomology modules. Keywords Associated graded rings · Rees Algebras · Local cohomology Mathematics Subject Classification (2010) Primary 13A30 · Secondary 13D40 · 13D07
1 Introduction Let (A, m) be a Cohen-Macaulay local ring of dimension d and let I be an m-primary ideal. The Hilbert function of A with respect to I is H I (A, n) = (I n /I n+1 ). Here, (−) denotes length as an A-module. A fruitful area of research has been to study the interplay between Hilbert functions and properties of algebras of A with respect to I , namely the blowup n /I n+1 , the Rees ring S (I ) = n = I associated graded ring GI (A) n≥0 n≥0 I and the n n extended Rees ring R(I ) = n∈Z I (here I = A for n ≤ 0 and R(I ) is considered as a subring of A[t, t −1 ]). See the texts [23, Section 6] and [24, Chapter 5] for nice surveys on this subject. Graded local cohomology has played an important role in this subject. For various applications, see [3, 4.4.3], [1, 8, 10, 20, 22] and [6]. Let H i (GI (A)) denote i th -local cohon n+1 . A line of inquiry mology module of GI (A) with respect to GI (A)+ = n>0 I /I in this subject is to find conditions on I such that GI (A) (or GI n (A) for all n 0) has high depth. This is equivalent to showing that H i (GI (A)) (or H i (GI n (A)) for all n 0) vanishes for some i < d. It is the belief of the author that local cohomology modules Tony J. Puthenpurakal
[email protected] 1
Department of Mathematics, IIT Bombay, Powai, Mumbai, 400 076, India
286
T.J. Puthenpurakal
H i (GI (A)) have some subtle properties for i < d. In this paper, we demonstrate this viewpoint with some success. See Section 2.1 for a general construction of Bockstein cohomology. This notion has been used in algebraic topology and in group cohomology since long time ago. But its appearance in commutative algebra is very recent; for instance, see [21]. Notice t −1 is a nonzero divisor on R(I ) and R(I )/t −1 R(I ) = GI (A). So, we have Bockstein operators β i : H i (GI (A))(−1) → H i+1 (GI (A)) for i ≥ 0. Since β i+1 (+1) ◦ β i = 0, we have Bockstein cohomology modules BH i (GI (A)) for i = 0, . . . , d. Despite being natural, Bockstein cohomology groups of assoc
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