Excellent Local Domain Research Articles

Finitely generated saturated multi-Rees algebras

We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao.

Multiplicities and mixed multiplicities of arbitrary filtrations

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

Completions of uncountable local rings with countable spectra

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results provide a way to construct uncountable local domains with countable spectra from a much broader class of complete local rings than was previously known. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.

The Minkowski equality of filtrations

Suppose that R is an analytically irreducible or excellent local domain with maximal ideal mR. We consider multiplicities and mixed multiplicities of R by filtrations of mR-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of mR-primary ideals.

Mixed multiplicities of divisorial filtrations

Suppose that R is an excellent local domain with maximal ideal mR. The theory of multiplicities and mixed multiplicities of mR-primary ideals extends to (possibly non Noetherian) filtrations of R by mR-primary ideals, and many of the classical theorems for mR-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for mR-primary ideals do not hold for filtrations.In this article, we consider multiplicities and mixed multiplicities of R by mR-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of mR-primary ideals, that are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of mR-primary ideals I⊂I′ with the same multiplicity then I and I′ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. The classical Minkowski inequalities for mR-primary ideals I1 and I2 hold quite generally for filtrations. If R has dimension two and there is equality in the Minkowski inequalities, then Teissier and Rees and Sharp have shown that there are powers I1a and I2b that have the same integral closure. This theorem does not hold for arbitrary filtrations. The Teissier-Rees-Sharp theorem has been extended by Katz to mR-primary ideals in arbitrary dimension. We show that the Teissier-Rees-Sharp theorem does hold for divisorial filtrations in an excellent local domain of dimension two.We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme X birationally dominating R, when R is an algebraic local domain (essentially of finite type over a field).

The vanishing conjecture for maps of Tor and derived splinters

We say an excellent local domain (S,\mathfrak n) satisfies the vanishing conditions for maps of Tor , if for every A\to R\to S with A regular and A\to R module-finite torsion-free extension, and every A -module M , the map \mathrm {Tor}^A_i(M, R)\to \mathrm {Tor}_i^A(M, S) vanishes for every i\geq 1 . Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions [HH95]. The main theorem of this paper shows that, in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of Bhatt [Bha12]. In particular, rational singularities in characteristic 0 satisfy the vanishing conditions. This greatly generalizes Hochster–Huneke’s result [HH95] and Boutot’s theorem [Bou87]. Moreover, our result leads to a new (and surprising) characterization of rational singularities in terms of splittings in module-finite extensions.

The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation

Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $\nu^*$ is a valuation of $K^*$ such that $\nu^*$ dominates $S$. Let $\nu$ be the restriction of $\nu^*$ to $K$. The associated graded ring ${\rm gr}_{\nu}(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,\nu)\rightarrow (K^*,\nu^*)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blow up of $R$, $S_1$ is a local ring of a blowup of $S$, $\nu^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{\nu^*}(S_1)$ is a finitely generated ${\rm gr}_{\nu}(R_1)$-algebra. We also investigate the role of splitting of the valuation $\nu$ in $K^*$ in finite generation of the extensions of associated graded rings along the valuation. We will say that $\nu$ does not split in $S$ if $\nu^*$ is the unique extension of $\nu$ to $K^*$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, $\nu^*$ has rational rank 1 and is not discrete and ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, then $\nu$ does not split in $S$. We give examples showing that such a strong statement is not true when $\nu$ does not satisfy these assumptions. We deduce that if $\nu$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $\nu$, then ${\rm gr}_{\nu}(R')$ is not a finitely generated ${\rm gr}_{\nu}(R)$-algebra.

Controlling the generic formal fiber of local domains and their polynomial rings

Let $T$ be a complete local ring with maximal ideal $M$, $C$ a countable set of incomparable prime ideals of $T$, and $B_1$ and $B_2$ sets of prime ideals of $T[[x_1,\ldots,x_n]]$ with cardinality less than that of $T$. We present necessary and sufficient conditions for the existence of a local domain $A$ with completion $T$, such that the generic formal fiber of $A$ has maximal elements equal to the ideals in $C$ and the generic formal fiber of $A[x_1,\ldots,x_n]_{(M\cap A,x_1,\ldots,x_n)}$ contains every element of $B_1$ but no element of $B_2$. If $T$ has characteristic $0$, we present necessary and sufficient conditions for the existence of an excellent local domain $A$ with the above properties.

Completions of Hypersurface Domains

Let V be a complete regular local (Noetherian) ring and let f ∈ V be a nonunit. We find necessary and sufficient conditions for to be the completion with respect to the maximal ideal of an integral domain of the form where S is a regular local ring whose completion with respect to its maximal ideal is V. In addition, if contains the rationals, we give necessary and sufficient conditions for to be the completion of an excellent local domain of the form , where f ∈ S, and S is a regular local ring whose completion is V.

Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman's methods. A key step is the proof of the following result, which answers a question of Colliot-Th\'el\`ene--Ojanguren--Parimala: For a Brauer class over $K$ of prime order $q$ different from the characteristic of $k$, if it is cyclic of degree $q$ over the completed field $K_v$ for every discrete valuation $v$ of $K$, then the same holds over $K$. This local-global principle for cyclicity is also established over function fields of $p$-adic curves with the same method.

A property of the absolute integral closure of an excellent local domain in mixed characteristic

A property of the absolute integral closure of an excellent local domain in mixed characteristic

Un cas de majoration affine pour la fonction d'approximation d'Artin

Let ( A , m , k ) be an henselian excellent local domain of equal characteristic and I be an homogenous ideal in A [ X , Y ] . We show that I has an Artin approximation function bounded by an affine function, extending the previously known results. To cite this article: M. Hickel, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Asymptotic vanishing conditions which force regularity in local rings of prime characteristic

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8} {\length(H_i(F^e(G_\dt)))}/{p^{ed}}$. We show that if $\gothic t_i(R) = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\Tor_i^R(k,R^+) = 0$ for some $i>0$, then $R$ is regular (where $R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i = 1$ or $2$ via completely different methods.

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p 1 ⊆ p 2 ⊆ ··· ⊆ p n be a chain of nonmaximal prime ideals of T such that T p n is a regular local ring. We construct a chain of excellent local domains A n ⊆ A n−1 ⊆ ··· ⊆ A 1 such that for each 1 ≤ i ≤ n, the completion of A i is T, the generic formal fiber of A i is local with maximal ideal p i , and if I is a nonzero ideal of A i then A i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht T Q, then there is a chain of excellent local domains B 0 ⊆ B 1 ⊆ ··· ⊆ B h ⊆ T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B i ) = i, the completion of B i is isomorphic to T[[X 1, X 2,…, X i ]] where the X j 's are indeterminants, and the formal fiber of Q ∩ B i is local.

On the vanishing of Tor of the absolute integral closure

Let R be an excellent local domain of positive characteristic with residue field k and let R + be its absolute integral closure. If Tor R 1( R +, k) vanishes, then R is weakly F-regular. If R has at most an isolated singularity or has dimension at most two, then R is regular.

Separable integral extensions and plus closure

We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring.

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings R = K[[x, y, z]]/(x3 + y3 + Z3), where K is a field of characteristic p and p 74 3. We use a Z3-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring K[[x, y]]. We show that Frobenius closure is the same as tight closure in certain classes of ideals when p =2 mod 3. Since IF C IR+ n R C 1*, we conclude that IR+ n R = 1* for these ideals. Using injective modules over the ring R?, the union of all peth roots of elements of R, we reduce the question of whether IF = I* for 23-graded ideals to the case of 23-graded irreducible modules. We classify the irreducible m-primary Z3-graded ideals. We then show that jF 1* for most irreducible rn-primary Z3-graded ideals in K[[x, y, z]]/(x3+y3+z3), where K is a field of characteristic p and p _ 2 mod 3. Hence 1* = IRMf n R for these ideals. In this paper we discuss the conjecture that I* = IR+ n R, where R+ denotes the integral closure of a domain R of characteristic p in an algebraic closure of its fraction field and I* denotes the tight closure of I. The ring R+ is characterized by the property that it is a domain integral over R and every monic polynomial with coefficients in R? factors into monic linear factors. This characterization can be used to prove the following property of R+: If W is a multiplicatively closed set of R, then (W-lR)+ W-1R+. Aside from providing a much more concrete description of tight closure, proving that I* = IR+ n R would solve the localization problem for tight closure. It is known that I* -IR+ n R for parameter ideals [Sml] and for rings in which every ideal of the normalization is tightly closed. Also, for those ideals I of an excellent local domain R such that R/I has finite phantom projective dimension, it is known that I* IR+ n R [Ab]. However, the conjecture is open even for two-dimensional normal Gorenstein domains. In particular, the conjecture is open for the cubical cone K[[x, y, z]]/(x3 + y3 + z3), where K is a field of characteristic p and p 5L 3, and more generally for rings of the form K[[x, y, z]]/(F(x, y, z)) where F is a homogeneous cubic polynomial. We consider tight closure, plus closure and Frobenius closure in the rings R K[[x, y, z]]/(x3 + y3 + Z3), where K is a field of characteristic p and p 5L 3. In Section 1 we use a Z3-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings K[[x, y]]. In Section 2 we show that the Frobenius closure of an ideal I, denoted IF, is the same as the tight closure in certain classes of ideals when p _ 2 mod 3. Since IF C IR+ n R C I*, Received by the editors August 27, 1997. 1991 Mathematics Subject Classification. Primary 13A35, 13A02, 13HI10.

A dimension theorem for real spectra

In this note, the author proves, in the context of excellent rings, two results on chains of specializations in the real spectrum and some corollaries about real dimension. The first result (Theorem I) is the following: let α0 be a point of SpecrA of dimension d0 in the closure of a constructible set S of real dimension d at α0; then there exists a chain of strict specializations of exactly d points αd→⋯→α0 entirely contained in S, specializing to α0. This theorem is in fact a full generalization, in that context, of the curve selection lemma'' (which corresponds to the existence of α1 in the case d0=1). The second result (2.1) says that if A is an excellent local domain, any ordering of the fraction field inducing an ordering in the residue field extends to a formal branch'' (that is, an irreducible component of the completion Â). Both results are simultaneously proved by induction on the Krull dimension of A [see also the author, C. R. Acad. Sci. Paris Ser. I Math. 302 (1986), no. 2, 67–69].