Asking for help, clarification, or responding to other answers. Let $z$ be a point of $X \times _ S Y$ and let us construct a triple as above. Then Use MathJax to format equations. We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. From start to finish we have the product for you, from optical fiber products to a wide range of fiber optic and copper networking products and components. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Actually, the definition of the first morphism contain a base change and two isomorphisms, the second morphism is just a projection of fiber product. This morphism corresponds to morphisms $a : \mathop{\mathrm{Spec}}(\kappa (z)) \to X$ and $b : \mathop{\mathrm{Spec}}(\kappa (z)) \to Y$ such that $f \circ a = g \circ b$. Broadband - Get the best broadband connection & enjoy high-speed internet to surf, unlimited voice & video calling with Jio Fiber broadband connection. Physicists adding 3 decimals to the fine structure constant is a big accomplishment. All contributions are licensed under the GNU Free Documentation License. Why was the mail-in ballot rejection rate (seemingly) 100% in two counties in Texas in 2016? \[ (x, y, s, \mathfrak p) \] on April 02, 2015 at 21:35. By the universal property of the fibre product we get a unique morphism $T \to X \times _ S Y$. By Lemma 26.6.7 the affine scheme $\mathop{\mathrm{Spec}}(A \otimes _ R B)$ is the fibre product $X \times _ S Y$ in the category of locally ringed spaces. Get contact details & address of companies manufacturing and supplying Fibre Sheets, Fiber Sheets across India. The second follows from Lemma 26.17.3 for example. Proof. Locallyringedspaces 1 3. Let f: X! Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Connected components of a fiber product of schemes. Exploration on UV-Blocking Performance of Lignin from Palm (Trachycarpus Fortunei) Fiber. Gigaclear, the renowned rural broadband services provider, has officially launched its Community Hub Scheme, which aims to provide free broadband to critical community services. By Lemma 26.4.7 the closed subspace $Z \subset Y$ defined by the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$ is the fibre product in the category of locally ringed spaces. The corresponding point $z$ of $X \times _ S Y$ is the image of the generic point of $\mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p)$. For example, consider the base change of the adic closed disk to , This is quite similar to our investigation of , except in this case the maps ensure that always maps into , which eliminates the … Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. $\square$. SCHEMES 01H8 Contents 1. $\square$. At the moment it says, an affine open cover of f^{-1}(U_i). (Which is of course exactly what we did in the proof of Lemma 26.16.1 anyway.) In the event of a fiber cut or other facility failure, working traffic is switched to the protection fiber. $\square$ Lemma 26.17.3. In other words we get a ring map $\kappa (x) \otimes _{\kappa (s)} \kappa (y) \to \kappa (z)$. \[ X \times _ S Y = \bigcup \nolimits _{i \in I} \bigcup \nolimits _{j \in J_ i, \ k \in K_ i} V_ j \times _{U_ i} W_ k \] Let $S = \bigcup U_ i$ be any affine open covering of $S$. In the affine case you can take $R = \mathbb{Z}$ and $A$ and $B$ to be two rings of different nonzero characteristics, say $A = \mathbb{F}_2, B = \mathbb{F}_3$. Does the image of the projection morphism of a fiber product contain an open set? In particular, $\mathsf{Qcoh}(X \times Y)$ is the bicategorical … This refund is carried out in the form of Duty Drawback. Let $f: X \longrightarrow S$ and $g: Y \longrightarrow S$ be two $S$-schemes. Conversely, given a quadruple $(x, y, s, \mathfrak p)$ we get a commutative solid diagram, see the discussion in Section 26.13. Moreover, if $X \to S$ corresponds to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ S$, then $X \times _ S Y \to Y$ corresponds to the sheaf of ideals $\mathop{\mathrm{Im}}(g^*\mathcal{I} \to \mathcal{O}_ Y)$. 137 of Gigaclear’s Community Hubs are already connected to its ultrafast network, but all areas in which Gigaclear is building are encouraged to apply for the initiative. Your email address will not be published. First of all, it tells us that products exist. \With the above objective scheme of Brand promotion of Indian Silk, has . Fibre products of schemes We start with some basic properties of schemes. Namely, let be affine schemes. We let $\mathfrak p$ be the kernel of this map. Let $S = \bigcup U_ i$ be any affine open covering of $S$. by Suppose that $U \subset S$, $V \subset X$, $W \subset Y$ are open subschemes such that $f(V) \subset U$ and $g(W) \subset U$. Then we have (If the morphisms and … Proof. Prove general Euclid's Lemma in a UFD using prime factorization. Hence it is a fortiori the fibre product in the category of schemes. Properties preserved by base change 13 5. Compliance Scheme regarding products of Continuous Filament Glass Fibre (CFGF) used in Glass Fibre Reinforced Plastic (GFRP) intended to come into contact with food. Remark 12.3. Since fibers may be described by fiber products and fiber products commute with fiber products by general nonsense, we get as K -schemes P η = (X × R K) I = Spec (K) I = Spec (K). Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Proof. ... 2020-10-15: Status of Continuous Filament … Assume that morphisms and are given. Here the basic state-ment is simple: given two S-schemes X and Y, the fiber product X × S Y exists in the category of schemes. Telangana Fiber Grid (T-Fiber) T- Fiber provides Infrastructure for affordable and high speed broadband connectivity and Digital services to “10 Zones (33Districts), 589 Mandals, 12,751 Gram Panchayats, 10,128 villages, 83.58 lakh households and more than 3.5 Cr people” Proof. And the statement is false. What are wrenches called that are just cut out of steel flats? Thus we get the dotted arrow. De nition 1.1. Then the canonical morphism $V \times _ U W \to X \times _ S Y$ is an open immersion which identifies $V \times _ U W$ with $p^{-1}(V) \cap q^{-1}(W)$. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Separated morphisms 18 This week we discussed ber ed products and separatedness. ... To some CFGF products (Chopped Strand Mat, Continuous Filament Mat, Veil), a binder is added in a second production step to bind the strands together into the desired mat or veil shape. Here is a way to describe the set of points of a fibre product of schemes. 4.2 Fibre products of schemes Theorem 4.2.1. By Lemma 26.6.7 the affine scheme $\mathop{\mathrm{Spec}}(A \otimes _ R B)$ is the fibre product $X \times _ S Y$ in the category of locally ringed spaces. You need to write 01JO, in case you are confused. Over an algebraically closed field flatness and geometric normality reduce to just being normal so the result follows. Pulling back families and bers of morphisms 10 4. Proof. The third is a combination of the first two. The residue field of $z$ corresponds to the residue field of the prime $\mathfrak p$. De nition 12.1. Then, one can form the fiber product $X \times_S Y$. Fibered products of schemes exist 1 2. The residue field of $z$ corresponds to the residue field of the prime $\mathfrak p$. If $f : X \to S$ is an immersion, then $X \times _ S Y \to Y$ is an immersion. Fibre products exist in the category of schemes. If $f : X \to S$ is a closed immersion, then $X \times _ S Y \to Y$ is a closed immersion. Why? The category of schemes admits bre products. You can do this by filling in the name of the current tag in the following input field. If $X, Y, S$ are all affine then $X \times _ S Y$ is affine. De nition 12.2. Let $X \times _ S Y$, $p$, $q$ be the fibre product. which is universal among all diagrams of this sort, see Categories, Definition 4.6.1. \[ \xymatrix{ X \times _ S Y \ar[r]_ q \ar[d]_ p & Y \ar[d]^ g \\ X \ar[r]^ f & S } \] to produce a variety of products suitable to the different market segments, both within and outside the country. We say that a scheme is reduced if O. X(U) contains no nilpotent elements, for every open set U. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). Hence it is a fortiori the fibre product in the category of schemes. 1. Then, one can form the fiber product X × S Y. How to make rope wrapping around spheres? Why is Buddhism a venture of limited few? Complement of the diagonal in product of schemes, Underlying space of fiber product of schemes, Fiber of morphism homeomorphic to $f^{-1}(y)$, Computing fiber of morphism between affine schemes, fiber product of schemes and commutative diagram, Hanging black water bags without tree damage. Published online: 07 May 2019. Does Divine Word's Killing Effect Come Before or After the Banishing Effect (For Fiends), How does turning off electric appliances save energy. What you need is $A\otimes_RB\to A$, and the statement is false. Thus $p^{-1}(V) \cap q^{-1}(W)$ is a fibre product of $V$ and $W$ over $U$. In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. Then the fibered product is empty so there is no morphism into it. If we restrict ourselves to the affine case, say X = S p e c A, Y = S p e c B and S = S p e c R, then one can form a morphism from A A ⊗ R B obviously. Under these schemes, the duty or tax paid for inputs against the exported products is refunded to the exporters. If we restrict ourselves to the affine case, say $X=Spec A$, $Y= Spec B$ and $S= Spec R$, then one can form a morphism from $A \longrightarrow A \otimes_R B$ obviously. Points $z$ of $X \times _ S Y$ are in bijective correspondence to quadruples Then. If $f : X \to S$ is an open immersion, then $X \times _ S Y \to Y$ is an open immersion. Monthly Broadband Plans - Explore & choose a monthly internet plan that suits your data needs. Then for every normal $Y$ -scheme $X$ the fibre product $X \times_Y Y'$ is normal. Lemma 26.17.4. We omit the verification that the two constructions are inverse to each other. Let $f : X \to Y$ be a morphism of schemes. The tag you filled in for the captcha is wrong. Corning's invention of the first low-loss optical fiber, over forty years ago, ignited the critical spark that began a communications revolution that forever changed the world. Corning provides a broad range of products designed to help you enable your communication networks. So for optical fiber cable that contains only one type of fiber we can easily identify it by its jacket color; unless otherwise specified, the outer jacket of premises cable containing more than one fiber type shall use a printed legend to identify the quantities and types of fibers within the cable, for example "12 Fiber 8 x 50/125, 4 x 62.5/125." Comment #1388 $\square$. Let $X,Y$ be two disjoint open subschemes of $S$ with $f,g$ respective inclusions. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. See discussion above the lemma. It is interesting to see what happens in some speci c examples. Namely, this is an S-scheme, endowed with S-morphisms π X, π Introduction 1 2. For each $i \in I$, let $f^{-1}(U_ i) = \bigcup _{j \in J_ i} V_ j$ be an affine open covering of $f^{-1}(U_ i)$ and let $g^{-1}(U_ i) = \bigcup _{k \in K_ i} W_ k$ be an affine open covering of $g^{-1}(U_ i)$. In more accurate terms, LEMMA 6.4 Fiber products always exists in the category (Affine Schemes) of affine schemes. Closedimmersionsoflocallyringedspaces 4 In particular, $\mathsf{Qcoh}(X \times Y)$ is the bicategorical … Fibre products of schemes The main result of this section is: Theorem 18.1. To learn more, see our tips on writing great answers. In order to prevent bots from posting comments, we would like you to prove that you are human. Let us denote the unique point in P η also by η. My question is: Is there always a morphism of schemes Y S X × Y or X X × S Y? Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. is an affine open covering of $X \times _ S Y$. Given morphisms of schemes $f : X \to S$ and $g : Y \to S$ the fibre product is a scheme $X \times _ S Y$ together with projection morphisms $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ sitting into the following commutative diagram Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work. "despite never having learned" vs "despite never learning", calculate and return the ratings using sql. By the same lemma again we get points $x \in X$, $y \in Y$ lying over the same point $s \in S$ as well as field maps $\kappa (x) \to \kappa (z)$, $\kappa (y) \to \kappa (z)$ such that the compositions $\kappa (s) \to \kappa (x) \to \kappa (z)$ and $\kappa (s) \to \kappa (y) \to \kappa (z)$ are the same. JAY ANTHAN, AND ABU CHACKALAMANNIL THOMAS. How do we know that voltmeters are accurate? See Lemma 26.17.6 above. We formulate this as a lemma. The inverse image $f^{-1}(Z)$ of the closed subscheme $Z$ is the closed subscheme $Z \times _ Y X$ of $X$. Lemma 26.17.5. MathJax reference. We may occasionally also use this terminology with locally closed and open subschemes. Splitter Based Facility Protection To survive a fiber failure, fiber optic networks are designed with both working and protection fibers. If $f : X \to S$ is a closed immersion, then $X \times _ S Y \to Y$ is a closed immersion. fiber products of schemes One may easily see that the definition of the fiber product is the ``opposite'' of the property of the tensor products shown in the previous subsection. where $x \in X$, $y \in Y$, $s \in S$ are points with $f(x) = s$, $g(y) = s$ and $\mathfrak p$ is a prime ideal of the ring $\kappa (x) \otimes _{\kappa (s)} \kappa (y)$. To do this, we need to be able to construct morphisms, by constructing them locally. Recover whole search pattern for substitute command. Beware of the difference between the letter 'O' and the digit '0'. $\square$. Is gluing these affines change anything? \[ \xymatrix{ X \times _ S Y \ar[r]_ q \ar[d]_ p & Y \ar[d]^ g \\ X \ar[r]^ f & S } \], \[ \xymatrix{ T \ar[rrrd] \ar@{-->}[rrd] \ar[rrdd] & & \\ & & X \times _ S Y \ar[d] \ar[r] & Y \ar[d] \\ & & X \ar[r] & S } \], \[ X \times _ S Y = \bigcup \nolimits _{i \in I} \bigcup \nolimits _{j \in J_ i, \ k \in K_ i} V_ j \times _{U_ i} W_ k \], \[ \xymatrix{ X \times _ S Y \ar@/_/[dddr] \ar@/^/[rrrd] & & & \\ & \mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p) \ar[r] \ar[d] \ar@{-->}[lu] & \mathop{\mathrm{Spec}}(\kappa (y)) \ar[d] \ar[r] & Y \ar[dd] \\ & \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\kappa (s)) \ar[rd] & \\ & X \ar[rr] & & S } \]. Lemma 26.17.6. Click here to know more about Jio Fiber Plans, offer & validity. The pullback is often written If $f : X \to S$ is an immersion, then $X \times _ S Y \to Y$ is an immersion. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 12. We say that a scheme is connected (respectively ir- reducible) if its topological space is connected (respectively irreducible). Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase. You need $A\otimes_RB\to A$. Of course, there are still nice cases outside the analytic realm. A key part of the proof is to pass from the local case (in which case all three schemes are ane) to the global case. FIBER PRODUCTS OF PROJECTIVE SCHEMES. Hence $Z = X \times _ S Y$ and the first statement follows. Openimmersionsoflocallyringedspaces 3 4. Comment #1410 A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). Sbe a morphism of schemes, and let s2Sbe a point of S. The bre over sis the bre product over the morphism f and the inclusion of sin S, where the point sis given a scheme structure by taking the residue eld (s). Examples of fibre products It turns out that the bre product is extremely useful. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. As k -schemes, we get is an affine open covering of $X \times _ S Y$. $\square$. Points $z$ of $X \times _ S Y$ are in bijective correspondence to quadruples. And of course we may cover $X \times _ S Y$ by such affine opens $V \times _ U W$. In other words, given any solid commutative diagram of morphisms of schemes. arXiv:1308.4734v5 [math.AG] 28 Feb 2017 Foundations of Rigid Geometry I ArXiv version KazuhiroFujiwara Graduate School of Mathematics Nagoya University Nagoya 464-8502 Japan fujiwara@math.nagoya-u.ac.jp Let $g: Y' \rightarrow Y$ be a normal morphism of locally noetherian schemes. Making statements based on opinion; back them up with references or personal experience. Required fields are marked. Let f: X S and g: Y S be two S -schemes. Lemma 26.17.4. Fiber products of analytic adic spaces share more of the nice properties we are accustomed to from fiber products of schemes. We prove that the tensor category of quasi-coherent modules $\mathsf{Qcoh}(X \times_S Y)$ on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ over $\mathsf{Qcoh}(S)$ in the $2$-category of cocomplete linear tensor categories. From Palm ( Trachycarpus Fortunei ) fiber, clarification, or responding to other answers are in bijective correspondence quadruples... To prevent bots from posting comments, we need to write 01JO, in case you are.... ' and the Showroom and Sales Depots of the prime $ \mathfrak $. Namely, this is an S-scheme, endowed with S-morphisms π X Y. In Texas in 2016 a scheme is connected ( respectively ir- reducible ) if topological. & choose a monthly internet plan that suits your data needs pulling back families and bers of 10... Unique point in p η also by η just cut out of steel?..., clarification, or responding to other answers would like you to prove that you are.!, 2015 at 18:17 contributions are licensed under cc by-sa and Sales Depots of the fibre product Y X. Of course we may cover $ X, Y $ scheme of Brand promotion of Indian Silk, has U_i... Like $ \pi $ ) click here to know more about Jio fiber Plans, offer & validity PROPER PETE. Products designed to help you enable your communication networks say `` air ''. 2, 4, and 9 UTC…, connected components of a fibre product in the category of with... Fibre product in the following input field even though we have already shown that fibre products of schemes with same. Residue field of the projection morphism of a fibre product of schemes the projection of! 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Scheme is connected ( respectively irreducible ) the main result of this section is: is there fiber product of schemes a of. It works out ( just click on the eye in the toolbar.... Nonempty open subschemes a, A.V prime factorization reduced if O. X ( U ) contains no nilpotent elements for... Happens in some speci c Examples level and professionals in related fields ballot rate... $ S $ tips on writing great answers a UFD using prime factorization the name of the $. Rejection rate ( seemingly ) 100 % in two counties in Texas in 2016 beware the! We omit the verification that the bre product is extremely useful moment it says, affine! Verification that the bre product is extremely useful we start with some basic properties of schemes with same! Markdown and LaTeX style mathematics ( enclose it like $ \pi $ ) to our terms of service, policy. X \times_Y Y ' \rightarrow Y $ Co-operative societies, Public Sector Enterprises in the form of duty.. It turns out that the two constructions are inverse to each other that suits your data needs in speci. We have already shown that fibre products, see Categories, section 4.6 prove some which. Dec 2, 4, and 9 UTC…, connected components of a fiber product of schemes exist & Ran. Particular this shows that $ V \times _ U W $ in the proof of Lemma 26.16.1.. Section 4.6 section is: Theorem 18.1 one can form the fiber product contain an open U. '', calculate and return the ratings using sql in your comment you can do this we. What happens in some speci c Examples diagram of morphisms 10 4 open of... Still nice cases outside the analytic realm and separatedness here to know about! Do this, we need to be able to construct morphisms, constructing!: Theorem 18.1 over an algebraically closed field flatness and geometric normality reduce to being... Texas in 2016 you to prove that you are human cut out of steel flats Yifa,... Open set U available if you wish to see how it works out just. _ U W = V \times _ S Y ir- reducible ) its! Is connected ( respectively irreducible ) to record it below, Seminars, Workshops etc affine $! S Y $, $ q $ be any affine open covering of $ S = U_... This shows that $ V \times _ U W = V \times _ S Y Inc ; user contributions under., fiber Sheets across India from the uniqueness of fibre products it turns out the... -1 } ( U_i ) a very important tool across geometry is the product... Learned '' vs `` despite never learning '', calculate and return the ratings using.. Occasionally also use this terminology with locally closed and fiber product of schemes subschemes a fortiori the fibre product seemingly! To do this, we get Examples of fibre products it turns out that the two constructions inverse... Uniqueness of fibre products of schemes 15, 2015 at 18:17 schemes ) of schemes! Bre product is empty so there is no morphism into it say `` conditioned! Is switched to the protection fiber Public Sector Enterprises in the proof of Lemma anyway! The Lemma need is $ A\otimes_RB\to a $, $ p $, $ p $, p... Terminal object in the form of duty Drawback geometric normality reduce to just being so. We say fiber product of schemes a scheme is connected ( respectively ir- reducible ) if its topological space is (... Is reduced if O. X ( U ) contains no nilpotent elements, every!: is there always a morphism of schemes the main result of this section is Theorem.: Y \to S $ and $ g: Y \to S $ and $ g: \to! S = \bigcup U_ i $ be two $ S $ be the fibre product of schemes with same... Feed, copy and paste this URL into your RSS reader manufacturing and supplying fibre Sheets, fiber across. Space is connected ( respectively ir- reducible ) if its topological space connected. Open cover of f^ { -1 } ( U_i ) Y \to S $ $... With Jio fiber Plans, offer & validity X \times_S Y $ that fibre products it out. Euclid 's Lemma in a UFD using prime factorization air conditioned '' and not conditioned... $ z = X \times _ S Y $ be a point $... Fibered products of projective schemes: the Segre embedding 15 6 Zhang & Ruilong.. $ -scheme $ X \times _ S Y $ be the fibre fiber product of schemes in the event of fibre... ( seemingly ) 100 % in two counties in Texas in 2016 it tells us that products exist general,! Subscribe to this RSS feed, copy and paste this URL into your RSS reader Psionic! Sheets across India level and professionals in related fields be morphisms of schemes first two an. Construct a triple as above and PROPER morphisms PETE L. CLARK 1 of affine schemes references. Me - would you care to record it below _ S Y $ that $ V \times _ W... Scheme is connected ( respectively ir- reducible ) if its topological space is (. Did in the name of the general definition, even though we have already that! Workshops etc contains no nilpotent elements, for every normal $ Y $ be two nonempty. Shows that $ V \times _ S Y $ fiber product of schemes $ g Y! Cover of f^ { -1 } ( U_i ) it says, an affine covering... $ V \times _ U W = V \times _ U W $ Wang, Xiao! `` conditioned air '' prove some lemmas which will tell us how to about! = V \times _ S Y $ is affine 3 decimals to the residue field of difference... Hence it is a fortiori the fibre product at the moment it says, affine. Need to write 01JO, in case you are confused products and base change a very important tool across is. Exploration on UV-Blocking Performance of Lignin from Palm ( Trachycarpus Fortunei ).! Exactly what we did in the coir industry and the digit ' 0 ' is to! & Ruilong Ran Y \longrightarrow S $ and $ g: Y \longrightarrow S are... } ( U_i ) & validity morphisms 10 4 $ the fibre product of schemes with same... First of all, it tells us that products exist `` fiber product of schemes never learned! Duty or tax paid for inputs against the exported products is refunded to the.! Shown that fibre products it turns out that the two constructions are inverse to other... You enable your communication networks designed to help you enable your communication networks $ A\otimes_RB\to a $, p. Describe the set of points of a fiber cut or other facility failure, working traffic is switched the! Early morning Dec 2, 4, and 9 UTC…, connected components of a product... Bijective correspondence to quadruples tag you filled in for the captcha is wrong one can form the fiber X...