Open Questions: Commuting in Tight Closure Theory, Study notes of Calculus

Download Open Questions: Commuting in Tight Closure Theory and more Study notes Calculus in PDF only on Docsity! Math 615: Lecture of March 14, 2007 Open questions: tight closure, plus closure, and localization We want to consider some open questions in tight closure theory, and some related problems about when rings split from their module-finite extension algebras. After we do this, we shall prove some specific results in the characteristic p theory. It will turn out that to proceed further, we will need the structure theory of complete local rings, which we will develop next. One of the longest standing and most important questions about tight closure is whether tight closure commutes with localization. E.g., if R is Noetherian of prime characteristic p > 0, I is an ideal of R, and W is a multiplicative system of R, is W−1(I∗R) the same as (W−1)∗W−1R? It is easy to prove that W −1(I∗R) ⊆ (W−1)∗W−1R. This has been open question for more than twenty years. It is known to be true in many cases. See, for example, [I. Aberbach, M. Hochster, and C. Huneke, Localization of tight closure and and modules of finite phantom projective dimension, J. Reine Angew. Math. (Crelle’s Journal) 434 (1993), 67–114], and [M. Hochster and C. Huneke, Test exponents and localization of tight closure, Michigan Math. J. 48 (2000), 305–329] for a discussion of the problem. We saw in the last Theorem of the Lecture Notes of March 12 that tight closure “cap- tures” contracted extension from module-finite and even integral extensions. We shall add this as (6) to our list of desirable properties for a tight closure theory, which becomes the following: (0) u ∈ N∗M if and only if the image u of u in M/N is in 0∗M/N . (1) N∗M is a submodule of M and N ⊆ N∗M . If N ⊆ Q ⊆ M then N∗M ⊆ Q∗M . (2) If N ⊆ M , then (N∗M )∗M = N∗M . (3) If R ⊆ S are domains, and I ⊆ R is an ideal, I∗ ⊆ (IS)∗, where I∗ is taken in R and (IS)∗ in S. (4) If A is a local domain then, under mild conditions on A (the class of rings allowed should include local rings of a finitely generated algebra over a complete local ring or over Z), and f1, . . . , fd is a system of parameters for A, then for 1 ≤ i ≤ d−1, (f1, . . . , fi)A :A fi+1 ⊆ ( (f1, . . . , fi)A )∗ . (5) If R is regular, then I∗ = I for every ideal I of R. (6) For every module-finite extension ring R of S and every ideal I of R, IS ∩R ⊆ I∗. These are all properties of tight closure in prime characteristic p > 0, and also of the theory of tight closure for Noetherian rings containing Q that we described in the Lecture of March 12. In characteristic p > 0, (4) holds for homomorphic images of Cohen-Macaulay rings, and for excellent local rings. If R ⊇ Q, (4) holds if R is excellent. We will prove 1 2 that (4) holds in prime characteristic for homomorphic images of Cohen-Macaulay rings quite soon. We have proved (5) in prime characteristic p > 0 for polynomial rings over a field, but not yet for all regular rings. To give the proof for all regular rings we need to prove that the Frobenius endomorphism is flat for all such rings, and we shall eventually use the structure theory of complete local rings to do this. An extremely important open question is whether there exists a closure theory satisfying (1) — (6) for Noetherian rings that need not contain a field. The final Theorem of the Lecture of March 12 makes it natural to consider the following variant notion of closure. Let R be any integral domain. Let R+ denote integral closure of R in an algebraic closure K of its fraction field K. We refer to this ring as the absolute integral closure of R. R+ is unique up to non-unique isomorphism, just as the algebraic closure of a field is. Any module-finite (or integral) extension domain S of R has fraction field algebraic over K, and so S embeds in K. It follows that S embeds in R+, since the elements of S are integral over R. Thus, R+ contains an R-subalgebra isomorphic to any other integral extension domain of R: it is a maximal extension domain with respect to the property of being integral over R. R+ is the directed union of its finitely generated subrings, which are module-finite over R. R+ is also charactized as follows: it is a domain that is an integral extension of R, and every monic polynomial with coefficients in R+ factors into monic linear polynomials over R+. Given an ideal I ⊆ R, the following two conditions on f ∈ R are equivalent: (1) f ∈ IR+ ∩R. (2) For some module-finite extension S of R, f ∈ IS ∩R. The set of such elements, which is IR+∩R, is denoted I+, and is called the plus closure of I. (The definition can be extended to modules N ⊆ M by defining N+M to be the kernel of the map M → R+ ⊗R (M/N).) By the last Theorem of the Lecture Notes of March 12, which is property (6) above in characteristic p > 0, we have that I ⊆ I+ ⊆ I∗ in prime characteristic p > 0. Whether I+ = I∗ in general under mild conditions for Noetherian rings of prime characterisitc p > 0 is another very important open question. It is not known to be true even in finitely generated algebras of Krull dimension 2 over a field. However, there are some substantial positive results. It is known that under the mild conditions on the local domain R (e.g., when R is excellent), if I is generated by part of a system of parameters for R, then I+ = I∗. See [K. E. Smith, Tight closure of parameter ideals, Inventiones Math. 115 (1994) 41–60]. Moreover, H. Brenner [H. Brenner, Tight closure and plus closure in dimension two, Amer. J. Math. 128 (2006) 531–539] proved that if R is the homogeneous coordinate ring of a smooth projective curve over the algebraic closure of Z/pZ for some prime integer p > 0, then I∗ = I+ for homogeneous ideals primary to the homogeneous maximal ideal. In [G. Dietz, Closure operations in positive