Abstract
The goal of this chapter is to extend the tight closure theory from the previous chapter to include all Noetherian rings containing a field. However, the theory becomes more involved, especially if one wants to maintain full functoriality. We opt in these notes to forego this cumbersome route (directing the interested reader to the joint paper [6] with Aschenbrenner), and only develop the theory minimally as to still obtain the desired applications. In particular, we will only focus on the local case.
Keywords
- Prime Ideal
- Local Ring
- Characteristic Zero
- Homomorphic Image
- Noetherian Ring
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References
Artin, M.: Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36, 23–58 (1969) 87, 100, 103, 104, 177
Aschenbrenner, M., Schoutens, H.: Lefschetz extensions, tight closure and big Cohen-Macaulay algebras. Israel J. Math. 161, 221–310 (2007) 4, 42, 97, 104, 109, 110
Becker, J., Denef, J., van den Dries, L., Lipshitz, L.: Ultraproducts and approximation in local rings I. Invent. Math. 51, 189–203 (1979) 3, 63, 102, 103, 104
Denef, J., Lipshitz, L.: Ultraproducts and approximation in local rings II. Math. Ann. 253, 1–28 (1980) 3, 103
Denef, J., Schoutens, H.: On the decidability of the existential theory of Fp[[t]]. In: Valuation theory and its applications, Vol. II (Saskatoon, 1999)
Olberding, B., Shapiro, J.: Prime ideals in ultraproducts of commutative rings. J. Algebra 285(2), 768–794 (2005) 58
Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam (1966) 15
Smith, K.: Tight closure of parameter ideals. Invent. Math. 115, 41–60 (1994) 92
Spivakovsky, M.: A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms. J. Amer.Math. Soc. 12, 381–444 (1999) 100
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Schoutens, H. (2010). Tight Closure in Characteristic Zero. Local Case. In: The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics(), vol 1999. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13368-8_7
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DOI: https://doi.org/10.1007/978-3-642-13368-8_7
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