Proof. The proof is completely analogous to the non-derived version in [ Reference Estrada, Guil Asensio and Trlifaj14 , Lemma 3·5], see also [ Reference Hrbek, Šťovíček and Trlifaj18 , Lemma 2·1].

Remark 3·1. We know by [ Reference Psaroudakis and Vitória34 , Lemma 4·5] that two silting objects $T_1$ and $T_2$ are equivalent if and only if $\mathrm{Add}(T_1)=\mathrm{Add}(T_2)$ . Similarly, two cosilting objects $C_1$ and $C_2$ are equivalent if and only if $\mathrm{Prod}(C_1)=\mathrm{Prod}(C_2)$ .

Proof. The characterisation of the silting complexes is a particular case of [ Reference Psaroudakis and Vitória34 , Proposition 4·13]. For the cosilting case, the argument is dual, the only modification occurs when we need to show that C induces a t-structure whose aisle is ${^{\perp_{\leq0}}C}$ (and consequently the coaisle is the smallest cosuspended and closed under products subcategory of $\mathbf{D}({R})$ containing C). Here we use the additional hypothesis that C is pure-injective and then the claim follows by [ Reference Laking and Vitória25 , Corollary 5·11] (the main result of [ Reference Neeman31 ] is essentially the reason why no extra assumption is needed on the silting side.)

Remark 3·3. (1) A t-structure $(\mathcal{Y},\mathcal{Z})$ is silting if and only if it extends to a TTF-triple $(\mathcal{X},\mathcal{Y},\mathcal{Z})$ which is non-degenerate, suspended and generated by a set of objects in $\mathbf{D}({R})$ , [ Reference Angeleri Hügel2 , Theorem 4·11]. If $(\mathcal{X},\mathcal{Y},\mathcal{Z})$ is induced by the silting object T, we will denote it by $(\mathcal{X}_T,\mathcal{Y}_T,\mathcal{Z}_T)$

(2) A silting object T is bounded, that is, isomorphic in $\mathbf{D}({R})$ to a bounded complex of projectives, if and only if $(\mathcal{X}_T,\mathcal{Y}_T,\mathcal{Z}_T)$ is intermediate, see [ Reference Angeleri Hügel and Hrbek3 , Theorem 2·11]. In particular, it follows from [ Reference Marks and Vitória26 , Theorem 3·6 and Proposition 3·10] that this TTF-triple is compactly generated.

(3) Dually, a t-structure $(\mathcal{U},\mathcal{V})$ in $\mathbf{D}({R^{\mathrm{op}}})$ is induced by a pure-injective cosilting object C if and only if it extends to a non-degenerate cosuspended TTF-triple $(\mathcal{U},\mathcal{V},\mathcal{W})$ which is homotopically smashing, [ Reference Laking24 , Theorem 4·6]. In this case we will write $(\mathcal{U},\mathcal{V},\mathcal{W})=(\mathcal{U}_C,\mathcal{V}_C,\mathcal{W}_C)$ .

(4) A cosilting object C is bounded, that is, isomorphic in $\mathbf{D}({R})$ to a bounded complex of injectives if and only if the associated TTF-triple $(\mathcal{U}_C,\mathcal{V}_C,\mathcal{W}_C)$ is cointermediate and cosuspended, see [ Reference Angeleri Hügel and Hrbek3 , Theorem 2·12]. As before, in this case $(\mathcal{U}_C,\mathcal{V}_C,\mathcal{W}_C)$ is homotopically smashing.

Remark 3·4. From [ Reference Saorín and Šťovíček37 ] it follows that all compactly generated t-structures in $\mathbf{D}({R})$ are homotopically smashing, hence in $\mathbf{D}({R})$ all cosilting objects of cofinite type are pure-injective. The converse is not true in general, but it holds if R is commutative noetherian [ Reference Hrbek and Nakamura17 , Corollary 2·14].

Remark 3·6. From the proof of [ Reference Angeleri Hügel and Hrbek3 , Lemma 3·2] it follows that in the above correspondences, the classes $\mathcal{Y}_{T}$ and $\mathcal{V}_{T^+}$ are dual definable classes, i.e. $Y\in \mathcal{Y}_T$ if and only if $Y^+\in \mathcal{V}_{T^+}$ and $V\in V_{T^+}$ if and only if $V^+\in \mathcal{Y}_T$ .

Proof. The equality (a) follows by the isomorphism

\begin{align*}\mathrm{Hom}_{\mathbf{D}({R})}(T,X[n])\cong H^n{\mathbf{R}}\mathrm{Hom}_R(T,X),\end{align*}

valid for any $n\in\mathbb{Z}$ .

For the equality (b), note that for all $n\in\mathbb{Z}$ we have the isomorphisms:

\begin{align*}\mathrm{Hom}_{\mathbf{D}({R^{\mathrm{op}}})}(X,T^+[n])&\cong H^n{\mathbf{R}}\mathrm{Hom}_{R^{\mathrm{op}}}(X,T^+)\\&=H^n{\mathbf{R}}\mathrm{Hom}_{R^{\mathrm{op}}}(X,{\mathbf{R}}\mathrm{Hom}_{\mathbb{Z}}(T,\mathbb{Q}/\mathbb{Z}))\\ &\cong H^n{\mathbf{R}}\mathrm{Hom}_{\mathbb{Z}}(T\otimes^{\mathbf L}_RX,\mathbb{Q}/Z)\\&=H^n(T\otimes^{\mathbf L}_RX)^+\end{align*}

and $H^n(T\otimes^{\mathbf L}_RX)^+=0$ if and only if $H^{-n}(T\otimes^{\mathbf L}_RX)=0$ .

Proof. The converse implication is obvious, so we need to prove only the direct one. Let $X\in\mathbf{D}({R^{\mathrm{op}}})$ with the property $T\otimes^{\mathbf L}_RX=0$ . Then $H^n(T\otimes^{\mathbf L}_RX)=0$ for all $n\in\mathbb{Z}$ and Lemma 3·8 implies that $X\in\mathcal{U}_{T^+}\cap\mathcal{V}_{T^+}=\{0\}$ .

Proof. Assume $\mathcal{C}$ is closed under products and let I be a set. Since $X \in \mathrm{susp}\,(\mathcal{C})$ , there is by the above description some $i \geq 0$ such that $X \in \mathcal{E}^+_i$ . We prove $X^I \in \mathcal{E}^+_i$ by induction on $i \geq 0$ . If $i=0$ then $X \in \mathrm{add}\{C[n] \mid C \in \mathcal{C}, n \geq 0\}$ . Since $\mathcal{C}$ is closed under products, clearly $X^I \in \mathrm{add}\{C[n] \mid C \in \mathcal{C}, n \geq 0\}$ . If $i\gt 0$ then there is a triangle $E_0 \to X \to E_{i-1} \to E_0[1]$ with $E_0 \in \mathcal{E}^+_0$ and $E_{i-1} \in \mathcal{E}^+_{i-1}$ . Then the triangle $E_0^I \to X^I \to E_{i-1}^I \to E_0^I[1]$ together with the induction hypothesis shows that $X^I \in \mathcal{E}^+_{i}$ . The case of coproduct closure and/or the cosuspended closure is handled the same way.

Proof. We prove only the silting result, the cosilting one follows by a completely analogous argument. Put $(\mathcal{Y},\mathcal{Z}) = (T^{\perp_{\gt 0}},T^{\perp_{\leq 0}})$ and our aim is to show that this constitutes a t-structure in $\mathbf{D}({R})$ . Without loss of generality, we may assume that T is a complex of projective R-modules concentrated in degrees $-n+1,-n+2,\ldots,-1,0$ for some $n\gt 0$ .

First, we show that for any $X \in \mathbf{D}({R})$ there is a triangle $Y \to X \to Z \to Y[1]$ with $Y \in \mathcal{Y}$ and $Z \in \mathcal{Z}$ . Put $X_0:=X$ and define inductively a sequence of triangles

\begin{align*}T^{(H_i)}[i] \xrightarrow{h_i} X_i \xrightarrow{f_i} X_{i+1} \to T^{(H_i)}[i+1],\end{align*}

where $H_i = \mathrm{Hom}_{\mathbf{D}({R})}(T[i],X_i)$ and $h_i$ is the canonical evaluation morphism. Since $\mathrm{Hom}_{\mathbf{D}({R})}(T[i],h_i)$ is surjective for each $i \geq 0$ and by (Sb1), we see that

\begin{align*}\mathrm{Hom}_{\mathbf{D}({R})}(T[j],X_i) = 0 \textrm{ for all } j=0,1,\ldots,i-1.\end{align*}

This construction defines an inductive system

\begin{align*}X_0 \xrightarrow{f_0} X_{1} \xrightarrow{f_1} X_{2} \xrightarrow{f_2} X_{3} \xrightarrow{f_3} \cdots\end{align*}

and we consider its homotopy colimit $Z = \mathrm{hocolim}_{i\geq 0}(X_i)$ . Recall that for any $j \geq 0$ we have $Z = \mathrm{hocolim}_{i\geq j}(X_i)$ , see [ Reference Neeman30 , Lemma 1·7·1]. Considering $X_j$ as the trivial homotopy colimit $X_j = \mathrm{hocolim}_{i\geq j}(X_j)$ of an inductive tower consisting of identities on $X_j$ , [ Reference Neeman30 , Lemma 1·6·6], we obtain a commutative square

where $f_{ji} = f_{i-1}\circ\cdots\circ f_{j+1}\circ f_j$ for $j\lt i$ and $f_{jj} =\mathrm{Id}_{X_j}$ . For each $0 \leq j \leq i$ , let $S_{ji}$ be the cone of $f_{ji}$ . Then $S_{ji}$ is an iterated extension of the objects $T[k+1]^{(H_k)}$ where $k=j,$ $j+1,\ldots,i-1$ . In particular, since $T \in \mathbf{D}({R})^{\leq 0}$ , we have $S_{ji} \in \mathbf{D}({R})^{\leq -j-1}$ . By [ Reference Beilinson, Bernstein and Deligne6 , 1·1·11], the square above extends to a diagram

in which all squares commute and all rows and columns extend to triangles. Since $S_{ji} \in \mathbf{D}({R})^{\leq -j-1}$ , we have $\coprod_{i \geq j}S_{ji} \in \mathbf{D}({R})^{\leq -j-1}$ , and therefore $C \in \mathbf{D}({R})^{\leq -j-1}$ . Using that $\mathrm{Hom}_{\mathbf{D}({R})}(T,\mathbf{D}({R})^{\leq -n})=0$ , we argue from the rightmost vertical triangle of the diagram that $\mathrm{Hom}_{\mathbf{D}({R})}(T[i],Z) \cong \mathrm{Hom}_{\mathbf{D}({R})}(T[i],X_{i+n+1}) = 0$ for any $i \geq 0$ . This shows that $Z \in \mathcal{Z} = T^{\perp_{\leq 0}}$ .

Now extend the map $X = X_0 \to Z$ to a triangle $Y \to X \to Z \to Y[1]$ and let us show that $Y \in \mathcal{Y} = T^{\perp_{\gt 0}}$ . The diagram above puts Y into a triangle

\begin{align*}\textstyle Y \to \coprod_{i \geq 0}S_{0i} \to \coprod_{i \geq 0}S_{0i} \to Y[1].\end{align*}

Put $C_{0i} = S_{0i}[-1]$ for all $i \gt 0$ , so that Y is the mapping cone of $\coprod_{i \geq 0}C_{0i} \to \coprod_{i \geq 0}C_{0i}$ .

Recall that $C_{0i}$ is an iterated extension of objects $T^{(H_0)},\ldots,T^{(H_{i-1})}[i-1]$ , and therefore $C_{0i} \in \mathrm{susp}(\mathrm{Add}(T)) \subseteq \mathcal{Y} = T^{\perp_{\gt 0}}$ , the latter inclusion follows from (Sb1) and the obvious closure properties of $\mathcal{Y}$ . It remains to show that also the coproduct $\coprod_{i \geq 0}C_{0i}$ belongs to $\mathcal{Y}$ . Note that by the assumption on T, we clearly have $\mathbf{D}({R})^{\leq -n} \subseteq \mathcal{Y}$ . Put $m = n+1$ . For any $i \gt m$ , there is a triangle $C_{0m} \to C_{0i} \to C_{mi} \to C_{0m}[1]$ , and we know that $C_{mi} \in \mathbf{D}({R})^{\leq -m}$ . Consider the coproduct triangle

\begin{align*}\textstyle \coprod_{i \gt m}C_{0m} \to \coprod_{i \gt m}C_{0i} \to \coprod_{i \gt m}C_{mi} \to \coprod_{i \gt m}C_{0m}[1].\end{align*}

Then $\coprod_{i \gt m}C_{mi} \in \mathbf{D}({R})^{\leq -m} = \mathbf{D}({R})^{\leq -n-1}$ . Since $\mathbf{D}({R})^{\leq -n} \subseteq \mathcal{Y}$ , we see that $\coprod_{i \gt m}C_{0i} \in \mathcal{Y}$ if and only if $\coprod_{i \gt m}C_{0m} \cong C_{0m}^{(\omega)} \in \mathcal{Y}$ . But the latter inclusion follows from Lemma 3·10 because $C_{0m} \in \mathrm{susp}\,(\mathrm{Add}(T)) \subseteq \mathcal{Y}$ . Therefore, $\coprod_{i \gt m}C_{0i} \in \mathcal{Y}$ . Since we already know that $\coprod_{i \leq m}C_{0i} \in \mathcal{Y}$ , we showed that $Y \in \mathcal{Y}$ .

Finally we need to show that $\mathrm{Hom}_{\mathbf{D}({R})}(\mathcal{Y},\mathcal{Z}) = 0$ . Take $X \in \mathcal{Y}$ , and consider the same triangle $Y \to X \to Z \to Y[1]$ as constructed above. Since $Y, X \in \mathcal{Y}$ , also $Z \in \mathcal{Y}$ . But then $Z \in \mathcal{Y} \cap \mathcal{Z} = T^{\perp_{\mathbb{Z}}}$ . It follows by (Sb2) that $Z = 0$ and so $X \cong Y$ . By the construction, Y belongs to the smallest suspended and coproduct closed subcategory of $\mathbf{D}({R})$ generated by T (since we constructed Y from T by using extensions, [ Reference Alonso Tarrío, López Jeremías and Saorín1 ], and coproducts). Since $\mathrm{Hom}_{\mathbf{D}({R})}(T,\mathcal{Z}) = 0$ , this shows $\mathrm{Hom}_{\mathbf{D}({R})}(\mathcal{Y},\mathcal{Z}) = 0$ , which in turn renders $(\mathcal{Y},\mathcal{Z})$ a t-structure.

Proof. We apply Theorem 3·11 together with [ Reference Wei40 , Corollary 3·7].

Proof. (I) (i) The ring homomorphism $\lambda$ induces a structure of R-module on S, and as a complex of R-modules it belongs to $\mathbf{D}^{\leq0}(R)$ . Since T is silting, it induces a t-structure $(T^{\perp_{\gt 0}}, T^{\perp_{\leq 0}})$ in $\mathbf{D}({A})$ . From $T\in{T^{\perp_{\gt 0}}}$ it follows by Lemma 4·1 that $T\otimes^{\mathbf L}_R S\in T^{\perp_{\gt 0}}$ .

For every $n\in\mathbb{Z}$ and every $X\in\mathbf{D}({B})$ , we have the adjunction isomorphism

\begin{align*}\mathrm{Hom}_{\mathbf{D}({A})}(T,X[n])\cong\mathrm{Hom}_{\mathbf{D}({B})}((T\otimes^{\mathbf L}_RS),X[n]).\end{align*}

Using this isomorphism for $n\gt 0$ and $X=T\otimes^{\mathbf L}_RS$ , together with $T\otimes^{\mathbf L}_RS\in T^{\perp_{\gt 0}}$ , we deduce that the B-complex $T\otimes^{\mathbf L}_RS$ belongs to $(T\otimes^{\mathbf L}_RS)^{\perp_{\gt 0}}$ .

Further, the same isomorphism, applied for $n=0$ and $X=\coprod_{i\in I}X_i$ , together with the fact that (the derived functor of) the restriction of the scalars functor $\mathbf{D}({B})\to \mathbf{D}({A})$ preserves all coproducts, shows that the class $(T\otimes^{\mathbf L}_RS)^{\perp_{\gt 0}}\subseteq \mathbf{D}({B})$ is closed under coproducts.

Finally, if $X\in\mathbf{D}({B})$ and $X\in(T\otimes^{\mathbf L}_RS)^{\perp_{\mathbb{Z}}}$ , it follows that $\mathrm{Hom}_{\mathbf{D}({A})}(T,X[n])=0$ for all $n\in\mathbb{Z}$ . Then X is acyclic as an A-complex, hence it is acyclic as a B-complex.

1. (ii) These equalities are consequences of the isomorphism

   \begin{align*}\mathrm{Hom}_{\mathbf{D}({B})}(T\otimes^{\mathbf L}_R S,Y)\cong \mathrm{Hom}_{\mathbf{D}({B})}(T\otimes^{\mathbf L}_A A \otimes^{\mathbf L}_R S,Y)\cong \mathrm{Hom}_{\mathbf{D}({R})}(T,{\mathbf{R}}\mathrm{Hom}_B(B,Y)).\end{align*}

2. (iii) Let $Y\in \mathcal{Y}_{T}$ . Then ${\mathbf{R}}\mathrm{Hom}_B(B,Y\otimes^{\mathbf L}_R S)=Y\otimes^{\mathbf L}_R S\in \mathcal{Y}_T$ . For all $i\gt 0$ we have

   \begin{align*} & \mathrm{Hom}_{\mathbf{D}({B})}(T\otimes^{\mathbf L}_R S, Y\otimes^{\mathbf L}_R S[i]) \cong \mathrm{Hom}_{\mathbf{D}({B})}(T\otimes^{\mathbf L}_A B, Y\otimes^{\mathbf L}_R S[i]) \\ \cong & \mathrm{Hom}_{\mathbf{D}({A})}(T, {\mathbf{R}}\mathrm{Hom}_B(B,Y\otimes^{\mathbf L}_R S)[i])\cong \mathrm{Hom}_{\mathbf{D}({A})}(T, Y\otimes^{\mathbf L}_R S[i])=0.\end{align*}

Let $X\in \mathcal{X}_T$ and $Y\in \mathcal{Y}_{T\otimes^{\mathbf L}_R S}$ . Then ${\mathbf{R}}\mathrm{Hom}_B(B,Y)\in \mathcal{Y}_T$ , hence the property $X\otimes^{\mathbf L}_R S=X\otimes^{\mathbf L}_A B\in \mathcal{X}_{T\otimes^{\mathbf L}_R S}$ can be obtained by applying the adjunction isomorphism.

1. (iv) If T is bounded, then obviously the same property holds true for $T\otimes^{\mathbf L}_RS\cong T\otimes^{\mathbf L}_AB$ .

(II) The functor ${\mathbf{R}}\mathrm{Hom}_{A}(B,-)$ preserves all products, hence it preserves the pure-injectivity property, [ Reference Saorín and Šťovíček37 , Lemma 5·3]. Therefore, using the natural isomorphisms

\begin{align*}\mathrm{Hom}_{\mathbf{D}({A})}(U\otimes^{\mathbf L}_S S,V)\cong \mathrm{Hom}_{\mathbf{D}({A})}(U\otimes^{\mathbf L}_B B,V)\cong \mathrm{Hom}_{\mathbf{D}({B})}(U,{\mathbf{R}}\mathrm{Hom}_A(B,V)),\end{align*}

a proof can be done by following the lines of the proof for (I).

Proof. (i) Since all cosilting objects of cofinite type are pure-injective, it follows that ${\mathbf{R}}\mathrm{Hom}_R(S,C)\in\mathbf{D}({B})$ is cosilting.

There exists a set of compact objects $\mathcal{P}\subseteq \mathcal{U}_C$ such that $\mathcal{V}_C=\mathcal{P}^{\perp_0}$ . Using the fact the the restriction functor preserves coproducts, it follows that $\mathcal{P}\otimes^{\mathbf L}_R S=\{P\otimes^{\mathbf L}_R S\mid P\in \mathcal{P}\}$ is a set of compact objects of $\mathbf{D}({B})$ .

Using Theorem 4·2(II·ii), we have $V\in \mathcal{V}_{{\mathbf{R}}\mathrm{Hom}_R(S,C)}$ if and only if $V\otimes^{\mathbf L}_S S={\mathbf{R}}\mathrm{Hom}_S(S,V)\in \mathcal{V}_C$ , and this is equivalent to

\begin{align*}0=\mathrm{Hom}_{\mathbf{D}({R})}(\mathcal{P}, {\mathbf{R}}\mathrm{Hom}_S(S,V))\cong \mathrm{Hom}_{\mathbf{D}({S})}(\mathcal{P}\otimes^{\mathbf L}_R S, V).\end{align*}

This implies that $\mathcal{V}_{{\mathbf{R}}\mathrm{Hom}_R(S,C)}=(\mathcal{P}\otimes^{\mathbf L}_R S)^{\perp_0},$ hence ${\mathbf{R}}\mathrm{Hom}_R(S,C)$ is of cofinite type.

(ii) The proof is similar to the proof used for the cosilting case.

Proof. Using Lemma 3·7, the equivalence is a consequence of the isomorphisms

\begin{align*}H^n((T\otimes^{\mathbf L}_R S)\otimes^{\mathbf L}_S (X\otimes^{\mathbf L}_R S))\cong H^n(T\otimes^{\mathbf L}_R X\otimes^{\mathbf L}_R S)\cong H^n(T\otimes^{\mathbf L}_R X)\otimes_R S,\end{align*}

that hold for all $X\in \mathbf{D}({R})$ .

Proof. (i) and (ii) follow from Lemma 4·4.

(iii) The inclusion $\mathcal{Y}_{T}\subseteq\{Y\in\mathbf{D}({A})\mid Y\otimes^{\mathbf L}_R S\in\mathcal{Y}_{T\otimes^{\mathbf L}_RS}\}$ already appeared in Theorem 4·2.

Conversely, since both T and $T\otimes^{\mathbf L}_R S$ are of finite type, the associated TTF-triples are compactly generated. Let $Y\in\mathbf{D}({A})$ be such that $Y\otimes^{\mathbf L}_R S\in\mathcal{Y}_{T\otimes^{\mathbf L}_RS}.$ If $C\in \mathcal{X}_{T}$ is a compact object then $C\otimes^{\mathbf L}_R S\in \mathcal{X}_{T\otimes^{\mathbf L}_R S}$ by Theorem 4·2, hence

\begin{align*}\mathrm{Hom}_{\mathbf{D}({A})}(C\otimes^{\mathbf L}_R S, Y\otimes^{\mathbf L}_R S)=0.\end{align*}

Using the isomorphism presented in Lemma 2·5 we get:

\begin{align*}\mathrm{Hom}_{\mathbf{D}({R})}(C, Y)\otimes_RS=0,\end{align*}

hence $\mathrm{Hom}_{\mathbf{D}({R})}(C, Y)=0$ , since S is faithfully flat. It follows $Y\in\mathcal{Y}_T$ .

Proof. We work in $\mathbf{D}({A})$ and replace T by its (bounded) projective resolution. From Lemma 4·1 it follows that $T\otimes^{\mathbf L}_R S\in T^{\perp_{\gt 0}}$ . By Proposition 4·2 we know that $T\otimes^{\mathbf L}_R S\in (T\otimes^{\mathbf L}_R S)^{\perp_{\gt 0}}$ , and that $T\otimes^{\mathbf L}_R S$ generates $\mathbf{D}({B})$ . Since S is flat, $T\otimes^{\mathbf L}_R S=T\otimes_R S$ is also a B-module, hence $T\otimes^{\mathbf L}_R S\in (T\otimes^{\mathbf L}_R S)^{\perp_{\lt 0}}$ . The conclusion follows from Corollary 3·12.

The last part is a consequence of Lemma 4·7 since all tilting cotorsion pairs are generated by families of strongly finitely presented modules.

Proof. The first part is proved in [ Reference Hrbek, Hu and Zhu16 , Proposition 5·12]. These bijections extend the bijections described in [ Reference Hrbek15 , Theorem 3·11].

The correspondence between bounded cosilting complexes and cointermediate, cosuspended TTF-triples is provided by [ Reference Marks and Vitória26 , Theorem 3·13], see also [ Reference Angeleri Hügel and Hrbek3 , Corollary 2·14]. For the correspondence between TTF-triples that are cointermediate and cosuspended and bounded Thomason filtrations, let $\mathbb{X}=(X_k)_{k\in \mathbb{Z}}$ be a bounded Thomason filtration. By [ Reference Hrbek15 , Theorem 3·11], the aisle of the corresponding t-structure is

\begin{align*}\mathcal{U}_{\mathbb{X}}=\{X\in \mathbf{D}({R})\mid \mathrm{Supp}(H^k(X))\subseteq X_k \textrm{ for all }n\in\mathbb{Z}\}.\end{align*}

Since $X_k=\varnothing$ for all $k\geq n$ , it follows that for every $X\in \mathcal{U}_{\mathbb{X}}$ we have $H^k(X)=0$ for all $k\geq n$ . It follows that

\begin{align*}\mathbf{D}({R})^{\geq n}=\{X\in\mathbf{D}({R})\mid H^i(X)=0 \textrm{ for all } i\lt n\}\subseteq \mathcal{U}_{\mathbb{X}}^{\perp_0}=\mathcal{V}_{\mathbb{X}}, \end{align*}

where $\mathcal{U}_{\mathbb{X}}$ is the coaisle of the t-structure associated to $\mathbb{X}$ . Conversely, it is easy to see that if $(\mathcal{U}, \mathcal{V},\mathcal{W})$ is a TTF-triple that is cointermediate and cosuspended TTF-triple then the associated Thomason filtration is bounded.

Proof. Since f is surjective, we have $f(\mathfrak{S}_1\setminus f^{-1}(Y))=\mathfrak{S}_2\setminus f(f^{-1}(Y))=\mathfrak{S}_2\setminus Y$ , hence $\mathfrak{S}_2\setminus Y$ is a closed set.

Proof. Since $\lambda^\star$ is continuous with respect to Thomason’s topology it follows that $\mathbb{Y}$ is a system of Thomason sets.

Observe that $\mathfrak{q}\in (\lambda^\star)^{-1}(X_n)$ if and only if $\lambda^{-1}(\mathfrak{q})\in X_n$ . Using this, it is easy to see that $\mathbb{Y}$ is a Thomason filtration.

For the last statement, we apply Lemma 5·3 and Lemma 5·2 to conclude that the terms of the sequence $\mathbb{X}$ are Thomason sets. As before, it is an easy exercise to prove that $\mathbb{X}$ is a Thomason filtration.

Remark 5·5. As we already mentioned in the introduction, $\boldsymbol{\Phi}$ induces a map

\begin{align*}\boldsymbol{\Phi}:\left\{\begin{array}{c}\hbox{equivalence classes of }\\ \hbox{ cosilting objects }\\ \hbox{of cofinite type in } \mathbf{D}({R})\end{array}\right\}\longrightarrow\left\{\begin{array}{c}\hbox{equivalence classes of }\\ \hbox{ cosilting objects } \\ \hbox{ of cofinite type in } \mathbf{D}({S})\end{array}\right\}. \end{align*}

If $\lambda$ is faithfully flat, this map is injective.

Proof. (i) Let $n\in \mathbb{Z}$ . From [ Reference Hrbek, Hu and Zhu16 , Lemma 3·7] and Theorem 4·2 (applied for $A=R$ and, consequently, $B=S$ ) it follows that

\begin{align*}Y_n&=\{\mathfrak{q}\in\mathrm{Spec}({S})\mid \kappa(\mathfrak{q})[-n]\in\mathcal{U}_{{\mathbf{R}}\mathrm{Hom}_R(S,C)}\}\\ &=\{\mathfrak{q}\in \mathrm{Spec}({S})\mid \kappa(\mathfrak{q})[-n]\otimes^{\mathbf L}_S S\in \mathcal{U}_C\}.\end{align*}

Note that for every $\mathfrak{q}\in \mathrm{Spec}({S})$ there exists a canonical morphism (of R-algebras) $\kappa(\lambda^\star(\mathfrak{q}))\to \kappa(\mathfrak{q})$ that is induced by the morphism $R/\lambda^{-1}(\mathfrak{q})\to S/\mathfrak{q}$ . This morphism is unital, and $\kappa(\lambda^\star(\mathfrak{q}))$ and $\kappa(\mathfrak{q})$ are fields. It follows that, as an R-module, $\kappa(\mathfrak{q})$ is a direct sum of copies of $\kappa(\lambda^\star(\mathfrak{q}))$ . Then, for every $V\in\mathbf{D}({R})$ we have $\mathrm{Hom}_{\mathbf{D}({R})}(\kappa(\mathfrak{q}),V)=0$ if and only if $\mathrm{Hom}_{\mathbf{D}({R})}(\kappa(\lambda^\star(\mathfrak{q})),V)=0$ .

Since $\kappa(\mathfrak{q})[-n]\in \mathcal{U}_{{\mathbf{R}}\mathrm{Hom}_R(S,C)}$ if and only if for all $i\leq 0$ we have

\begin{align*}0= \mathrm{Hom}_{\mathbf{D}({S})}(\kappa(\mathfrak{q})[-n],{\mathbf{R}}\mathrm{Hom}_R(S,C)[i])\cong \mathrm{Hom}_{\mathbf{D}({R})}(\kappa(\mathfrak{q})[-n],C[i]),\end{align*}

we conclude that $\kappa(\mathfrak{q})[-n]\in \mathcal{U}_{{\mathbf{R}}\mathrm{Hom}_R(S,C)}$ if and only if for all $i\leq 0$ we have $\mathrm{Hom}_{\mathbf{D}({R})}(\kappa(\lambda^\star(\mathfrak{q}))[-n],C[i])=0$ . Then $Y_n$ is Thomason saturated with respect to $\lambda$ .

(ii) For this case we apply (i) and the isomorphism $(T\otimes^{\mathbf L}_RS)^+\cong {\mathbf{R}}\mathrm{Hom}_R(S,T^+)$ .

Proof. (i) From Lemma 5·6 it follows that for all $n\in \mathbb{Z}$ we have $Y_n=(\lambda^\star)^{-1}(X_n)$ , where $\mathbb{X}=(X_n)_{n\in\mathbb{Z}}$ is the Thomason filtration associated to C, and the proof is complete.

(ii) This is also a consequence of Lemma 5·6.

Proof. (i) $\Rightarrow$ (ii) follows from Lemma 5·6.

(ii) $\Rightarrow$ (i) From the proof of Theorem 5·4. it follows that $\mathbb{X}=(\lambda^\star(Y_n))_{n\in \mathbb{Z}}$ is a Thomason filtration. Let $C\in\mathbf{D}({R})$ be a cosilting complex that corresponds to $\mathbb{X}$ . It follows from Theorem 5·7 that ${\mathbf{R}}\mathrm{Hom}_R(S,C)$ is equivalent to $\overline{C}$ .

Proof. The implication (i) $\Rightarrow$ (ii) is proved in Lemma 5·6. Conversely, since $\mathbb{Y}=(Y_n)_{n\in \mathbb{Z}}$ is a (bounded) Thomason filtration that is saturated with respect to $\lambda$ , there exists a bounded cosilting complex $C\in \mathbf{D}({R})$ such that ${\mathbf{R}}\mathrm{Hom}_R(S,C)$ is equivalent to $\overline{T}^+$ . Since C is bounded, it follows that C is equivalent to a complex of the form $T^+$ with $T\in \mathbf{D}({R})$ a bounded silting complex. Applying Theorem 3·5(ii), it follows that the bounded silting complexes $\overline{T}$ and $T\otimes^{\mathbf L}_R S$ are equivalent.

Remark 5·11. The above corollary is also valid for all silting complexes of finite type $\overline{T}\in \mathbf{D}({S})$ if we assume that in $\mathbf{D}({R})$ all cosilting complexes of cofinite type are of the form $T^+$ , with $T\in \mathbf{D}({R})$ a silting complex of finite type (e.g., if R is noetherian, cf. [ Reference Angeleri Hügel and Hrbek3 , Theorem 3·8]).

Proof. Since the Thomason filtration of ${\mathbf{R}}\mathrm{Hom}_R(S,T^+)\cong (T\otimes^{\mathbf L}_R S)^+$ is bounded and Thomason saturated with respect to $\lambda$ , we can apply Proposition 5·10 to observe that there exists a bounded complex of projectives $\widetilde{T}$ in $\mathbf{D}({R})$ such that $\widetilde{T}$ is silting and $(\widetilde{T}\otimes^{\mathbf L}_R S)^+$ is equivalent to $(T\otimes^{\mathbf L}_R S)^+$ . It follows that $(\widetilde{T}\otimes^{\mathbf L}_R S)^+$ and $(T\otimes^{\mathbf L}_R S)^+$ , induce the same t-structure whose aisle and coaisle are

\begin{align*}\mathcal{V}_{(T\otimes^{\mathbf L}_RS)^+}={^{\perp_{\leq0}}[(T\otimes^{\mathbf L}_RS)^+]}={^{\perp_{\leq0}}[(\widetilde{T}\otimes^{\mathbf L}_RS)^+]},\end{align*}

respectively

\begin{align*}\mathcal{W}_{(T\otimes^{\mathbf L}_RS)^+}={^{\perp_{\gt 0}}[(T\otimes^{\mathbf L}_RS)^+]}={^{\perp_{\gt 0}}[(\widetilde{T}\otimes^{\mathbf L}_RS)^+]}.\end{align*}

Using Corollary 4·4 twice, we get

\begin{align*}{^{\perp_{\leq0}}(T^+)}=\{X\in\mathbf{D}({R})\mid X\otimes^{\mathbf L}_RS\in\mathcal{V}_{(T\otimes^{\mathbf L}_RS)^+}\}={^{\perp_{\leq0}}(\widetilde{T}^+)}\end{align*}

and

\begin{align*}{^{\perp_{\gt 0}}(T^+)}=\{X\in\mathbf{D}({R})\mid X\otimes^{\mathbf L}_RS\in\mathcal{W}_{(T\otimes^{\mathbf L}_RS)^+}\}={^{\perp_{\gt 0}}(\widetilde{T}^+)}.\end{align*}

Therefore the pair $({^{\perp_{\leq0}}(T^+)}, {^{\perp_{\gt 0}}(T^+)})$ coincides with the cosilting t-structure induced by $(\widetilde T)^+$ . Hence $T^+$ is cosilting too and it is obviously equivalent to $(\widetilde T)^+$ .

Proof. For the first isomorphism we observe that $(R\mathbb{A}_n/I)\otimes_R S\cong (R\mathbb{A}_n\otimes_R S)/(I\otimes_R S)\cong S\mathbb{A}_n/K$ , where K is the ideal in $S\mathbb{A}_n$ generated by all paths of length 2. The commutativity of the diagram can be checked by direct computations.

Proof. Since $T\otimes_R S$ is bounded silting, we can use Lemma 5·6 and Corollary 5·10 to observe that there exists a silting complex $P\in D(R)$ such that $P\otimes^{\mathbf L}_R S$ is equivalent to $T\otimes^{\mathbf L}_R S$ . Moreover, since $\lambda$ is faithfully flat, we can assume that P is concentrated in $-n+1,\dots,-1,0$ . We use [ Reference Marks and Vitória26 , Theorem 2·10] to observe that $\widehat{P}$ is an $(n-1)$ -tilting module over A(R), and that $\widehat{P\otimes^{\mathbf L}_R S}\cong \widehat{P}\otimes_R S$ and $\widehat{T\otimes^{\mathbf L}_R S}$ are equivalent $(n-1)$ -tilting modules over A(S). From Proposition 4·8 it follows that $\widehat{T}^{(I)}\in \mathcal{K}_{\widehat{P}}\bigcap \mathcal{L}_{\widehat{P}}$ for all sets I. Then $\mathrm{Ext}^{i}_{A(R)}(T,\mathrm{Add}(T))=0$ for all $i\gt 0$ . Since T is a complex of projectives, we can use Lemma 6·1 to obtain the conclusion.

Proof. By Proposition 6·3, we already know that (Sb1) from Theorem 3·11 holds for T. It remains to show that (Sb2), that is, we need to check $\mathrm{Loc}(T) = \mathbf{D}({R})$ . We have $T \otimes^{\mathbf L}_R S \in \mathrm{Loc}(T)$ and by the assumption, $T \otimes^{\mathbf L}_R S$ is a silting in $\mathbf{D}({S})$ . Since $T \otimes^{\mathbf L}_R S$ is silting, we have $(T \otimes^{\mathbf L}_R S)^{\perp_{\mathbb{Z}}}=0$ , hence $\mathrm{Loc}(T \otimes^{\mathbf L}_R S)=\mathbf{D}({S})$ . We obtain that $S\in \mathrm{Loc}(T)$ , and therefore $\mathrm{Loc}(T) = \mathbf{D}({R})$ by the assumption we made on S.

Remark 6·6. Following [ Reference Psaroudakis and Vitória34 ], a silting object T is called a tilting object if in addition we have $\mathrm{Add}(T) \subseteq T^{\perp_{\lt 0}}$ . If $T \in \mathbf{D}({R})$ is bounded silting, then T is tilting precisely if the realization functor from the bounded derived category of the heart of the silting t-structure to the bounded derived category of R is an equivalence [ Reference Psaroudakis and Vitória34 , Corollary 5·2]. Any n-tilting module is a tilting object as an object of the derived category.

We don’t know if the descent (or even the ascent) properties we established for bounded silting complexes restrict to tilting complexes, in general. In particular, our approach which used the transfer to the category $\mathrm{Rep}(A(R))$ does not offer information about the groups $\mathrm{Hom}(T, T^{(I)}[k])$ for $k\lt 0$ .

Proof. This is essentially proved in [ Reference Hrbek, Šťovíček and Trlifaj18 , Lemma 4·1].

Proof. Clearly, if X is a bounded complex of projective R-modules then $X \otimes_R S$ is a bounded complex of projective S-modules for any flat ring morphism $R \to S$ .

Let $R \to S$ be faithfully flat and assume without loss of generality that $X \otimes_R S$ is isomorphic to a complex of projective S-modules concentrated in degrees $-n,-n+1,\ldots,0$ . Since $R \to S$ is faithfully flat and $X \otimes_R S$ is bounded, we see immediately that homology of X vanishes outside of degrees $-n,-n+1,\ldots,0$ . There is a complex P of projective R-modules concentrated in non-positive degrees which is isomorphic to X in $\mathbf{D}({R})$ . Since $P \otimes_R S$ is a complex of projective S-modules isomorphic in $\mathbf{D}({S})$ to $X \otimes_R S$ , by the assumption on $X \otimes_R S$ , the cokernel of the differential map $X^{-n-1} \otimes_R S \to X^{-n} \otimes_R S$ , which is the same as $\mathrm{Coker}(X^{-n-1} \to X^{-n}) \otimes_R S$ , is a projective S-module. Then by the result of Raynaud and Gruson [ 42 , Theorem 10·95·6, Tag 05A9], the cokernel of the differential $X^{-n-1} \to X^{-n}$ is a projective R-module. It follows that the soft truncation $\tau^{\geq -n}X$ , which is quasi-isomorphic to X, is a bounded complex of projective R-modules.

Proof. Combine Theorem 4·2, Lemma 6·7 and Lemma 6·8, and apply the Affine Communication Lemma 2·2.

Proof. Recall from [ 42 , Lemma 35·4·8.] that the map $\lambda\,:\, R \to S$ is a pure monomorphism of R-modules. It follows that for any object $X \in \mathbf{D}({R})$ the morphism $X \otimes^{\mathbf L}_R \lambda\,:\, X \to X \otimes^{\mathbf L}_R S$ is a pure monomorphism in $\mathbf{D}({R})$ by [ Reference Angeleri Hügel and Hrbek3 , Lemma 2·6]. If X is pure-injective, then $X \otimes^{\mathbf L}_R \lambda$ is a split monomorphism and so X is a direct summand in $X \otimes^{\mathbf L}_R S$ . Since $X \otimes^{\mathbf L}_R S \in \mathrm{Loc}(S)$ , we have $X \in \mathrm{Loc}(S)$ . Then $\mathcal{I} \subseteq \mathrm{Loc}(S)$ and so $\mathrm{Loc}(S) = \mathbf{D}({R})$ .

Remark 6·11. The condition $\mathrm{Loc}(\mathcal{I}) = \mathbf{D}({R})$ holds in the following hypotheses:

• (i) R noetherian, this follows from [ Reference Rickard35 , Theorem 3·3];

• (ii) R admits a finite injective resolution, in particular, if R has finite global dimension;

• (iii) more generally, if R is isomorphic to a bounded complex of pure-injectives, in particular, if R has finite pure global dimension, in particular, if R is countable or of cardinality $\aleph_n$ for some $n \gt 0$ [ Reference Gruson and Jensen20 , Proposition 10·5];

• (iv) if $\mathbf{D}({R})$ is a compactly generated triangulated category of finite pure global dimension in the sense of [ Reference Beligiannis7 ], in particular, if $\mathbf{D}({R})$ satisfies the “Brown representability for homology of morphisms“, see [ Reference Christensen, Keller and Neeman10 ];

• (v) if R is such that every localising subcategory of $\mathbf{D}({R})$ is cohomological in the sense of Krause, see e.g. [ Reference Krause and Stevenson23 , Section 3·2].

We also remark that in [ Reference Christensen and Iyengar11 , Proposition 4·2] it is proved that if $R \to S$ is a faithfully flat homomorphism of commutative rings such that S is of projective dimension at most 1 over R, and such that the projective dimension of flat R-modules are bounded by some positive integer, then $\mathrm{Loc}(S) = \mathbf{D}({R})$ .

Remark 6·12. Combining the above, we see that if $\mathrm{Loc}(\mathcal{I}) = \mathbf{D}({R})$ holds for all commutative rings then the property of being a bounded silting object is even an ad-property. We do not know any ring for which $\mathrm{Loc}(\mathcal{I}) = \mathbf{D}({R})$ fails. An example for which injective modules do not generate $\mathbf{D}({R})$ is in [ Reference Rickard35 ]. If R is von Neumann regular then $\mathrm{Loc}(\mathcal{I}) = \mathbf{D}({R})$ precisely if injectives generate in the sense of [ Reference Rickard35 ].