Section 38.33 (0F3T): Nagata compactification—The Stacks project (original) (raw)

38.33 Nagata compactification

In this section we prove the theorem announced in Section 38.32.

Lemma 38.33.1. Let XtoSX \to SXtoS be a morphism of schemes. If X=UcupVX = U \cup VX=UcupV is an open cover such that UtoSU \to SUtoS and VtoSV \to SVtoS are separated and UcapVtoUtimesSVU \cap V \to U \times _ S VUcapVtoUtimesSV is closed, then XtoSX \to SXtoS is separated.

Proof. Omitted. Hint: check that Delta:XtoXtimesSX\Delta : X \to X \times _ S XDelta:XtoXtimesSX is closed by using the open covering of XtimesSXX \times _ S XXtimesSX given by UtimesSUU \times _ S UUtimesSU, UtimesSVU \times _ S VUtimesSV, VtimesSUV \times _ S UVtimesSU, and VtimesSVV \times _ S VVtimesSV. square\squaresquare

Lemma 38.33.2. Let XXX be a quasi-compact and quasi-separated scheme. Let UsubsetXU \subset XUsubsetX be a quasi-compact open.

1. If Z1,Z2subsetXZ_1, Z_2 \subset XZ1,Z2subsetX are closed subschemes of finite presentation such that Z1capZ2capU=emptysetZ_1 \cap Z_2 \cap U = \emptyset Z_1capZ2capU=emptyset, then there exists a UUU-admissible blowing up X′toXX' \to XX′toX such that the strict transforms of Z1Z_1Z1 and Z2Z_2Z_2 are disjoint.
2. If T1,T2subsetUT_1, T_2 \subset UT_1,T2subsetU are disjoint constructible closed subsets, then there is a UUU-admissible blowing up X′toXX' \to XX′toX such that the closures of T1T_1T1 and T2T_2T_2 are disjoint.

Proof. Proof of (1). The assumption that ZitoXZ_ i \to XZitoX is of finite presentation signifies that the quasi-coherent ideal sheaf mathcalIi\mathcal{I}_ imathcalIi of ZiZ_ iZi is of finite type, see Morphisms, Lemma 29.21.7. Denote ZsubsetXZ \subset XZsubsetX the closed subscheme cut out by the product mathcalI1mathcalI2\mathcal{I}_1 \mathcal{I}_2mathcalI1mathcalI2. Observe that ZcapUZ \cap UZcapU is the disjoint union of Z1capUZ_1 \cap UZ1capU and Z2capUZ_2 \cap UZ2capU. By Divisors, Lemma 31.34.5 there is a UcapZU \cap ZUcapZ-admissible blowup Z′toZZ' \to ZZ′toZ such that the strict transforms of Z1Z_1Z1 and Z2Z_2Z2 are disjoint. Denote YsubsetZY \subset ZYsubsetZ the center of this blowing up. Then YtoXY \to XYtoX is a closed immersion of finite presentation as the composition of YtoZY \to ZYtoZ and ZtoXZ \to XZtoX (Divisors, Definition 31.34.1 and Morphisms, Lemma 29.21.3). Thus the blowing up X′toXX' \to XX′toX of YYY is a UUU-admissible blowing up. By general properties of strict transforms, the strict transform of Z1,Z2Z_1, Z_2Z1,Z2 with respect to X′toXX' \to XX′toX is the same as the strict transform of Z1,Z2Z_1, Z_2Z_1,Z_2 with respect to Z′toZZ' \to ZZ′toZ, see Divisors, Lemma 31.33.2. Thus (1) is proved.

Proof of (2). By Properties, Lemma 28.24.1 there exists a finite type quasi-coherent sheaf of ideals mathcalJisubsetmathcalOU\mathcal{J}_ i \subset \mathcal{O}_ UmathcalJisubsetmathcalOU such that Ti=V(mathcalJi)T_ i = V(\mathcal{J}_ i)Ti=V(mathcalJi) (set theoretically). By Properties, Lemma 28.22.2 there exists a finite type quasi-coherent sheaf of ideals mathcalIisubsetmathcalOX\mathcal{I}_ i \subset \mathcal{O}_ XmathcalIisubsetmathcalOX whose restriction to UUU is mathcalJi\mathcal{J}_ imathcalJi. Apply the result of part (1) to the closed subschemes Zi=V(mathcalIi)Z_ i = V(\mathcal{I}_ i)Zi=V(mathcalIi) to conclude. square\squaresquare

Lemma 38.33.3. Let f:XtoYf : X \to Yf:XtoY be a proper morphism of quasi-compact and quasi-separated schemes. Let VsubsetYV \subset YVsubsetY be a quasi-compact open and U=f−1(V)U = f^{-1}(V)U=f−1(V). Let TsubsetVT \subset VTsubsetV be a closed subset such that f∣U:UtoVf|_ U : U \to Vf∣U:UtoV is an isomorphism over an open neighbourhood of TTT in VVV. Then there exists a VVV-admissible blowing up Y′toYY' \to YY′toY such that the strict transform f′:X′toY′f' : X' \to Y'f′:X′toY′ of fff is an isomorphism over an open neighbourhood of the closure of TTT in Y′Y'Y′.

Proof. Let T′subsetVT' \subset VT′subsetV be the complement of the maximal open over which f∣Uf|_ Uf∣U is an isomorphism. Then T′,TT', TT′,T are closed in VVV and TcapT′=emptysetT \cap T' = \emptyset TcapT′=emptyset. Since VVV is a spectral topological space, we can find constructible closed subsets Tc,T′cT_ c, T'_ cTc,T′c with TsubsetTcT \subset T_ cTsubsetTc, T′subsetT′cT' \subset T'_ cT′subsetT′c such that TccapT′c=emptysetT_ c \cap T'_ c = \emptyset TccapT′c=emptyset (choose a quasi-compact open WWW of VVV containing T′T'T′ not meeting TTT and set Tc=VsetminusWT_ c = V \setminus WTc=VsetminusW, then choose a quasi-compact open W′W'W′ of VVV containing TcT_ cTc not meeting T′T'T′ and set T′c=VsetminusW′T'_ c = V \setminus W'T′c=VsetminusW′). By Lemma 38.33.2 we may, after replacing YYY by a VVV-admissible blowing up, assume that TcT_ cTc and T′cT'_ cT′c have disjoint closures in YYY. Set Y0=YsetminusoverlineT′cY_0 = Y \setminus \overline{T}'_ cY0=YsetminusoverlineT′c, V0=VsetminusT′cV_0 = V \setminus T'_ cV0=VsetminusT′c, U0=UtimesVV0U_0 = U \times _ V V_0U0=UtimesVV0, and X0=XtimesYY0X_0 = X \times _ Y Y_0X0=XtimesYY0. Since U0toV0U_0 \to V_0U0toV0 is an isomorphism, we can find a V0V_0V0-admissible blowing up Y′0toY0Y'_0 \to Y_0Y′0toY0 such that the strict transform X′0X'_0X′0 of X0X_0X0 maps isomorphically to Y′0Y'_0Y′0, see Lemma 38.31.3. By Divisors, Lemma 31.34.3 there exists a VVV-admissible blow up Y′toYY' \to YY′toY whose restriction to Y0Y_0Y0 is Y′0toY0Y'_0 \to Y_0Y′_0toY0. If f′:X′toY′f' : X' \to Y'f′:X′toY′ denotes the strict transform of fff, then we see what we want is true because f′f'f′ restricts to an isomorphism over Y′0Y'_0Y′_0. square\squaresquare

Lemma 38.33.4. Let SSS be a quasi-compact and quasi-separated scheme. Let UtoX_1U \to X_1UtoX1 and UtoX2U \to X_2UtoX2 be open immersions of schemes over SSS and assume UUU, X1X_1X1, X2X_2X_2 of finite type and separated over SSS. Then there exists a commutative diagram

\[ \xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 } \]

of schemes over SSS where Xi′toXiX_ i' \to X_ iXi′toXi is a UUU-admissible blowup, Xi′toXX_ i' \to XXi′toX is an open immersion, and XXX is separated and finite type over SSS.

Proof. Throughout the proof all schemes will be separated of finite type over SSS. This in particular implies these schemes are quasi-compact and quasi-separated and the morphisms between them are quasi-compact and separated. See Schemes, Sections 26.19 and 26.21. We will use that if UtoWU \to WUtoW is an immersion of such schemes over SSS, then the scheme theoretic image ZZZ of UUU in WWW is a closed subscheme of WWW and UtoZU \to ZUtoZ is an open immersion, UsubsetZU \subset ZUsubsetZ is scheme theoretically dense, and UsubsetZU \subset ZUsubsetZ is dense topologically. See Morphisms, Lemma 29.7.7.

Let X12subsetX1timesSX2X_{12} \subset X_1 \times _ S X_2X12subsetX1timesSX2 be the scheme theoretic image of UtoX1timesSX2U \to X_1 \times _ S X_2UtoX1timesSX2. The projections pi:X12toXip_ i : X_{12} \to X_ ipi:X12toXi induce isomorphisms pi−1(U)toUp_ i^{-1}(U) \to Upi−1(U)toU by Morphisms, Lemma 29.6.8. Choose a UUU-admissible blowup XiitoXiX_ i^ i \to X_ iXiitoXi such that the strict transform X12iX_{12}^ iX12i of X12X_{12}X12 is isomorphic to an open subscheme of XiiX_ i^ iXii, see Lemma 38.31.3. Let mathcalIisubsetmathcalOXi\mathcal{I}_ i \subset \mathcal{O}_{X_ i}mathcalIisubsetmathcalOXi be the corresponding finite type quasi-coherent sheaf of ideals. Recall that X12itoX12X_{12}^ i \to X_{12}X12itoX12 is the blowup in pi−1mathcalIimathcalOX12p_ i^{-1}\mathcal{I}_ i \mathcal{O}_{X_{12}}pi−1mathcalIimathcalOX12, see Divisors, Lemma 31.33.2. Let X12′X_{12}'X12′ be the blowup of X12X_{12}X12 in p1−1mathcalI1p2−1mathcalI2mathcalOX12p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}p1−1mathcalI1p2−1mathcalI2mathcalOX12, see Divisors, Lemma 31.32.12 for what this entails. We obtain in particular a commutative diagram

\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_{12}^2 \ar[d] \\ X_{12}^1 \ar[r] & X_{12} } \]

where all the morphisms are UUU-admissible blowing ups. Since X12isubsetXiiX_{12}^ i \subset X_ i^ iX12isubsetXii is an open we may choose a UUU-admissible blowup Xi′toXiiX_ i' \to X_ i^ iXi′toXii restricting to X12′toX12iX_{12}' \to X_{12}^ iX12′toX12i, see Divisors, Lemma 31.34.3. Then X12′subsetXi′X_{12}' \subset X_ i'X12′subsetXi′ is an open subscheme and the diagram

\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_ i' \ar[d] \\ X_{12}^ i \ar[r] & X_ i^ i } \]

is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that X′12toX1′timesSX2′X'_{12} \to X_1' \times _ S X_2'X′12toX1′timesSX2′ is an immersion and proper (use that X′12toX12X'_{12} \to X_{12}X′12toX12 is proper and X12toX1timesSX2X_{12} \to X_1 \times _ S X_2X12toX1timesSX2 is closed and X1′timesSX2′toX1timesSX2X_1' \times _ S X_2' \to X_1 \times _ S X_2X1′timesSX2′toX1timesSX2 is separated and apply Morphisms, Lemma 29.41.7). Thus X′12toX1′timesSX2′X'_{12} \to X_1' \times _ S X_2'X′12toX1′timesSX2′ is a closed immersion. It follows that if we define XXX by glueing X1′X_1'X1′ and X2′X_2'X2′ along the common open subscheme X12′X_{12}'X12′, then XtoSX \to SXtoS is of finite type and separated (Lemma 38.33.1). As compositions of UUU-admissible blowups are UUU-admissible blowups (Divisors, Lemma 31.34.2) the lemma is proved. square\squaresquare

Lemma 38.33.5. Let XtoSX \to SXtoS and YtoSY \to SYtoS be morphisms of schemes. Let UsubsetXU \subset XUsubsetX be an open subscheme. Let VtoXtimesSYV \to X \times _ S YVtoXtimesSY be a quasi-compact morphism whose composition with the first projection maps into UUU. Let ZsubsetXtimesSYZ \subset X \times _ S YZsubsetXtimesSY be the scheme theoretic image of VtoXtimesSYV \to X \times _ S YVtoXtimesSY. Let X′toXX' \to XX′toX be a UUU-admissible blowup. Then the scheme theoretic image of VtoX′timesSYV \to X' \times _ S YVtoX′timesSY is the strict transform of ZZZ with respect to the blowing up.

Proof. Denote Z′toZZ' \to ZZ′toZ the strict transform. The morphism Z′toX′Z' \to X'Z′toX′ induces a morphism Z′toX′timesSYZ' \to X' \times _ S YZ′toX′timesSY which is a closed immersion (as Z′Z'Z′ is a closed subscheme of X′timesXZX' \times _ X ZX′timesXZ by definition). Thus to finish the proof it suffices to show that the scheme theoretic image Z′′Z''Z′′ of VtoZ′V \to Z'VtoZ′ is Z′Z'Z′. Observe that Z′′subsetZ′Z'' \subset Z'Z′′subsetZ′ is a closed subscheme such that VtoZ′V \to Z'VtoZ′ factors through Z′′Z''Z′′. Since both VtoXtimesSYV \to X \times _ S YVtoXtimesSY and VtoX′timesSYV \to X' \times _ S YVtoX′timesSY are quasi-compact (for the latter this follows from Schemes, Lemma 26.21.14 and the fact that X′timesSYtoXtimesSYX' \times _ S Y \to X \times _ S YX′timesSYtoXtimesSY is separated as a base change of a proper morphism), by Morphisms, Lemma 29.6.3 we see that Zcap(UtimesSY)=Z′′cap(UtimesSY)Z \cap (U \times _ S Y) = Z'' \cap (U \times _ S Y)Zcap(UtimesSY)=Z′′cap(UtimesSY). Thus the inclusion morphism Z′′toZ′Z'' \to Z'Z′′toZ′ is an isomorphism away from the exceptional divisor EEE of Z′toZZ' \to ZZ′toZ. However, the structure sheaf of Z′Z'Z′ does not have any nonzero sections supported on EEE (by definition of strict transforms) and we conclude that the surjection mathcalOZ′tomathcalOZ′′\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}mathcalOZ′tomathcalOZ′′ must be an isomorphism. square\squaresquare

Lemma 38.33.6. Let SSS be a quasi-compact and quasi-separated scheme. Let UUU be a scheme of finite type and separated over SSS. Let VsubsetUV \subset UVsubsetU be a quasi-compact open. If VVV has a compactification VsubsetYV \subset YVsubsetY over SSS, then there exists a VVV-admissible blowing up Y′toYY' \to YY′toY and an open VsubsetV′subsetY′V \subset V' \subset Y'VsubsetV′subsetY′ such that VtoUV \to UVtoU extends to a proper morphism V′toUV' \to UV′toU.

Proof. Consider the scheme theoretic image ZsubsetYtimesSUZ \subset Y \times _ S UZsubsetYtimesSU of the “diagonal” morphism VtoYtimesSUV \to Y \times _ S UVtoYtimesSU. If we replace YYY by a VVV-admissible blowing up, then ZZZ is replaced by the strict transform with respect to this blowing up, see Lemma 38.33.5. Hence by Lemma 38.31.3 we may assume ZtoYZ \to YZtoY is an open immersion. If V′subsetYV' \subset YV′subsetY denotes the image, then we see that the induced morphism V′toUV' \to UV′toU is proper because the projection YtimesSUtoUY \times _ S U \to UYtimesSUtoU is proper and V′congZV' \cong ZV′congZ is a closed subscheme of YtimesSUY \times _ S UYtimesSU. square\squaresquare

The following lemma is formulated in the Noetherian case only. The version for quasi-compact and quasi-separated schemes is true as well, but will be trivially implied by the main theorem in this section.

Lemma 38.33.7. Let SSS be a Noetherian scheme. Let UUU be a scheme of finite type and separated over SSS. Let U=U1cupU2U = U_1 \cup U_2U=U1cupU2 be opens such that U1U_1U1 and U2U_2U2 have compactifications over SSS and such that U1capU2U_1 \cap U_2U_1capU_2 is dense in UUU. Then UUU has a compactification over SSS.

Proof. Choose a compactification UisubsetXiU_ i \subset X_ iUisubsetXi for i=1,2i = 1, 2i=1,2. We may assume UiU_ iUi is scheme theoretically dense in XiX_ iXi. We may assume there is an open VisubsetXiV_ i \subset X_ iVisubsetXi and a proper morphism psii:VitoU\psi _ i : V_ i \to Upsii:VitoU extending textid:UitoUi\text{id} : U_ i \to U_ itextid:UitoUi, see Lemma 38.33.6. Picture

\[ \xymatrix{ U_ i \ar[r] \ar[d] & V_ i \ar[r] \ar[dl]^{\psi _ i} & X_ i \\ U } \]

If i,j=1,2\{ i, j\} = \{ 1, 2\} i,j=1,2 denote Zi=UsetminusUj=Uisetminus(U1capU2)Z_ i = U \setminus U_ j = U_ i \setminus (U_1 \cap U_2)Zi=UsetminusUj=Uisetminus(U1capU2) and Zj=UsetminusUi=Ujsetminus(U1capU2)Z_ j = U \setminus U_ i = U_ j \setminus (U_1 \cap U_2)Zj=UsetminusUi=Ujsetminus(U_1capU_2). Thus we have

\[ U = U_1 \amalg Z_2 = Z_1 \amalg U_2 = Z_1 \amalg (U_1 \cap U_2) \amalg Z_2 \]

Denote Zi,isubsetViZ_{i, i} \subset V_ iZi,isubsetVi the inverse image of ZiZ_ iZi under psii\psi _ ipsii. Observe that psii\psi _ ipsii is an isomorphism over an open neighbourhood of ZiZ_ iZi. Denote Zi,jsubsetViZ_{i, j} \subset V_ iZi,jsubsetVi the inverse image of ZjZ_ jZj under psii\psi _ ipsii. Observe that psii:Zi,jtoZj\psi _ i : Z_{i, j} \to Z_ jpsii:Zi,jtoZj is a proper morphism. Since ZiZ_ iZi and ZjZ_ jZj are disjoint closed subsets of UUU, we see that Zi,iZ_{i, i}Zi,i and Zi,jZ_{i, j}Zi,j are disjoint closed subsets of ViV_ iVi.

Denote overlineZi,i\overline{Z}_{i, i}overlineZi,i and overlineZi,j\overline{Z}_{i, j}overlineZi,j the closures of Zi,iZ_{i, i}Zi,i and Zi,jZ_{i, j}Zi,j in XiX_ iXi. After replacing XiX_ iXi by a ViV_ iVi-admissible blowup we may assume that overlineZi,i\overline{Z}_{i, i}overlineZi,i and overlineZi,j\overline{Z}_{i, j}overlineZi,j are disjoint, see Lemma 38.33.2. We assume this holds for both X1X_1X1 and X2X_2X2. Observe that this property is preserved if we replace XiX_ iXi by a further ViV_ iVi-admissible blowup.

Set V12=V1timesUV2V_{12} = V_1 \times _ U V_2V12=V1timesUV2. We have an immersion V12toX1timesSX2V_{12} \to X_1 \times _ S X_2V12toX1timesSX2 which is the composition of the closed immersion V12=V1timesUV2toV1timesSV2V_{12} = V_1 \times _ U V_2 \to V_1 \times _ S V_2V12=V1timesUV2toV1timesSV2 (Schemes, Lemma 26.21.9) and the open immersion V1timesSV2toX1timesSX2V_1 \times _ S V_2 \to X_1 \times _ S X_2V1timesSV2toX1timesSX2. Let X12subsetX1timesSX2X_{12} \subset X_1 \times _ S X_2X12subsetX1timesSX2 be the scheme theoretic image of V12toX1timesSX2V_{12} \to X_1 \times _ S X_2V_12toX1timesSX_2. The projection morphisms

\[ p_1 : X_{12} \to X_1 \quad \text{and}\quad p_2 : X_{12} \to X_2 \]

are proper as X1X_1X1 and X2X_2X2 are proper over SSS. If we replace X1X_1X1 by a V1V_1V1-admissible blowing up, then X12X_{12}X12 is replaced by the strict transform with respect to this blowing up, see Lemma 38.33.5.

Denote psi:V12toU\psi : V_{12} \to Upsi:V12toU the compositions psi=psi1circp1∣V12=psi2circp2∣V12\psi = \psi _1 \circ p_1|_{V_{12}} = \psi _2 \circ p_2|_{V_{12}}psi=psi1circp1∣V12=psi2circp2∣V12. Consider the closed subscheme

\[ Z_{12, 2} = (p_1|_{V_{12}})^{-1}(Z_{1, 2}) = (p_2|_{V_{12}})^{-1}(Z_{2, 2}) = \psi ^{-1}(Z_2) \subset V_{12} \]

The morphism p1∣V12:V12toV1p_1|_{V_{12}} : V_{12} \to V_1p1∣V12:V12toV1 is an isomorphism over an open neighbourhood of Z1,2Z_{1, 2}Z1,2 because psi2:V2toU\psi _2 : V_2 \to Upsi2:V2toU is an isomorphism over an open neighbourhood of Z2Z_2Z2 and V12=V1timesUV2V_{12} = V_1 \times _ U V_2V12=V1timesUV2. By Lemma 38.33.3 there exists a V1V_1V1-admissible blowing up X1′toX1X_1' \to X_1X1′toX1 such that the strict transform p′1:X′12toX′1p'_1 : X'_{12} \to X'_1p′1:X′12toX′1 of p1p_1p1 is an isomorphism over an open neighbourhood of the closure of Z1,2Z_{1, 2}Z1,2 in X′1X'_1X′1. After replacing X1X_1X1 by X′1X'_1X′1 and X12X_{12}X12 by X′12X'_{12}X′12 we may assume that p1p_1p1 is an isomorphism over an open neighbourhood of overlineZ1,2\overline{Z}_{1, 2}overlineZ1,2.

The reduction of the previous paragraph tells us that

\[ X_{12} \cap (\overline{Z}_{1, 2} \times _ S \overline{Z}_{2, 1}) = \emptyset \]

where the intersection taken in X1timesSX2X_1 \times _ S X_2X1timesSX2. Namely, the inverse image p1−1(overlineZ1,2)p_1^{-1}(\overline{Z}_{1, 2})p1−1(overlineZ1,2) in X12X_{12}X12 maps isomorphically to overlineZ1,2\overline{Z}_{1, 2}overlineZ1,2. In particular, we see that Z12,2Z_{12, 2}Z12,2 is dense in p1−1(overlineZ1,2)p_1^{-1}(\overline{Z}_{1, 2})p1−1(overlineZ1,2). Thus p2p_2p2 maps p1−1(overlineZ1,2)p_1^{-1}(\overline{Z}_{1, 2})p1−1(overlineZ1,2) into overlineZ2,2\overline{Z}_{2, 2}overlineZ2,2. Since overlineZ2,2capoverlineZ2,1=emptyset\overline{Z}_{2, 2} \cap \overline{Z}_{2, 1} = \emptyset overlineZ2,2capoverlineZ2,1=emptyset we conclude.

Consider the schemes

\[ W_ i = U \coprod \nolimits _{U_ i} (X_ i \setminus \overline{Z}_{i, j}), \quad i = 1, 2 \]

obtained by glueing. Let us apply Lemma 38.33.1 to see that WitoSW_ i \to SWitoS is separated. First, UtoSU \to SUtoS and XitoSX_ i \to SXitoS are separated. The immersion UitoUtimesS(XisetminusoverlineZi,j)U_ i \to U \times _ S (X_ i \setminus \overline{Z}_{i, j})UitoUtimesS(XisetminusoverlineZi,j) is closed because any specialization uileadstouu_ i \leadsto uuileadstou with uiinUiu_ i \in U_ iuiinUi and uinUsetminusUiu \in U \setminus U_ iuinUsetminusUi can be lifted uniquely to a specialization uileadstoviu_ i \leadsto v_ iuileadstovi in ViV_ iVi along the proper morphism psii:VitoU\psi _ i : V_ i \to Upsii:VitoU and then viv_ ivi must be in Zi,jZ_{i, j}Zi,j. Thus the image of the immersion is closed, whence the immersion is a closed immersion.

On the other hand, for any valuation ring AAA over SSS with fraction field KKK and any morphism gamma:mathopmathrmSpec(K)to(U1capU2)\gamma : \mathop{\mathrm{Spec}}(K) \to (U_1 \cap U_2)gamma:mathopmathrmSpec(K)to(U1capU2) over SSS, there is an iii and an extension of gamma\gamma gamma to a morphism hi:mathopmathrmSpec(A)toWih_ i : \mathop{\mathrm{Spec}}(A) \to W_ ihi:mathopmathrmSpec(A)toWi. Namely, for both i=1,2i = 1, 2i=1,2 there is a morphism gi:mathopmathrmSpec(A)toXig_ i : \mathop{\mathrm{Spec}}(A) \to X_ igi:mathopmathrmSpec(A)toXi extending gamma\gamma gamma by the valuative criterion of properness for XiX_ iXi over SSS, see Morphisms, Lemma 29.42.1. Thus we only are in trouble if gi(mathfrakmA)inoverlineZi,jg_ i(\mathfrak m_ A) \in \overline{Z}_{i, j}gi(mathfrakmA)inoverlineZi,j for i=1,2i = 1, 2i=1,2. This is impossible by the emptyness of the intersection of X12X_{12}X12 and overlineZ1,2timesSoverlineZ2,1\overline{Z}_{1, 2} \times _ S \overline{Z}_{2, 1}overlineZ1,2timesSoverlineZ2,1 we proved above.

Consider a diagram

\[ \xymatrix{ W_1' \ar[d] \ar[r] & W & W_2' \ar[l] \ar[d] \\ W_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & W_2 } \]

as in Lemma 38.33.4. By the previous paragraph for every solid diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_\gamma \ar[d] & W \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar@{..>}[ru] \ar[r] & S } \]

where mathopmathrmIm(gamma)subsetU1capU2\mathop{\mathrm{Im}}(\gamma ) \subset U_1 \cap U_2mathopmathrmIm(gamma)subsetU1capU2 there is an iii and an extension hi:mathopmathrmSpec(A)toWih_ i : \mathop{\mathrm{Spec}}(A) \to W_ ihi:mathopmathrmSpec(A)toWi of gamma\gamma gamma. Using the valuative criterion of properness for W′itoWiW'_ i \to W_ iW′itoWi, we can then lift hih_ ihi to h′i:mathopmathrmSpec(A)toW′ih'_ i : \mathop{\mathrm{Spec}}(A) \to W'_ ih′i:mathopmathrmSpec(A)toW′i. Hence the dotted arrow in the diagram exists. Since WWW is separated over SSS, we see that the arrow is unique as well. This implies that WtoSW \to SWtoS is universally closed by Morphisms, Lemma 29.42.2. As WtoSW \to SWtoS is already of finite type and separated, we win. square\squaresquare

reference

Theorem 38.33.8. Let SSS be a quasi-compact and quasi-separated scheme. Let XtoSX \to SXtoS be a separated, finite type morphism. Then XXX has a compactification over SSS.

Proof. We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. There exists a closed immersion XtoX′X \to X'XtoX′ with X′toSX' \to SX′toS of finite presentation and separated. See Limits, Proposition 32.9.6. If we find a compactification of X′X'X′ over SSS, then taking the scheme theoretic image of XXX in this will give a compactification of XXX over SSS. Thus we may assume XtoSX \to SXtoS is separated and of finite presentation. We may write S=mathopmathrmlimnolimitsSiS = \mathop{\mathrm{lim}}\nolimits S_ iS=mathopmathrmlimnolimitsSi as a directed limit of a system of Noetherian schemes with affine transition morphisms. See Limits, Proposition 32.5.4. We can choose an iii and a morphism XitoSiX_ i \to S_ iXitoSi of finite presentation whose base change to SSS is XtoSX \to SXtoS, see Limits, Lemma 32.10.1. After increasing iii we may assume XitoSiX_ i \to S_ iXitoSi is separated, see Limits, Lemma 32.8.6. If we can find a compactification of XiX_ iXi over SiS_ iSi, then the base change of this to SSS will be a compactification of XXX over SSS. This reduces us to the case discussed in the next paragraph.

Assume SSS is Noetherian. We can choose a finite affine open covering X=bigcupi=1,ldots,nUiX = \bigcup _{i = 1, \ldots , n} U_ iX=bigcupi=1,ldots,nUi such that U1capldotscapUnU_1 \cap \ldots \cap U_ nU1capldotscapUn is dense in XXX. This follows from Properties, Lemma 28.29.4 and the fact that XXX is quasi-compact with finitely many irreducible components. For each iii we can choose an nigeq0n_ i \geq 0nigeq0 and an immersion UitomathbfAniSU_ i \to \mathbf{A}^{n_ i}_ SUitomathbfAniS by Morphisms, Lemma 29.39.2. Hence UiU_ iUi has a compactification over SSS for i=1,ldots,ni = 1, \ldots , ni=1,ldots,n by taking the scheme theoretic image in mathbfPniS\mathbf{P}^{n_ i}_ SmathbfPniS. Applying Lemma 38.33.7 (n−1)(n - 1)(n−1) times we conclude that the theorem is true. square\squaresquare