2.1

Unless otherwise specified, a module means a left module. Let $B$ be a ring. $Hom_{B}$ or $Ext_{B}$ mean the $Hom$ or $Ext$ for left $B$ -modules. $B^{op}$ denotes the opposite ring of $B$ , so a $B^{op}$ -module is nothing but a right $B$ -module. Let $BMod$ denote the category of $B$ -modules. $B^{op} Mod$ is also denoted by $ModB$ . For a left (resp. right) Noetherian ring $B$ , $B{\rm mod}$ (resp.  ${\rm mod}B$ ) denotes the full subcategory of $BMod$ (resp.  $ModB$ ) consisting of finitely generated left (resp. right) $B$ -modules.

2.2

For derived categories, we employ standard notation found in [Reference HartshorneHart].

For an abelian category ${\mathcal{A}}$ , $D({\mathcal{A}})$ denotes the unbounded derived category of ${\mathcal{A}}$ . For a plump subcategory (that is, a full subcategory which is closed under kernels, cokernels, and extensions) ${\mathcal{B}}$ of ${\mathcal{A}}$ , $D_{{\mathcal{B}}}({\mathcal{A}})$ denotes the triangulated subcategory of $D({\mathcal{A}})$ consisting of objects $\mathbb{F}$ such that $H^{i}(\mathbb{F})\in {\mathcal{B}}$ for any $i$ . For a ring $B$ , we denote $D(BMod)$ by $D(B)$ , and $D_{B mod}(BMod)$ by $D_{fg}(B)$ (if $B$ is left Noetherian).

2.3

Throughout the paper, let $R$ denote a commutative Noetherian ring. If $R$ is semilocal (resp. local) and $\mathfrak{m}$ its Jacobson radical, then we say that $(R,\mathfrak{m})$ is semilocal (resp. local). We say that $(R,\mathfrak{m},k)$ is semilocal (resp. local) if $(R,\mathfrak{m})$ is semilocal (resp. local) and $k=R/\mathfrak{m}$ .

2.4

We set $\hat{\mathbb{R}}:=\mathbb{R}\cup \{\infty ,-\infty \}$ and consider that $-\infty <\mathbb{R}<\infty$ . As a convention, for a subset $\unicode[STIX]{x1D6E4}$ of $\hat{\mathbb{R}}$ , $\inf \unicode[STIX]{x1D6E4}$ means $\inf (\unicode[STIX]{x1D6E4}\cup \{\infty \})$ , which exists uniquely as an element of $\hat{\mathbb{R}}$ . Similarly for $\sup$ .

2.5

For an ideal $I$ of $R$ and $M\in modR$ , we define

$$\begin{eqnarray}depth_{R}(I,M):=\inf \{i\in \mathbb{Z}\mid Ext_{R}^{i}(R/I,M)\neq 0\},\end{eqnarray}$$

and call it the $I$ -depth of $M$ [Reference MatsumuraMat, Section 16]. It is also called the $M$ -grade of $I$ [Reference Brodmann and SharpBS, (6.2.4)]. When $(R,\mathfrak{m})$ is semilocal, we denote $depth(\mathfrak{m},M)$ by $depth_{R}M$ or $depthM$ , and call it the depth of $M$ .

2.6

For a subset $F$ of $X= SpecR$ , we define $codimF= codim_{X}F$ , the codimension of $F$ in $X$ , by $\inf \{htP\mid P\in F\}$ . So $htI= codimV(I)$ for an ideal $I$ of $R$ . For $M\in modR$ , we define $codimM:= codimSupp_{R}M= htannM$ , where $ann$ denotes the annihilator. For $n\geqslant 0$ , we denote the set $ht^{-1}(n)=\{P\in SpecR\mid htP=n\}$ by $R^{\langle n\rangle }$ . For a subset $\unicode[STIX]{x1D6E4}$ of $\mathbb{Z}$ , $R^{\langle \unicode[STIX]{x1D6E4}\rangle }$ means $ht^{-1}(\unicode[STIX]{x1D6E4})=\bigcup _{n\in \unicode[STIX]{x1D6E4}}R^{\langle n\rangle }$ . Moreover, we use notation such as $R^{\langle {\leqslant}3\rangle }$ , which stands for $R^{\langle \{n\in \mathbb{Z}\mid n\leqslant 3\}\rangle }$ , the set of primes of height at most $3$ . For $M\in modR$ , the set of minimal primes of $M$ is denoted by $MinM$ .

We define $M^{[n]}:=\{P\in SpecR\mid depthM_{P}=n\}$ . Similarly, we use notation such as $M^{[{<}n]}(=\{P\in SpecR\mid depthM_{P}<n\})$ .

2.7

Let $M,N\in modR$ . We say that $M$ satisfies the $(S_{n}^{N})^{R}$ -condition or $(S_{n}^{N})$ -condition if for any $P\in SpecR$ , $depth_{R_{P}}M_{P}\geqslant \min (n,\dim _{R_{P}}N_{P})$ . The $(S_{n}^{R})^{R}$ -condition or $(S_{n}^{R})$ -condition is simply denoted by $(S_{n}^{\prime })^{R}$ or $(S_{n}^{\prime })$ . We say that $M$ satisfies the $(S_{n})^{R}$ -condition or $(S_{n})$ -condition if $M$ satisfies the $(S_{n}^{M})$ -condition. $(S_{n})$ (resp.  $(S_{n}^{\prime })$ ) is equivalent to say that for any $P\in M^{[{<}n]}$ , $M_{P}$ is a Cohen–Macaulay (resp. maximal Cohen–Macaulay) $R_{P}$ -module. That is, $depthM_{P}=\dim M_{P}$ (resp.  $depthM_{P}=\dim R_{P}$ ). We consider that $(S_{n}^{N})^{R}$ is a class of modules, and also write $M\in (S_{n}^{N})^{R}$ (or $M\in (S_{n}^{N})$ ).

Proof. (1) follows from the depth lemma:

$$\begin{eqnarray}\forall P\quad depth_{R_{P}}M_{P}\geqslant \min (depth_{R_{P}}L_{P},depth_{R_{P}}N_{P}),\end{eqnarray}$$

and the fact that maximal Cohen–Macaulay modules are closed under extensions. (2) is similar. ◻

Corollary 2.9. Let

$$\begin{eqnarray}0\rightarrow M\rightarrow L_{n}\rightarrow \cdots \rightarrow L_{1}\end{eqnarray}$$

be an exact sequence in ${\rm mod}R$ , and assume that $L_{i}$ satisfies $(S_{i}^{\prime })$ for $1\leqslant i\leqslant n$ . Then $M$ satisfies $(S_{n}^{\prime })$ .

Lemma 2.10. (cf. [Reference Iyama and WemyssIW, (3.4)])

Let

$$\begin{eqnarray}\mathbb{L}:0\rightarrow L_{s}\xrightarrow[{}]{\unicode[STIX]{x2202}_{s}}L_{s-1}\xrightarrow[{}]{\unicode[STIX]{x2202}_{s-1}}\rightarrow \cdots \rightarrow L_{1}\xrightarrow[{}]{\unicode[STIX]{x2202}_{1}}L_{0}\end{eqnarray}$$

be a complex in ${\rm mod}R$ such that

1. (1) For each $i\in \mathbb{Z}$ with $1\leqslant i\leqslant s$ , $L_{i}\in (S_{i}^{\prime })$ .

2. (2) For each $i\in \mathbb{Z}$ with $1\leqslant i\leqslant s$ , $codimH_{i}(\mathbb{L})\geqslant s-i+1$ .

Then $\mathbb{L}$ is acyclic.

Lemma 2.12. Let $(R,\mathfrak{m})$ be a Noetherian local ring, and $N\in modR$ . Assume that $N$ satisfies the $(S_{n})$ condition. If $P\in MinN$ with $\dim R/P<n$ , then we have

$$\begin{eqnarray}\dim R/P= depthN=\dim N<n.\end{eqnarray}$$

If, moreover, $N$ satisfies $(S_{n}^{\prime })$ , then $depthN=\dim R$ .

2.15

Let $M,N\in modR$ . We say that $M$ satisfies the $(S_{n}^{\prime })_{N}$ -condition, or $M\in (S_{n}^{\prime })_{N}=(S_{n}^{\prime })_{N}^{R}$ , if $M\in (S_{n}^{\prime })$ and $Supp_{R}M\subset Supp_{R}N$ .

2.17

There is another case that $(S_{n})$ implies $(S_{n}^{\prime })$ . An $R$ -module $N$ is said to be full if $Supp_{R}N= SpecR$ . A finitely generated faithful $R$ -module is full.

Proof. We only prove (2). Let $P\in SpecR$ , and $depth_{R_{P}}M_{P}<n$ . Then by Lemma 2.19 and [Reference Brodmann and SharpBS, (6.2.7)], there exists some $Q\in SpecS$ such that $\unicode[STIX]{x1D711}^{-1}(Q)=P$ and

$$\begin{eqnarray}depth_{S_{Q}}M_{Q}=\inf _{\unicode[STIX]{x1D711}^{-1}(Q^{\prime })=P} depth_{S_{Q^{\prime }}}M_{Q^{\prime }}= depth_{S_{P}}M_{P}= depth_{R_{P}}M_{P}<n.\end{eqnarray}$$

Then $htQ= depth_{R_{P}}M_{P}$ . So it suffices to show $htP= htQ$ . By assumption, $R_{P}$ is quasi-unmixed. So $R_{P}$ is equidimensional and universally catenary [Reference MatsumuraMat, (31.6)]. By [Reference GrothendieckGro, (13.3.6)], $htP= htQ$ , as desired.◻

2.21

We say that $R$ satisfies $(R_{n})$ (resp.  $(T_{n})$ ) if $R_{P}$ is regular (resp. Gorenstein) for any prime $P$ of $R$ with $htP\leqslant n$ .

3.1

Let ${\mathcal{A}}$ be an abelian category, and ${\mathcal{C}}$ its additive subcategory closed under direct summands. Let $n\geqslant 0$ . We define

$$\begin{eqnarray}\text{}^{\bot _{n}}{\mathcal{C}}:=\{a\in {\mathcal{A}}\mid Ext_{{\mathcal{A}}}^{i}(a,c)=0,1\leqslant i\leqslant n\}.\end{eqnarray}$$

Let $a\in {\mathcal{A}}$ . A sequence

(1) $$\begin{eqnarray}\mathbb{C}:0\rightarrow a\rightarrow c^{0}\rightarrow c^{1}\rightarrow \cdots \rightarrow c^{n-1}\end{eqnarray}$$

is said to be an $(n,{\mathcal{C}})$ -pushforward if it is exact with $c^{i}\in {\mathcal{C}}$ . If in addition,

$$\begin{eqnarray}\mathbb{C}^{\dagger }:0\leftarrow a^{\dagger }\leftarrow (c^{0})^{\dagger }\leftarrow (c^{1})^{\dagger }\leftarrow \cdots \leftarrow (c^{n-1})^{\dagger }\end{eqnarray}$$

is exact for any $c\in {\mathcal{C}}$ , where $(?)^{\dagger }= Hom_{{\mathcal{A}}}(?,c)$ , we say that $\mathbb{C}$ is a universal $(n,{\mathcal{C}})$ -pushforward.

If $a\in {\mathcal{A}}$ has an $(n,{\mathcal{C}})$ -pushforward, we say that $a$ is an $(n,{\mathcal{C}})$ -syzygy, and we write $a\in Syz(n,{\mathcal{C}})$ . If $a\in {\mathcal{A}}$ has a universal $(n,{\mathcal{C}})$ -pushforward, we say that $a\in UP_{{\mathcal{A}}}(n,{\mathcal{C}})= UP(n,{\mathcal{C}})$ . Obviously, $UP_{{\mathcal{A}}}(n,{\mathcal{C}})\subset Syz_{{\mathcal{A}}}(n,{\mathcal{C}})$ .

3.2

We write ${\mathcal{X}}_{n,m}({\mathcal{C}})={\mathcal{X}}_{n,m}:=\text{}^{\bot _{n}}{\mathcal{C}}\cap UP(m,{\mathcal{C}})$ for $n,m\geqslant 0$ . Also, for $a\neq 0$ , we define

$$\begin{eqnarray}\displaystyle & {\mathcal{C}}dima=\inf \!\big\{m\in \mathbb{Z}_{{\geqslant}0}\mid \text{there is a resolution } & \displaystyle \nonumber\\ \displaystyle & \qquad \qquad \qquad \qquad \qquad 0\rightarrow c_{m}\rightarrow c_{m-1}\rightarrow \cdots \rightarrow c_{0}\rightarrow a\rightarrow 0\big\}. & \displaystyle \nonumber\end{eqnarray}$$

We define ${\mathcal{C}}dim\,0=-\infty$ . We define ${\mathcal{Y}}_{n}({\mathcal{C}})={\mathcal{Y}}_{n}:=\{a\in {\mathcal{A}}\mid {\mathcal{C}}dim\,a<n\}$ . A sequence $\mathbb{E}$ is said to be ${\mathcal{C}}$ -exact if it is exact, and ${\mathcal{A}}(\mathbb{E},c)$ is also exact for each $c\in {\mathcal{C}}$ . Letting a ${\mathcal{C}}$ -exact sequence an exact sequence, ${\mathcal{A}}$ is an exact category, which we denote by ${\mathcal{A}}_{{\mathcal{C}}}$ in order to distinguish it from the abelian category ${\mathcal{A}}$ (with the usual exact sequences).

3.3

Let ${\mathcal{C}}_{0}\subset {\mathcal{A}}$ be a subset. Then $\text{}^{\bot _{n}}{\mathcal{C}}_{0}$ , $UP(n,{\mathcal{C}}_{0})$ , ${\mathcal{X}}_{n,m}({\mathcal{C}}_{0})$ , $\{$ , and ${\mathcal{Y}}_{n}({\mathcal{C}}_{0})={\mathcal{Y}}_{n}$ mean $\text{}^{\bot _{n}}{\mathcal{C}}$ , $UP(n,{\mathcal{C}})$ , ${\mathcal{X}}_{n,m}({\mathcal{C}})$ , ${\mathcal{C}}dim$ , and ${\mathcal{Y}}_{n}({\mathcal{C}})$ , respectively, where ${\mathcal{C}}= add{\mathcal{C}}_{0}$ , the smallest additive subcategory containing ${\mathcal{C}}_{0}$ and closed under direct summands. A ${\mathcal{C}}_{0}$ -exact sequence means a ${\mathcal{C}}$ -exact sequence. A sequence $\mathbb{E}$ in ${\mathcal{A}}$ is ${\mathcal{C}}_{0}$ -exact if and only if it is exact, and for any $c\in {\mathcal{C}}_{0}$ , ${\mathcal{A}}(\mathbb{E},c)$ is exact. If $c\in {\mathcal{A}}$ , $\text{}^{\bot _{n}}c$ , $UP(n,c)$ and so on mean $\text{}^{\bot _{n}} addc$ , $UP(n,addc)$ and so on.

3.4

By definition, any object of ${\mathcal{C}}$ is an injective object in ${\mathcal{A}}_{{\mathcal{C}}}$ .

3.5

Let ${\mathcal{E}}$ be an exact category, and ${\mathcal{I}}$ an additive subcategory of ${\mathcal{E}}$ . Then for $e\in {\mathcal{E}}$ , we define

$$\begin{eqnarray}\displaystyle & & \displaystyle Push_{{\mathcal{E}}}(n,{\mathcal{I}}):=\big\{e\in {\mathcal{E}}\mid \text{There exists an exact sequence}\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad \qquad 0\rightarrow e\rightarrow c^{0}\rightarrow c^{1}\rightarrow \cdots \rightarrow c^{n-1}\text{ with }c^{i}\in {\mathcal{I}}\big\}.\nonumber\end{eqnarray}$$

Note that $Push_{{\mathcal{E}}}(0,{\mathcal{I}})$ is the whole ${\mathcal{E}}$ . Thus $Push_{{\mathcal{A}}_{{\mathcal{C}}}}(n,{\mathcal{C}})= UP_{{\mathcal{A}}}(n,{\mathcal{C}})$ .

If $a\in {\mathcal{E}}$ is a direct summand of an object of ${\mathcal{I}}$ , then $a$ admits an exact sequence

$$\begin{eqnarray}0\rightarrow a\rightarrow c^{0}\rightarrow c^{1}\rightarrow \cdots\end{eqnarray}$$

with $c^{i}\in {\mathcal{I}}$ , and hence $a\in \bigcap _{n\geqslant 0} Push_{{\mathcal{E}}}(n,{\mathcal{I}})$ .

Lemma 3.6. Let ${\mathcal{E}}$ be an exact category. Let ${\mathcal{I}}$ be an additive subcategory of ${\mathcal{E}}$ consisting of injective objects. Let

$$\begin{eqnarray}0\rightarrow a\xrightarrow[{}]{f}a^{\prime }\xrightarrow[{}]{g}a^{\prime \prime }\rightarrow 0\end{eqnarray}$$

be an exact sequence in ${\mathcal{E}}$ and $m\geqslant 0$ . Then

1. (1) If $a\in Push(m,{\mathcal{I}})$ and $a^{\prime \prime }\in Push(m,{\mathcal{I}})$ , then $a^{\prime }\in Push(m,{\mathcal{I}})$ .

2. (2) If $a^{\prime }\in Push(m+1,{\mathcal{I}})$ and $a^{\prime \prime }\in Push(m,{\mathcal{I}})$ , then $a\in Push(m+1,{\mathcal{I}})$ .

3. (3) If $a\in Push(m+1,{\mathcal{I}})$ , $a^{\prime }\in Push(m,{\mathcal{I}})$ , then $a^{\prime \prime }\in Push(m,{\mathcal{I}})$ .

Proof. Let $i:{\mathcal{E}}{\hookrightarrow}{\mathcal{A}}$ be the Gabriel–Quillen embedding [Reference Thomason and TrobaughTT]. We consider that ${\mathcal{E}}$ is a full subcategory of ${\mathcal{A}}$ closed under extensions, and a sequence in ${\mathcal{E}}$ is exact if and only if it is so in ${\mathcal{A}}$ .

We prove (1). We use induction on $m$ . The case that $m=0$ is trivial, and so we assume that $m>0$ . Let

$$\begin{eqnarray}0\rightarrow a\rightarrow c\rightarrow b\rightarrow 0\end{eqnarray}$$

be an exact sequence such that $c\in {\mathcal{I}}$ and $b\in Push(m-1,{\mathcal{I}})$ . Let

$$\begin{eqnarray}0\rightarrow a^{\prime \prime }\rightarrow c^{\prime \prime }\rightarrow b^{\prime \prime }\rightarrow 0\end{eqnarray}$$

be an exact sequence such that $c^{\prime \prime }\in {\mathcal{I}}$ and $b^{\prime \prime }\in Push(m-1,{\mathcal{I}})$ . As ${\mathcal{C}}(a^{\prime },c)\rightarrow {\mathcal{C}}(a,c)$ is surjective, we can form a commutative diagram with exact rows and columns

in ${\mathcal{A}}$ . As ${\mathcal{E}}$ is closed under extensions in ${\mathcal{A}}$ , this diagram is a diagram in ${\mathcal{E}}$ . By induction hypothesis, $b^{\prime }\in Push(m-1,{\mathcal{I}})$ . Hence $a^{\prime }\in Push(m,{\mathcal{I}})$ .

We prove (2). Let $0\rightarrow a^{\prime }\rightarrow c\rightarrow b^{\prime }\rightarrow 0$ be an exact sequence in ${\mathcal{E}}$ such that $c\in {\mathcal{I}}$ and $b^{\prime }\in Push(m,{\mathcal{I}})$ . Then we have a commutative diagram in ${\mathcal{E}}$ with exact rows and columns

Applying (1), which we have already proved, $b\in Push(m,{\mathcal{I}})$ , since $a^{\prime \prime }$ and $b^{\prime }$ lie in $Push(m,{\mathcal{I}})$ . So $a\in Push(m+1,{\mathcal{I}})$ , as desired.

We prove (3). Let $0\rightarrow a\rightarrow c\rightarrow b\rightarrow 0$ be an exact sequence in ${\mathcal{E}}$ such that $c\in {\mathcal{I}}$ and $b\in Push(m,{\mathcal{I}})$ . Take the push-out diagram

Then $u\in Push(m,{\mathcal{I}})$ by (1), which we have already proved. Since $c\in {\mathcal{I}}$ , the middle row splits. Then by the exact sequence $0\rightarrow a^{\prime \prime }\rightarrow u\rightarrow c\rightarrow 0$ and (2), we have that $a^{\prime \prime }\in Push(m,{\mathcal{I}})$ , as desired.◻

Proof. The ‘if’ part is obvious by Lemma 3.6(1), considering the exact sequence

(2) $$\begin{eqnarray}0\rightarrow a\rightarrow a\oplus a^{\prime }\rightarrow a^{\prime }\rightarrow 0.\end{eqnarray}$$

We prove the ‘only if’ part by induction on $m$ . If $m=0$ , then there is nothing to prove. Let $m>0$ . Then by induction hypothesis, $a^{\prime }\in Push(m-1,{\mathcal{I}})$ . Then applying Lemma 3.6(2) to the exact sequence (2), we have that $a\in Push(m,{\mathcal{I}})$ . $a^{\prime }\in Push(m,{\mathcal{I}})$ is proved similarly.◻

Corollary 3.8. Let

$$\begin{eqnarray}0\rightarrow a\xrightarrow[{}]{f}a^{\prime }\xrightarrow[{}]{g}a^{\prime \prime }\rightarrow 0\end{eqnarray}$$

be a ${\mathcal{C}}$ -exact sequence in ${\mathcal{A}}$ and $m\geqslant 0$ . Then

1. (1) If $a\in UP(m,{\mathcal{C}})$ and $a^{\prime \prime }\in UP(m,{\mathcal{C}})$ , then $a^{\prime }\in UP(m,{\mathcal{C}})$ .

2. (2) If $a^{\prime }\in UP(m+1,{\mathcal{C}})$ and $a^{\prime \prime }\in UP(m,{\mathcal{C}})$ , then $a\in UP(m+1,{\mathcal{C}})$ .

3. (3) If $a\in UP(m+1,{\mathcal{C}})$ , $a^{\prime }\in UP(m,{\mathcal{C}})$ , then $a^{\prime \prime }\in UP(m,{\mathcal{C}})$ .

3.9

We define $\text{}^{\bot _{}}{\mathcal{C}}=\text{}^{\bot _{\infty }}{\mathcal{C}}:=\bigcap _{i\geqslant 0}\text{}^{\bot _{i}}{\mathcal{C}}$ and $UP(\infty ,{\mathcal{C}}):=\bigcap _{j\geqslant 0} UP(j,{\mathcal{C}})$ . Obviously, ${\mathcal{C}}\subset UP(\infty ,{\mathcal{C}})$ .

Lemma 3.10. We have

$$\begin{eqnarray}\displaystyle & & \displaystyle UP(\infty ,{\mathcal{C}})=\big\{a\in {\mathcal{A}}\mid \text{There exists some }{\mathcal{C}}\text{-exact sequence }\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \qquad 0\rightarrow a\rightarrow c^{0}\rightarrow c^{1}\rightarrow c^{2}\rightarrow \cdots \text{ with }c^{i}\in {\mathcal{C}}\text{for }i\geqslant 0\big\}.\nonumber\end{eqnarray}$$

Proof. Let $a\in UP(\infty ,{\mathcal{C}})$ , and take any ${\mathcal{C}}$ -exact sequence

$$\begin{eqnarray}0\rightarrow a\rightarrow c^{0}\rightarrow a^{1}\rightarrow 0\end{eqnarray}$$

with $c^{0}\in {\mathcal{C}}$ . Then $a^{1}\in UP(\infty ,{\mathcal{C}})$ by Corollary 3.8, and we can continue infinitely.◻

3.11

We define ${\mathcal{Y}}_{\infty }:=\bigcup _{i\geqslant 0}{\mathcal{Y}}_{i}$ . So $a\in {\mathcal{Y}}_{\infty }$ if and only if ${\mathcal{C}}dim\,a<\infty$ . We also define ${\mathcal{X}}_{i,j}:=\text{}^{\bot _{i}}{\mathcal{C}}\cap UP(j,{\mathcal{C}})$ for $0\leqslant i,j\leqslant \infty$ .

3.12

Let $0\leqslant i,j\leqslant \infty$ . We say that $a\in {\mathcal{A}}$ lies in ${\mathcal{Z}}_{i,j}$ if there is a short exact sequence

$$\begin{eqnarray}0\rightarrow y\rightarrow x\rightarrow a\rightarrow 0\end{eqnarray}$$

in ${\mathcal{A}}$ such that $x\in {\mathcal{X}}_{i,j}$ and $y\in {\mathcal{Y}}_{i}$ .

3.13

We define $\infty \pm r=\infty$ for $r\in \mathbb{R}$ .

Proof. By assumption, there is an exact sequence

$$\begin{eqnarray}0\rightarrow y\xrightarrow[{}]{\unicode[STIX]{x1D704}}x^{\prime }\xrightarrow[{}]{\unicode[STIX]{x1D711}}z\rightarrow 0\end{eqnarray}$$

such that ${\mathcal{C}}dim\,y<i$ and $x^{\prime }\in {\mathcal{X}}_{i,j}$ . As $j\geqslant 1$ , there is an ${\mathcal{C}}$ -exact sequence

$$\begin{eqnarray}0\rightarrow x^{\prime }\xrightarrow[{}]{h}c\rightarrow x^{\prime \prime \prime }\rightarrow 0\end{eqnarray}$$

such that $c\in {\mathcal{C}}$ . Then we have a commutative diagram with exact rows and columns

As the top row is exact, $y\in {\mathcal{Y}}_{i}$ , and $c\in {\mathcal{C}}$ , $y^{\prime }\in {\mathcal{Y}}_{i+1}$ . By assumption, $c\in {\mathcal{X}}_{i+1,\infty }$ and $x\in {\mathcal{X}}_{i+1,j-1}$ . So $c\oplus x\in {\mathcal{X}}_{i+1,j-1}$ . As the middle row is ${\mathcal{C}}$ -exact and $x^{\prime }\in {\mathcal{X}}_{i,j}$ , we have that $x^{\prime \prime }\in {\mathcal{X}}_{i+1,j-1}$ by Corollary 3.8. The right column shows that $z^{\prime }\in {\mathcal{Z}}_{i+1,j-1}$ , as desired.◻

Lemma 3.15. Let $0\leqslant i,j\leqslant \infty$ , and assume that $i\geqslant 1$ and ${\mathcal{C}}\subset \text{}^{\bot _{i}}{\mathcal{C}}$ . Let

(3) $$\begin{eqnarray}0\rightarrow z\xrightarrow[{}]{f}x\xrightarrow[{}]{g}z^{\prime }\rightarrow 0\end{eqnarray}$$

be a short exact sequence in ${\mathcal{A}}$ with $z^{\prime }\in {\mathcal{Z}}_{i,j}$ and $x\in {\mathcal{X}}_{i,j+1}$ . Then $z\in {\mathcal{Z}}_{i-1,j+1}$ .

Proof. Take an exact sequence $0\rightarrow y^{\prime }\rightarrow x^{\prime \prime }\xrightarrow[{}]{h}z^{\prime }\rightarrow 0$ such that $x^{\prime \prime }\in {\mathcal{X}}_{i,j}$ and $y^{\prime }\in {\mathcal{Y}}_{i}$ . Taking the pull-back of (3) by $h$ , we get a commutative diagram with exact rows and columns

By induction, we can prove easily that $\text{}^{\bot _{i}}{\mathcal{C}}\subset \text{}^{\bot _{i+1-l}}{\mathcal{Y}}_{l}$ . In particular, $\text{}^{\bot _{i}}{\mathcal{C}}\subset \text{}^{\bot _{1}}{\mathcal{Y}}_{i}$ , and $Ext_{{\mathcal{A}}}^{1}(x,y^{\prime })=0$ . Hence the middle column splits, and we can replace $a$ by $x\oplus y^{\prime }$ . By the definition of ${\mathcal{Y}}_{i}$ , there is an exact sequence

$$\begin{eqnarray}0\rightarrow y\rightarrow c\rightarrow y^{\prime }\rightarrow 0\end{eqnarray}$$

of ${\mathcal{A}}$ such that $y\in {\mathcal{Y}}_{i-1}$ and $c\in {\mathcal{C}}$ . Then adding $1_{x}$ to this sequence, we get

$$\begin{eqnarray}0\rightarrow y\rightarrow x\oplus c\rightarrow x\oplus y^{\prime }\rightarrow 0\end{eqnarray}$$

is exact. Pulling back this exact sequence with $j:z\rightarrow a=x\oplus y^{\prime }$ , we get a commutative diagram with exact rows and columns

As $x^{\prime \prime }\in \text{}^{\bot _{1}}{\mathcal{C}}$ , the middle column is ${\mathcal{C}}$ -exact. As $x^{\prime \prime }\in {\mathcal{X}}_{i,j}$ and $x\oplus c\in {\mathcal{X}}_{i,j+1}$ , we have that $x^{\prime }\in {\mathcal{X}}_{i-1,j+1}$ . As the top row shows, $z\in {\mathcal{Z}}_{i-1,j+1}$ , as desired. ◻

Proof. (1) $\Rightarrow$ (2). There is an exact sequence $0\rightarrow y\rightarrow x_{0}\xrightarrow[{}]{\unicode[STIX]{x1D700}}z\rightarrow 0$ with $x_{0}\in {\mathcal{X}}_{n,m}$ and $y\in {\mathcal{Y}}_{n}$ . So there is an exact sequence

$$\begin{eqnarray}0\rightarrow x_{n}\xrightarrow[{}]{d_{n}}x_{n-1}\xrightarrow[{}]{d_{n-1}}\cdots \xrightarrow[{}]{d_{2}}x_{1}\rightarrow y\rightarrow 0\end{eqnarray}$$

with $x_{i}\in {\mathcal{C}}$ for $1\leqslant i\leqslant n$ . As ${\mathcal{C}}\subset {\mathcal{X}}_{n,\infty }$ , we are done.

(2) $\Rightarrow$ (1). Let $z_{i}= Imd_{i}$ for $i=1,\ldots ,n$ , and $z_{0}:=z$ . Then by descending induction on $i$ , we can prove $z_{i}\in {\mathcal{Z}}_{n-i,m+i}$ for $i=n,n-1,\ldots ,0$ , using Lemma 3.14 easily.

(1) $\Rightarrow$ (3) is also proved easily, using Lemma 3.15.

(3) $\Rightarrow$ (2) is trivial.◻

4.1

In the rest of this paper, let $\unicode[STIX]{x1D6EC}$ be a module-finite $R$ -algebra, which need not be commutative. A $\unicode[STIX]{x1D6EC}$ -bimodule means a $\unicode[STIX]{x1D6EC}\otimes _{R}\unicode[STIX]{x1D6EC}^{op}$ -module. Let $C\in mod\unicode[STIX]{x1D6EC}$ be fixed. Set $\unicode[STIX]{x1D6E4}:= End_{\unicode[STIX]{x1D6EC}^{op}}C$ . Note that $\unicode[STIX]{x1D6E4}$ is also a module-finite $R$ -algebra. We denote $(?)^{\dagger }:= Hom_{\unicode[STIX]{x1D6EC}^{op}}(?,C): mod\unicode[STIX]{x1D6EC}\rightarrow (\unicode[STIX]{x1D6E4}{\rm mod})^{op}$ , and $(?)^{\ddagger }:= Hom_{\unicode[STIX]{x1D6E4}}(?,C):\unicode[STIX]{x1D6E4}{\rm mod}\rightarrow ({\rm mod}\unicode[STIX]{x1D6EC})^{op}$ .

4.2

We denote $Syz_{{\rm mod}\unicode[STIX]{x1D6EC}}(n,C)$ , $UP_{{\rm mod}\unicode[STIX]{x1D6EC}}(n,C)$ , and $Cdim_{{\rm mod}\unicode[STIX]{x1D6EC}}M$ respectively by $Syz_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ , $UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ , and $Cdim_{\unicode[STIX]{x1D6EC}^{op}}M$ .

4.3

Note that for $M\in mod\unicode[STIX]{x1D6EC}$ and $N\in \unicode[STIX]{x1D6E4}{\rm mod}$ , we have standard isomorphisms

(5) $$\begin{eqnarray}Hom_{\unicode[STIX]{x1D6EC}^{op}}(M,N^{\ddagger })\cong Hom_{\unicode[STIX]{x1D6E4}\otimes _{R}\unicode[STIX]{x1D6EC}^{op}}(N\otimes _{R}M,C)\cong Hom_{\unicode[STIX]{x1D6E4}}(N,M^{\dagger }).\end{eqnarray}$$

The first isomorphism sends $f:M\rightarrow N^{\ddagger }$ to the map $(n\otimes m\mapsto f(m)(n))$ . Its inverse is given by $g:N\otimes _{R}M\rightarrow C$ to $(m\mapsto (n\mapsto g(n\otimes m)))$ . This shows that $(?)^{\dagger }$ has $((?)^{\ddagger })^{op}:(\unicode[STIX]{x1D6E4}{\rm mod})^{op}\rightarrow mod\unicode[STIX]{x1D6EC}$ as a right adjoint. Hence $((?)^{\dagger })^{op}$ is right adjoint to $(?)^{\ddagger }$ . We denote the unit of adjunction $\operatorname{Id}\rightarrow (?)^{\dagger \ddagger }=(?)^{\ddagger }(?)^{\dagger }$ by $\unicode[STIX]{x1D706}$ . Note that for $M\in mod\unicode[STIX]{x1D6EC}$ , the map $\unicode[STIX]{x1D706}_{M}:M\rightarrow M^{\dagger \ddagger }$ is given by $\unicode[STIX]{x1D706}_{M}(m)(\unicode[STIX]{x1D713})=\unicode[STIX]{x1D713}(m)$ for $m\in M$ and $\unicode[STIX]{x1D713}\in M^{\dagger }= Hom_{\unicode[STIX]{x1D6EC}^{op}}(M,C)$ . We denote the unit of adjunction $N\rightarrow N^{\ddagger \dagger }$ by $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{N}$ for $N\in \unicode[STIX]{x1D6E4}{\rm mod}$ . When we view $\unicode[STIX]{x1D707}$ as a morphism $N^{\ddagger \dagger }\rightarrow N$ (in the opposite category $(\unicode[STIX]{x1D6E4}{\rm mod})^{op}$ ), then it is the counit of adjunction.

Proof. We have a commutative diagram

with exact rows.

We only prove (3). We may assume that $n\geqslant 1$ . So $\unicode[STIX]{x1D706}_{M}$ is an isomorphism and $\unicode[STIX]{x1D706}_{L}$ is injective. By the five lemma, $\unicode[STIX]{x1D706}_{N}$ is injective, and the case that $n=1$ has been done. If $n\geqslant 2$ , then $\unicode[STIX]{x1D706}_{L}$ is also an isomorphism and $Ext_{\unicode[STIX]{x1D6E4}}^{1}(M^{\dagger },C)=0$ , and so $\unicode[STIX]{x1D706}_{N}$ is an isomorphism. Moreover, for $1\leqslant i\leqslant n-2$ , $Ext_{\unicode[STIX]{x1D6E4}}^{i}(L^{\dagger },C)$ and $Ext_{\unicode[STIX]{x1D6E4}}^{i+1}(M^{\dagger },C)$ vanish. so $Ext_{\unicode[STIX]{x1D6E4}}^{i}(N^{\dagger },C)=0$ for $1\leqslant i\leqslant n-2$ , and hence $N\in TF(n,C)$ .

(1) and (2) are also proved similarly. ◻

Proof. If $n=0$ , then $Syz_{\unicode[STIX]{x1D6EC}^{op}}(n,C)= TF_{\unicode[STIX]{x1D6EC}^{op}}(0,C)= UP_{\unicode[STIX]{x1D6EC}^{op}}(0,C)= mod\unicode[STIX]{x1D6EC}$ . So we may assume that $n\geqslant 1$ .

Let $M\in Syz_{\unicode[STIX]{x1D6EC}^{op}}(1,C)$ . Then there is an injection $\unicode[STIX]{x1D711}:M\rightarrow N$ with $N\in addC$ . Then

is a commutative diagram. So $\unicode[STIX]{x1D706}_{M}$ is injective, and $M\in TF_{\unicode[STIX]{x1D6EC}^{op}}(1,C)$ . This shows $UP_{\unicode[STIX]{x1D6EC}^{op}}(1,C)\subset Syz_{\unicode[STIX]{x1D6EC}^{op}}(1,C)\subset TF_{\unicode[STIX]{x1D6EC}^{op}}(1,C)$ . So (2) $\Rightarrow$ (1).

We prove (2). First, we prove $UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)\subset TF_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ for $n\geqslant 1$ . We use induction on $n$ . The case $n=1$ is already done above.

Let $n\geqslant 2$ and $M\in UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ . Then by the definition of $UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ , there is a $C$ -exact sequence

$$\begin{eqnarray}0\rightarrow M\rightarrow L\rightarrow N\rightarrow 0\end{eqnarray}$$

such that $L\in addC$ and $N\in UP_{\unicode[STIX]{x1D6EC}^{op}}(n-1,C)$ . By induction hypothesis, $N\in TF_{\unicode[STIX]{x1D6EC}^{op}}(n-1,C)$ . Hence $M\in TF_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ by Lemma 4.6. We have proved that $UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)\subset TF_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ .

Next we show that $TF_{\unicode[STIX]{x1D6EC}^{op}}(n,C)\subset UP_{\unicode[STIX]{x1D6EC}^{op}}(n,C)$ for $n\geqslant 1$ . We use induction on $n$ .

Let $n=1$ . Let $\unicode[STIX]{x1D70C}:F\rightarrow M^{\dagger }$ be any surjective $\unicode[STIX]{x1D6E4}$ -linear map with $F\in add\unicode[STIX]{x1D6E4}$ . Then the map $\unicode[STIX]{x1D70C}^{\prime }:M\rightarrow F^{\ddagger }$ which corresponds to $\unicode[STIX]{x1D70C}$ by the adjunction (5) is

$$\begin{eqnarray}\unicode[STIX]{x1D70C}^{\prime }:M\xrightarrow[{}]{\unicode[STIX]{x1D706}_{M}}M^{\dagger \ddagger }\xrightarrow[{}]{\unicode[STIX]{x1D70C}^{\ddagger }}F^{\ddagger },\end{eqnarray}$$

which is injective by assumption. Then $\unicode[STIX]{x1D70C}$ is the composite

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:F\xrightarrow[{}]{\unicode[STIX]{x1D707}_{F}}F^{\ddagger \dagger }\xrightarrow[{}]{(\unicode[STIX]{x1D70C}^{\prime })^{\dagger }}M^{\dagger },\end{eqnarray}$$

which is a surjective map by assumption. So $(\unicode[STIX]{x1D70C}^{\prime })^{\dagger }$ is also surjective, and hence $\unicode[STIX]{x1D70C}^{\prime }:M\rightarrow F^{\ddagger }$ gives a $(1,C)$ -universal pushforward.

Now let $n\geqslant 2$ . By what we have proved, $M$ has a $(1,C)$ -universal pushforward $h:M\rightarrow L$ . Let $N= Cokerh$ . Then we have a $C$ -exact sequence

$$\begin{eqnarray}0\rightarrow M\rightarrow L\rightarrow N\rightarrow 0\end{eqnarray}$$

with $L\in addC$ . As $M\in TF(n,C)$ , $N\in TF(n-1,C)$ by Lemma 4.6. By induction hypothesis, $N\in UP(n-1,C)$ . So by the definition of $UP(n,C)$ , we have that $M\in UP(n,C)$ , as desired.◻

Proof. Let

$$\begin{eqnarray}F_{1}\xrightarrow[{}]{h}F_{0}\rightarrow N\rightarrow 0\end{eqnarray}$$

be an exact sequence in $\unicode[STIX]{x1D6E4}{\rm mod}$ such that $F_{i}\in add\unicode[STIX]{x1D6E4}$ . Then

$$\begin{eqnarray}0\rightarrow N^{\ddagger }\rightarrow F_{0}^{\ddagger }\xrightarrow[{}]{h^{\ddagger }}F_{1}^{\ddagger }\end{eqnarray}$$

is exact, and $F_{i}^{\ddagger }\in addC$ . This shows that $N^{\ddagger }\in Syz(2,C)$ .◻

4.9

We denote by $(S_{n}^{\prime })_{C}=(S_{n}^{\prime })_{C}^{\unicode[STIX]{x1D6EC}^{op},R}$ the class of $M\in mod\unicode[STIX]{x1D6EC}$ such that $M$ viewed as an $R$ -module lies in $(S_{n}^{\prime })_{C}^{R}$ ; see (2.15).

4.11

For an additive category ${\mathcal{C}}$ and its additive subcategory ${\mathcal{X}}$ , we denote by ${\mathcal{C}}/{\mathcal{X}}$ the quotient of ${\mathcal{C}}$ divided by the ideal consisting of morphisms which factor through objects of ${\mathcal{X}}$ .

4.12

For each $M\in mod\unicode[STIX]{x1D6EC}$ , take a presentation

(6) $$\begin{eqnarray}\mathbb{F}(M):F_{1}(M)\xrightarrow[{}]{\unicode[STIX]{x2202}}F_{0}(M)\xrightarrow[{}]{\unicode[STIX]{x1D700}}M\rightarrow 0\end{eqnarray}$$

with $F_{i}\in add\unicode[STIX]{x1D6EC}_{\unicode[STIX]{x1D6EC}}$ . We denote

$$\begin{eqnarray}Coker(\unicode[STIX]{x2202}^{\dagger })= Coker(1_{C}\otimes \unicode[STIX]{x2202}^{t})=C\otimes _{\unicode[STIX]{x1D6EC}}TrM\end{eqnarray}$$

by $Tr_{C}M$ , where $(?)^{t}= Hom_{\unicode[STIX]{x1D6EC}^{op}}(?,\unicode[STIX]{x1D6EC})$ and $Tr$ is the transpose (see [Reference Assem, Simson and SkowrońskiASS, (V.2)]), and we call it the $C$ -transpose of $M$ . $Tr_{C}$ is an additive functor from $\operatorname{mod̲}\unicode[STIX]{x1D6EC}:= mod\unicode[STIX]{x1D6EC}/add\unicode[STIX]{x1D6EC}_{\unicode[STIX]{x1D6EC}}$ to $\unicode[STIX]{x1D6E4}_{C}\operatorname{mod̲}:=\unicode[STIX]{x1D6E4}{\rm mod}/addC$ .

Proof. (0) is obvious by assumption.

We consider that $\mathbb{F}(M)$ is a complex with $M$ at degree zero. Then consider

$$\begin{eqnarray}\mathbb{Q}(M):=\mathbb{F}(M)^{\dagger }[2]:F_{1}(M)^{\dagger }\xleftarrow[{}]{\unicode[STIX]{x2202}^{\dagger }}F_{0}(M)^{\dagger }\xleftarrow[{}]{\unicode[STIX]{x1D700}^{\dagger }}M^{\dagger }\leftarrow 0\end{eqnarray}$$

where $F_{1}(M)^{\dagger }$ is at degree zero. As this complex is quasi-isomorphic to $Tr_{C}(M)$ , there is a spectral sequence

$$\begin{eqnarray}E_{1}^{p,q}= Ext_{\unicode[STIX]{x1D6E4}}^{q}(\mathbb{Q}(M)^{-p},C)\Rightarrow Ext_{\unicode[STIX]{x1D6E4}}^{p+q}(Tr_{C}M,C).\end{eqnarray}$$

In general, $Ker\unicode[STIX]{x1D706}_{M}=E_{2}^{1,0}\cong E_{\infty }^{1,0}\subset E^{1}$ . If $n\geqslant 1$ , then $E_{1}^{0,1}=0$ , and $E_{\infty }^{1,0}=E^{1}$ . Moreover, as $E_{1}^{0,1}=0$ , $Coker\unicode[STIX]{x1D706}_{M}\cong E_{2}^{2,0}\cong E_{\infty }^{2,0}\subset E^{2}$ . So (1) follows.

If $n\geqslant 2$ , then $E_{1}^{0,2}=E_{1}^{1,1}=0$ by assumption, so $E_{\infty }^{2,0}=E^{2}$ , and (i) of (2) follows. Note that $E_{1}^{p,q}=0$ for $p\geqslant 3$ . Moreover, $E_{1}^{p,q}=0$ for $p=0,1$ and $1\leqslant q\leqslant n$ . So for $1\leqslant i\leqslant n-1$ , we have

$$\begin{eqnarray}E_{1}^{2,i}\cong E_{\infty }^{2,i}{\hookrightarrow}E^{i+2},\end{eqnarray}$$

and the inclusion is an isomorphism if $1\leqslant i\leqslant n-2$ . So (ii) and (iii) of (2) follow.◻

5.1

Let $R=(R,\mathfrak{m})$ be semilocal, where $\mathfrak{m}$ is the Jacobson radical of $R$ .

5.2

We say that a dualizing complex $\mathbb{I}$ over $R$ is normalized if for any maximal ideal $\mathfrak{n}$ of $R$ , $Ext_{R}^{0}(R/\mathfrak{n},\mathbb{I})\neq 0$ . We follow the definition of [Reference HartshorneHart].

5.3

For a left or right $\unicode[STIX]{x1D6EC}$ -module $M$ , $\dim M$ or $\dim _{\unicode[STIX]{x1D6EC}}M$ denotes the dimension $\dim _{R}M$ of $M$ , which is independent of the choice of $R$ . We call $depth_{R}(\mathfrak{m},M)$ , which is also independent of $R$ , the global depth, $\unicode[STIX]{x1D6EC}$ -depth, or depth of $M$ , and denote it by $depth_{\unicode[STIX]{x1D6EC}}M$ or $depthM$ . $M$ is called globally Cohen–Macaulay or GCM for short, if $\dim M= depthM$ . $M$ is GCM if and only if it is Cohen–Macaulay as an $R$ -module, and all the maximal ideals of $R/ann_{R}M$ have the same height. This notion is independent of $R$ , and depends only on $\unicode[STIX]{x1D6EC}$ and $M$ . $M$ is called a globally maximal Cohen–Macaulay (GMCM for short) if $\dim \unicode[STIX]{x1D6EC}= depthM$ . We say that the algebra $\unicode[STIX]{x1D6EC}$ is GCM if the $\unicode[STIX]{x1D6EC}$ -module $\unicode[STIX]{x1D6EC}$ is GCM. However, in what follows, if $R$ happens to be local, then GCM and Cohen–Macaulay (resp. GMSM and maximal Cohen–Macaulay) (over $R$ ) are the same thing, and used interchangeably.

5.4

Assume that $(R,\mathfrak{m})$ is complete semilocal, and $\unicode[STIX]{x1D6EC}\neq 0$ . Let $\mathbb{I}$ be a normalized dualizing complex of $R$ . The lowest nonvanishing cohomology group $Ext_{R}^{-s}(\unicode[STIX]{x1D6EC},\mathbb{I})$ ( $Ext_{R}^{i}(\unicode[STIX]{x1D6EC},\mathbb{I})=0$ for $i<-s$ ) is denoted by $K_{\unicode[STIX]{x1D6EC}}$ , and is called the canonical module of $\unicode[STIX]{x1D6EC}$ . Note that $K_{\unicode[STIX]{x1D6EC}}$ is a $\unicode[STIX]{x1D6EC}$ -bimodule. Hence it is also a $\unicode[STIX]{x1D6EC}^{op}$ -bimodule. In this sense, $K_{\unicode[STIX]{x1D6EC}}=K_{\unicode[STIX]{x1D6EC}^{op}}$ . If $\unicode[STIX]{x1D6EC}=0$ , then we define $K_{\unicode[STIX]{x1D6EC}}=0$ .

5.5

Let $S$ be the center of $\unicode[STIX]{x1D6EC}$ . Then $S$ is module-finite over $R$ , and $\mathbb{I}_{S}=\mathbf{R}\text{Hom}_{R}(S,\mathbb{I})$ is a normalized dualizing complex of $S$ . This shows that $\mathbf{R}\text{Hom}_{R}(\unicode[STIX]{x1D6EC},\mathbb{I})\cong \mathbf{R}\text{Hom}_{S}(\unicode[STIX]{x1D6EC},\mathbb{I}_{S})$ , and hence the definition of $K_{\unicode[STIX]{x1D6EC}}$ is also independent of $R$ .

Lemma 5.6. The number $s$ in (5.4) is nothing but $d:=\dim \unicode[STIX]{x1D6EC}$ . Moreover,

$$\begin{eqnarray}Ass_{R}K_{\unicode[STIX]{x1D6EC}}= Assh_{R}\unicode[STIX]{x1D6EC}:=\{P\in Min_{R}\unicode[STIX]{x1D6EC}\mid \dim R/P=\dim \unicode[STIX]{x1D6EC}\}.\end{eqnarray}$$

5.8

Assume that $(R,\mathfrak{m})$ is semilocal which need not be complete. We say that a finitely generated $\unicode[STIX]{x1D6EC}$ -bimodule $K$ is a canonical module of $\unicode[STIX]{x1D6EC}$ if $\hat{K}$ is isomorphic to the canonical module $K_{\hat{\unicode[STIX]{x1D6EC}}}$ as a $\hat{\unicode[STIX]{x1D6EC}}$ -bimodule, where $\hat{?}$ denotes the $\mathfrak{m}$ -adic completion. It is unique up to isomorphisms, and denoted by $K_{\unicode[STIX]{x1D6EC}}$ . We say that $K\in mod\unicode[STIX]{x1D6EC}$ is a right canonical module of $\unicode[STIX]{x1D6EC}$ if $\hat{K}$ is isomorphic to $K_{\hat{\unicode[STIX]{x1D6EC}}}$ in ${\rm mod}\hat{\unicode[STIX]{x1D6EC}}$ . If $K_{\unicode[STIX]{x1D6EC}}$ exists, then $K$ is a right canonical module if and only if $K\cong K_{\unicode[STIX]{x1D6EC}}$ in ${\rm mod}\unicode[STIX]{x1D6EC}$ .

These definitions are independent of $R$ , in the sense that the (right) canonical module over $R$ and that over the center of $\unicode[STIX]{x1D6EC}$ are the same thing. The right canonical module of $\unicode[STIX]{x1D6EC}^{op}$ is called the left canonical module. A $\unicode[STIX]{x1D6EC}$ -bimodule $\unicode[STIX]{x1D714}$ is said to be a weakly canonical bimodule if $_{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D714}$ is left canonical, and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6EC}}$ is right canonical. The canonical module $K_{\unicode[STIX]{x1D6EC}^{op}}$ of $\unicode[STIX]{x1D6EC}^{op}$ is canonically identified with $K_{\unicode[STIX]{x1D6EC}}$ .

5.9

If $R$ has a normalized dualizing complex $\mathbb{I}$ , then $\hat{\mathbb{I}}$ is a normalized dualizing complex of $\hat{R}$ , and so it is easy to see that $K_{\unicode[STIX]{x1D6EC}}$ exists and agrees with $Ext^{-d}(\unicode[STIX]{x1D6EC},\mathbb{I})$ , where $d=\dim \unicode[STIX]{x1D6EC}(:=\dim _{R}\unicode[STIX]{x1D6EC})$ . In this case, for any $P\in SpecR$ , $\mathbb{I}_{P}$ is a dualizing complex of $R_{P}$ . So if $R$ has a dualizing complex and $(K_{\unicode[STIX]{x1D6EC}})_{P}\neq 0$ , then $(K_{\unicode[STIX]{x1D6EC}})_{P}$ , which is the lowest nonzero cohomology group of $\mathbf{R}\text{Hom}_{R_{P}}(\unicode[STIX]{x1D6EC}_{P},\mathbb{I}_{P})$ , is the $R_{P}$ -canonical module of $\unicode[STIX]{x1D6EC}_{P}$ . See also Theorem 7.5 below.

5.11

A $\unicode[STIX]{x1D6EC}$ -module $M$ is said to be $\unicode[STIX]{x1D6EC}$ -full over $R$ if $Supp_{R}M= Supp_{R}\unicode[STIX]{x1D6EC}$ .

5.13

Let $(R,\mathfrak{m})$ be local, and $\mathbb{I}$ be a normalized dualizing complex. By the local duality,

$$\begin{eqnarray}K_{\unicode[STIX]{x1D6EC}}^{\vee }= Ext^{-d}(\unicode[STIX]{x1D6EC},\mathbb{I})^{\vee }\cong H_{\mathfrak{ m}}^{d}(\unicode[STIX]{x1D6EC})\end{eqnarray}$$

(as $\unicode[STIX]{x1D6EC}$ -bimodules), where $E_{R}(R/\mathfrak{m})$ is the injective hull of the $R$ -module $R/\mathfrak{m}$ , and $(?)^{\vee }$ is the Matlis dual $Hom_{R}(?,E_{R}(R/\mathfrak{m}))$ .

5.14

Let $(R,\mathfrak{m})$ be semilocal, and $\mathbb{I}$ be a normalized dualizing complex. Note that $\mathbf{R}\text{Hom}_{R}(?,\mathbb{I})$ induces a contravariant equivalence between $D_{fg}(\unicode[STIX]{x1D6EC}^{op})$ and $D_{fg}(\unicode[STIX]{x1D6EC})$ . Let $\mathbb{J}\in D_{fg}(\unicode[STIX]{x1D6EC}\otimes _{R}\unicode[STIX]{x1D6EC}^{op})$ be $\mathbf{R}\text{Hom}_{R}(\unicode[STIX]{x1D6EC},\mathbb{I})$ .

$$\begin{eqnarray}\mathbf{R}\text{Hom}_{R}(?,\mathbb{I}):D_{fg}(\unicode[STIX]{x1D6EC}^{op})\rightarrow D_{ fg}(\unicode[STIX]{x1D6EC})\end{eqnarray}$$

is identified with

$$\begin{eqnarray}\mathbf{R}\text{Hom}_{\unicode[STIX]{x1D6EC}^{op}}(?,\mathbf{R}\text{Hom}_{R}(\text{}_{\unicode[STIX]{x1D6EC}}\unicode[STIX]{x1D6EC}_{R},\mathbb{I}))=\mathbf{R}\text{Hom}_{\unicode[STIX]{x1D6EC}^{op}}(?,\mathbb{J})\end{eqnarray}$$

and similarly,

$$\begin{eqnarray}\mathbf{R}\text{Hom}_{R}(?,\mathbb{I}):D_{fg}(\unicode[STIX]{x1D6EC})\rightarrow D_{fg}(\unicode[STIX]{x1D6EC}^{op})\end{eqnarray}$$

is identified with $\mathbf{R}\text{Hom}_{\unicode[STIX]{x1D6EC}}(?,\mathbb{J})$ . Note that a left or right $\unicode[STIX]{x1D6EC}$ -module $M$ is GMCM if and only if $\mathbf{R}\text{Hom}_{R}(M,\mathbb{I})$ is concentrated in degree $-d$ , where $d=\dim \unicode[STIX]{x1D6EC}$ .

5.15

$\mathbb{J}$ above is a dualizing complex of $\unicode[STIX]{x1D6EC}$ in the sense of Yekutieli [Reference YekutieliYek, (3.3)].

5.16

$\unicode[STIX]{x1D6EC}$ is GCM if and only if $K_{\unicode[STIX]{x1D6EC}}[d]\rightarrow \mathbb{J}$ is an isomorphism. If so, $M\in mod\unicode[STIX]{x1D6EC}$ is GMCM if and only if $\mathbf{R}\text{Hom}_{R}(M,\mathbb{I})$ is concentrated in degree $-d$ if and only if $Ext_{\unicode[STIX]{x1D6EC}^{op}}^{i}(M,K_{\unicode[STIX]{x1D6EC}})=0$ for $i>0$ . Also, in this case, as $K_{\unicode[STIX]{x1D6EC}}[d]$ is a dualizing complex, it is of finite injective dimension both as a left and a right $\unicode[STIX]{x1D6EC}$ -module. To prove these, we may take the completion, and may assume that $R$ is complete. All the assertions are independent of $R$ , so taking the Noether normalization, we may assume that $R$ is local. By (5.14), the assertions follow.

5.17

For any $M\in mod\unicode[STIX]{x1D6EC}$ which is GMCM,

$$\begin{eqnarray}M\cong \mathbf{R}\text{Hom}_{R}(\mathbf{R}\text{Hom}_{R}(M,\mathbb{I}),\mathbb{I})\cong \mathbf{R}\text{Hom}_{R}(Ext_{\unicode[STIX]{x1D6EC}^{op}}^{-d}(M,K_{\unicode[STIX]{x1D6EC}}[d]),\mathbb{I})[-d].\end{eqnarray}$$

Hence $M^{\dagger }:= Hom_{\unicode[STIX]{x1D6EC}^{op}}(M,K_{\unicode[STIX]{x1D6EC}})$ is also a GMCM $\unicode[STIX]{x1D6EC}$ -module, and hence

$$\begin{eqnarray}Hom_{\unicode[STIX]{x1D6EC}}(M^{\dagger },K_{\unicode[STIX]{x1D6EC}})\rightarrow \mathbf{R}\text{Hom}_{\unicode[STIX]{x1D6EC}}(M^{\dagger },\mathbb{J})=\mathbf{R}\text{Hom}_{R}(M^{\dagger },\mathbb{I})\end{eqnarray}$$

is an isomorphism (in other words, $Ext_{\unicode[STIX]{x1D6EC}}^{i}(M^{\dagger },K_{\unicode[STIX]{x1D6EC}})=0$ for $i>0$ ). So the canonical map

(7) $$\begin{eqnarray}M\rightarrow Hom_{\unicode[STIX]{x1D6EC}}(Hom_{\unicode[STIX]{x1D6EC}^{op}}(M,K_{\unicode[STIX]{x1D6EC}}),K_{\unicode[STIX]{x1D6EC}})= Hom_{\unicode[STIX]{x1D6EC}}(M^{\dagger },K_{\unicode[STIX]{x1D6EC}})\end{eqnarray}$$

$m\mapsto (\unicode[STIX]{x1D711}\mapsto \unicode[STIX]{x1D711}m)$ is an isomorphism. This isomorphism is true without assuming that $R$ has a dualizing complex (but assuming the existence of a canonical module), passing to the completion. Note that if $\unicode[STIX]{x1D6EC}=R$ , $K_{R}$ exists, and $R$ is Cohen–Macaulay, then $K_{R}$ is a dualizing complex of $R$ .

Similarly, for $N\in \unicode[STIX]{x1D6EC}{\rm mod}$ which is GMCM,

$$\begin{eqnarray}N\rightarrow Hom_{\unicode[STIX]{x1D6EC}^{op}}(Hom_{\unicode[STIX]{x1D6EC}}(N,K_{\unicode[STIX]{x1D6EC}}),K_{\unicode[STIX]{x1D6EC}})\end{eqnarray}$$

$n\mapsto (\unicode[STIX]{x1D711}\mapsto \unicode[STIX]{x1D711}n)$ is an isomorphism.

5.18

In particular, letting $M=\unicode[STIX]{x1D6EC}$ , if $\unicode[STIX]{x1D6EC}$ is GCM, we have that $K_{\unicode[STIX]{x1D6EC}}= Hom_{\unicode[STIX]{x1D6EC}^{op}}(\unicode[STIX]{x1D6EC},K_{\unicode[STIX]{x1D6EC}})$ is GMCM. Moreover,

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}\rightarrow End_{\unicode[STIX]{x1D6EC}^{op}}K_{\unicode[STIX]{x1D6EC}}\end{eqnarray}$$

is an $R$ -algebra isomorphism, where $a\in \unicode[STIX]{x1D6EC}$ goes to the left multiplication by $a$ . Similarly,

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}\rightarrow (End_{\unicode[STIX]{x1D6EC}}K_{\unicode[STIX]{x1D6EC}})^{op}\end{eqnarray}$$

is an isomorphism of $R$ -algebras.

5.19

Let $(R,\mathfrak{m})$ be a $d$ -dimensional complete local ring, and $\dim \unicode[STIX]{x1D6EC}=d$ . Then by the local duality,

$$\begin{eqnarray}H_{\mathfrak{m}}^{d}(K_{\unicode[STIX]{x1D6EC}})^{\vee }\cong Ext_{R}^{-d}(K_{\unicode[STIX]{x1D6EC}},\mathbb{I})\cong Ext_{\unicode[STIX]{x1D6EC}^{op}}^{-d}(K_{\unicode[STIX]{x1D6EC}},\mathbb{J})\cong End_{\unicode[STIX]{x1D6EC}^{op}}K_{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$

where $\mathbb{J}= Hom_{R}(\unicode[STIX]{x1D6EC},\mathbb{I})$ and $(?)^{\vee }= Hom_{R}(?,E_{R}(R/\mathfrak{m}))$ .

6.1

We say that $\unicode[STIX]{x1D714}$ is an $R$ -semicanonical right $\unicode[STIX]{x1D6EC}$ -module (resp.  $R$ -semicanonical left $\unicode[STIX]{x1D6EC}$ -module, weakly $R$ -semicanonical $\unicode[STIX]{x1D6EC}$ -bimodule, $R$ -semicanonical $\unicode[STIX]{x1D6EC}$ -bimodule) if for any $P\in SpecR$ , $R_{P}\otimes _{R}\unicode[STIX]{x1D714}$ is the right canonical module (resp. left canonical module, weakly canonical module, canonical module) of $R_{P}\otimes _{R}\unicode[STIX]{x1D6EC}$ for any $P\in Supp_{R}\unicode[STIX]{x1D714}$ . If we do not mention what $R$ is, then one may take $R$ to be the center of $\unicode[STIX]{x1D6EC}$ . An $R$ -semicanonical right $\unicode[STIX]{x1D6EC}^{op}$ -module (resp.  $R$ -semicanonical left $\unicode[STIX]{x1D6EC}^{op}$ -module, weakly $R$ -semicanonical $\unicode[STIX]{x1D6EC}^{op}$ -bimodule, $R$ -semicanonical $\unicode[STIX]{x1D6EC}^{op}$ -bimodule) is nothing but an $R$ -semicanonical left $\unicode[STIX]{x1D6EC}$ -module (resp.  $R$ -semicanonical right $\unicode[STIX]{x1D6EC}$ -module, weakly $R$ -semicanonical $\unicode[STIX]{x1D6EC}$ -bimodule, $R$ -semicanonical $\unicode[STIX]{x1D6EC}$ -bimodule).

6.2

Let $C\in mod\unicode[STIX]{x1D6EC}$ (resp.  $\unicode[STIX]{x1D6EC}{\rm mod}$ , $(\unicode[STIX]{x1D6EC}\otimes _{R}\unicode[STIX]{x1D6EC}^{op}){\rm mod}$ , $(\unicode[STIX]{x1D6EC}\otimes _{R}\unicode[STIX]{x1D6EC}^{op}){\rm mod}$ ). We say that $C$ is an $n$ -canonical right $\unicode[STIX]{x1D6EC}$ -module (resp.  $n$ -canonical left $\unicode[STIX]{x1D6EC}$ -module, weakly $n$ -canonical $\unicode[STIX]{x1D6EC}$ -bimodule, $n$ -canonical $\unicode[STIX]{x1D6EC}$ -bimodule) over $R$ if $C\in (S_{n}^{\prime })^{R}$ , and for each $P\in R^{\langle {<}n\rangle }$ , we have that $C_{P}$ is an $R_{P}$ -semicanonical right $\unicode[STIX]{x1D6EC}_{P}$ -module (resp.  $R_{P}$ -semicanonical left $\unicode[STIX]{x1D6EC}_{P}$ -module, weakly $R_{P}$ -semicanonical $\unicode[STIX]{x1D6EC}_{P}$ -bimodule, $R_{P}$ -semicanonical $\unicode[STIX]{x1D6EC}_{P}$ -bimodule). If we do not mention what $R$ is, it may mean $R$ is the center of $\unicode[STIX]{x1D6EC}$ .

6.4

As in section 4, let $C\in mod\unicode[STIX]{x1D6EC}$ , and set $\unicode[STIX]{x1D6E4}= End_{\unicode[STIX]{x1D6EC}^{op}}C$ , $(?)^{\dagger }= Hom_{\unicode[STIX]{x1D6EC}^{op}}(?,C)$ , and $(?)^{\ddagger }= Hom_{\unicode[STIX]{x1D6E4}}(?,C)$ . Moreover, we set $\unicode[STIX]{x1D6EC}_{1}:=(End_{\unicode[STIX]{x1D6E4}}C)^{op}$ . The $R$ -algebra map $\unicode[STIX]{x1D6F9}_{1}:\unicode[STIX]{x1D6EC}\rightarrow \unicode[STIX]{x1D6EC}_{1}$ is induced by the right action of $\unicode[STIX]{x1D6EC}$ on $C$ .

6.9

Let $C$ be a $1$ -canonical right $\unicode[STIX]{x1D6EC}$ -module over $R$ . Define $Q:=\prod _{P\in Min_{R}C}R_{P}$ . If $P\in Min_{R}C$ , then $C_{P}=K_{\unicode[STIX]{x1D6EC}_{P}}$ . Hence $\unicode[STIX]{x1D6F7}_{P}:\unicode[STIX]{x1D6EC}_{P}\rightarrow (\unicode[STIX]{x1D6EC}_{1})_{P}$ is an isomorphism by Lemma 6.7. So $1_{Q}\otimes \unicode[STIX]{x1D6F9}_{1}:Q\otimes _{R}\unicode[STIX]{x1D6EC}\rightarrow Q\otimes _{R}\unicode[STIX]{x1D6EC}_{1}$ is also an isomorphism. As $Ass_{R}\unicode[STIX]{x1D6EC}_{1}= Min_{R}C$ , we have that $\unicode[STIX]{x1D6EC}_{1}\subset Q\otimes _{R}\unicode[STIX]{x1D6EC}_{1}$ .