The Zariski topology on sets of semistar operations

The Zariski topology on sets of semistar operations Dario Spirito (joint work with Carmelo Finocchiaro and Marco Fontana) Università di Roma Tre February 13th, 2015 Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 1 / 28

Indice 1 Definitions and examples 2 The Zariski topology 3 Spectral and eab operations Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 2 / 28

Definitions and examples Assumptions and notations All rings R will be commutative and unitary; we also assume that R is an integral domain with quotient field K. We will not assume that R is Noetherian. An overring is a ring comprised between R and K. Zar(R) is the set of valuation overrings of R. We denote by F(R) the set of R-submodules of K. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 3 / 28

Definitions and examples Definition Definition A map : F(R) F(R), I I is a semistar operation if, for every I, J F(R), it is extensive: I I ; it is order-preserving: if I J then I J ; it is idempotent: (I ) = I ; for every x K, x I = (xi ). If I = I, we say that I is -closed. The first three properties makes sense in every partially ordered set, giving the general concept of closure operation. Related kinds of closure operations: star and semiprime operations. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 4 / 28

Definitions and examples The order structure Definition Given two semistar operations 1, 2, we say that 1 2 if I 1 I 2 every R-submodule I. for The set SStar(R) of all semistar operations, with this order, is a complete lattice. The infimum of { α } α A is the operation such that I = I α. α A There is no general formula for the supremum of { α } α A ; however, I = I if and only if I = I α for every α. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 5 / 28

Definitions and examples Finite type Definition Let be a semistar operation. Then, define f as the map I I f := {F : F I, F is finitely generated}. f is always a semistar operation. f. I = I f if I is finitely generated. is a semistar operation of finite type (or a finite-type operation) if = f. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 6 / 28

Definitions and examples Why operations of finite type? Finite-type closures depends only on finitely-generated fractional ideals (in particular, on ideals inside the ring). For example, if 1 and 2 are of finite type, and we want to know whether 1 = 2, we only need to check it at ideals. More uniform behaviour (no jumps ). If is of finite type and L is R-flat, then (IL) = I L. Existence of -maximal ideals: say that I is a quasi- -ideal if I = I R. Then, I is contained in a maximal quasi- -ideal, which moreover is prime. We can control the supremum: I sup(a) = {I 1 n : 1,..., n A} Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 7 / 28

Definitions and examples Examples If T is an overring, the extension T : I IT is a semistar operation of finite type. If T = R we get the identity (denoted by d): I d = I for every submodule I. If T = K we get the trivial extension K : I K = K for every I (0). Given a set of overrings, := inf{ T : T }, i.e., I = T If = {R P : P Y } (with Y Spec(R)) is a set of localizations, then we define s Y as : I s Y = IR P, P Y IT. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 8 / 28

Definitions and examples Examples (2) The b-operation (or integral closure): if I is an ideal, I b is the set of x K such that for some a i I i. x n + a 1 x n 1 + + a n 1 x + a n = 0 Equivalently, b = Zar(R). Since we need only a finite amount of data (enough to generate the a i ) the b-operation is of finite type. The v-operation: I v = (R : (R : I )), where (F : G) := {x K : xg F }. More generally, it can be done by using any submodule L in place of R. This is usually not of finite type: if V is a valuation domain with non-finitely generated maximal ideal M, then M v = V but I v = I for every finitely generated ideal. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 9 / 28

Definitions and examples A Noetherian example Let R be a Noetherian domain such that dim(r) 2. Consider the set := {V Zar(R) : V is a DVR}. If I is an ideal of R, I = I b ; in particular, and b coincide over finitely generated ideals: hence, ( ) f = b f = b. If W is a non-discrete valuation overring (for example, if dim(w ) 2), then there is (at most) one V above W. Hence, W = V W, while W b = W : hence, b. Therefore, ( ) f, that is, is not of finite type. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 10 / 28

The Zariski topology The Zariski topology Definition We define the Zariski topology on SStar(R) to be the topology whose subbasic open set are those in the form V I := { SStar(R) : 1 I } as I ranges among the R-submodule of K. We get the same topology if we use the sets of the form V I,y := { SStar(R) : y I } since V I,y = V y 1 I. However, if we want reduce to generalize the topology (for example, to semiprime operations or to rings with zerodivisors), then we have to use the V I,y. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 11 / 28

The Zariski topology Basic properties SStar(R) is T 0. Link with the order structure: if O is open, O and, then O. In particular, the closure of { } is { : }. There is a unique closed point (d) and a unique generic point ( K ). SStar(R) is not T 1 nor T 2 (Haussdorff). [Unless R = K.] SStar(R) is compact. Limited functoriality: if A B is an extension of integral domains, there is a continuous map SStar(B) SStar(A), which is injective if B is an overring of A. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 12 / 28

The Zariski topology The Zariski topology on SStar f (R) SStar f (R) is dense in SStar(R). The map Ψ f :SStar(R) is a topological retraction. If U F := V F SStar f (R), then SStar f (R) f {U F : F is a finitely generated R-submodule of K} is a subbasis of the induced topology. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 13 / 28

The Zariski topology The Zariski topology on SStar f (R) (2) The map ι : Over(R) T is a topological embedding. SStar f (R) T The topology on Over(R) is generated by the sets BF := Over(R[F ]), as F varies among the finite subsets of K. If Λ SStar f (R) is compact, then inf Λ is of finite type. If Over(R) is compact, then is of finite type. If R is Noetherian, dim(r) 2, then {V Zar(R) : V is a DVR} is not compact. The converse does not hold. SStar f (R) is a spectral space. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 14 / 28

The Zariski topology Spectral spaces Definition A spectral space is a topological space that is homemorphic to the prime spectrum of a commutative ring R, endowed with the Zariski topology. Spec(R) (with the Zariski toplogy) is a spectral space. Zar(R) and Over(R) are spectral spaces. Every finite poset (endowed with the order topology) is a spectral space. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 15 / 28

The Zariski topology Topological characterization of spectral spaces Spectral spaces can be characterized topologically [Hochster]: X is a spectral space if and only if the following properties hold: X is compact and T 0 ; every irreducible closed subset of X has a generic point (i.e., it is the closure of a single point); there is a basis of compact subsets that is closed by finite intersections. If X = SStar(R) or X = SStar f (R), then the first and the third point are easy (we use as a base the family of finite intersections of the sets U F ). What about the second property? Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 16 / 28

The Zariski topology Ultrafilters Definition Let X be a set. A filter on X is a subset Y of P(X ) such that if A Y and A B then B Y; if A, B Y then A B Y; / Y. An ultrafilter is a maximal filter. Every filter is contained in an ultrafilter (consequence of Zorn s lemma). Ultrafilters can be used to prove Tychonoff s theorem. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 17 / 28

The Zariski topology Spectral spaces and ultrafilters Through Hochster s theorem, we can characterize spectral spaces in terms of ultrafilters: Proposizione ([Finocchiaro, 2014]) Let X be a T 0 space. Then, X is a spectral space if and only if there is a subbasis S of X such that, for every ultrafilter U on X, the set is nonempty. X S (U ) := {x X : [ B S, x B B U ]} The subbase S matters. Very non-constructive criterion. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 18 / 28

The Zariski topology SStar f (R) as a spectral space S := {U F : F is finitely generated}. The natural candidate is := sup{inf(u F ) : U F U }} inf(u F ) is of finite type since U F is compact. Since all is of finite type, we can control the supremum. We can show that X S (U ). Teorema SStar f (R) is a spectral space. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 19 / 28

The Zariski topology What kind of ring? Suppose SStar f (R) Spec(D). SStar f (R) has a minimum and a maximum; hence D is local and has a unique minimal prime (so we can take it as a domain). dim(d) Spec(R). Spec(R) < : write Spec(R) = {P 1,..., P n } in a way such that P i is a minimal element of {P i,..., P n }. Then, if i := {P 1,..., P k }, we have a descending chain K = s > s 1 > s 2 > > s n = d, and thus dim(d) n = Spec(R). Spec(R) = : we can do the previous reasoning with arbitrary large finite subsets. In particular, if Spec(R) is infinite, dim(d) = and, being local, D cannot not Noetherian. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 20 / 28

Spectral and eab operations More kinds of semistar operations Definition A semistar operation is: stable if (I J) = I J for every I, J F(R); spectral if I = IR P for some Spec(R). P By flatness, a spectral operation is stable. The converse is not true: V a valuation domain with non-finitely generated maximal ideal, = v. However, a stable operation of finite type is spectral. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 21 / 28

Spectral and eab operations More kinds of semistar operations (2) Definition A semistar operation is: eab if, whenever F, G, H are finitely generated, (FG) (FH) implies G H ; valutative if I = IV for some Zar(R). V Valutative operations are eab. Not all eab operations are valutative: V a valuation domain with non-finitely generated maximal ideal, = v. However, an eab operation of finite type is valutative. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 22 / 28

Spectral and eab operations Spectral and eab: differences The relations stable/spectral and eab/valuative are different: spectral is equivalent to stable and semifinite, but valuative is not equivalent to eab and semifinite [Fontana and Loper, 2009]. While valutative implies f valutative, spectral does not imply f spectral [Anderson and Cook, 2000]. The supremum of a family of finite-type spectral operations is spectral. Does the same holds for valutative operations? Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 23 / 28

Spectral and eab operations Finite type, spectral and eab: analogies Like Over(R) is embedded in SStar f (R), the set of localizations of R is embedded iton SStar f,sp (R), while Zar(R) is embedded into SStar f,eab (R). Like for Ψ f : SStar(R) SStar f (R), we can define retractions Ψ sp : SStar(R) SStar f,sp (R) and Ψ a : SStar(R) SStar f,eab (R). However, while Ψf ( ) and Ψ sp ( ), we have Ψ a ( ) f, and we can t in general compare Ψ a ( ) with. SStar f,sp (R) is a spectral space. The proof follows the same path of the proof for SStar f (R). Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 24 / 28

Spectral and eab operations Finite type, spectral and eab: analogies (2) If is of finite type and eab or spectral, then there is a ring R (X ) such that R[X ] R (X ) K(X ) and I = IR (X ) K for every I F(R). If Spec(R) or Zar(R), then s (respectively, ) is of finite type if and only if is compact. New proof of the fact that Zar(R) is compact. If, Λ Spec(R) are compact, then = Λ if and only if = Λ Y := {P Spec(R) : P Q for some Q Y } is the generization of Y. An analogous criterion holds for valutative operations, but in the other way: = Λ if and only if = Λ. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 25 / 28

Spectral and eab operations The Kronecker function ring Definition The Kronecker function ring of R is { } f Kr(R) := K(X ) : f, g R[X ], c(f ) c(g)b g where c(f ) (the content of f ) is the ideal of R generated by the coefficients of f. Kr(R) is always a Bézout domain. In particular, Spec(Kr(R)) Zar(Kr(R)). With the previous notation, Kr(R) = R b (X ) Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 26 / 28

Spectral and eab operations The Kronecker function ring (2) Teorema The map Φ : Zar(R) Zar(Kr(R)), V Kr(V ) is an homeomorphism. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 27 / 28

Spectral and eab operations The Kronecker function ring (2) Teorema The map Φ : Zar(R) Zar(Kr(R)), V Kr(V ) is an homeomorphism. Let be a valutative operation of finite type. = for a unique Zar(R) such that is compact and = ; Φ( ) Zar(Kr(R)) is compact and Φ( ) = Φ( ); Φ( ) corresponds to Y Spec(Kr(R)), which is compact and such that Y = Y ; Y generates s Y. SStar f,eab (R) SStar f,sp (Kr(R)) Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 27 / 28

Spectral and eab operations Bibliography Anderson, D. D. and Cook, S. J. (2000), Two star-operations and their induced lattices. Comm. Algebra, 28(5):2461 2475. Finocchiaro, C. A. (2014), Spectral spaces and ultrafilters. Comm. Algebra, 42(4):1496 1508. Finocchiaro, C. A. and Spirito, D. (2014), Some topological considerations on semistar operations. J. Algebra, 409:199 218. Fontana, M. and Loper, K. A. (2009), Cancellation properties in ideal systems: a classification of e.a.b. semistar operations. J. Pure Appl. Algebra, 213(11):2095 2103. Hochster, M. (1969), Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142:43 60. Dario Spirito (Univ. Roma Tre) Topology and semistar operations February 13th, 2015 28 / 28