The Stacks project

13.39 Admissible subcategories

A reference for this section is [Section 1, Bondal-Kapranov].

Definition 13.39.1. Let $\mathcal{D}$ be an additive category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory. The right orthogonal $\mathcal{A}^\perp $ of $\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\mathcal{D}$ such that $\mathop{\mathrm{Hom}}\nolimits (A, X) = 0$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. The left orthogonal ${}^\perp \mathcal{A}$ of $\mathcal{A}$ is the full subcategory consisting of the objects $X$ of $\mathcal{D}$ such that $\mathop{\mathrm{Hom}}\nolimits (X, A) = 0$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.

Lemma 13.39.2. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory invariant under all shifts. Consider a distinguished triangle

\[ X \to Y \to Z \to X[1] \]

of $\mathcal{D}$. The following are equivalent

  1. $Z$ is in $\mathcal{A}^\perp $, and

  2. $\mathop{\mathrm{Hom}}\nolimits (A, X) = \mathop{\mathrm{Hom}}\nolimits (A, Y)$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.

Proof. By Lemma 13.4.2 the functor $\mathop{\mathrm{Hom}}\nolimits (A, -)$ is homological and hence we get a long exact sequence as in (13.3.5.1). Assume (1) and let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. Then we consider the exact sequence

\[ \mathop{\mathrm{Hom}}\nolimits (A[1], Z) \to \mathop{\mathrm{Hom}}\nolimits (A, X) \to \mathop{\mathrm{Hom}}\nolimits (A, Y) \to \mathop{\mathrm{Hom}}\nolimits (A, Z) \]

Since $A[1] \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ we see that the first and last groups are zero. Thus we get (2). Assume (2) and let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. Then we consider the exact sequence

\[ \mathop{\mathrm{Hom}}\nolimits (A, X) \to \mathop{\mathrm{Hom}}\nolimits (A, Y) \to \mathop{\mathrm{Hom}}\nolimits (A, Z) \to \mathop{\mathrm{Hom}}\nolimits (A[-1], X) \to \mathop{\mathrm{Hom}}\nolimits (A[-1], Y) \]

and we conclude that $\mathop{\mathrm{Hom}}\nolimits (A, Z) = 0$ as desired. $\square$

Lemma 13.39.3. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory invariant under all shifts. Then both the right orthogonal $\mathcal{A}^\perp $ and the left orthogonal ${}^\perp \mathcal{A}$ of $\mathcal{A}$ are strictly full, saturated1, triangulated subcagories of $\mathcal{D}$.

Proof. It is immediate from the definitions that the orthogonals are preserved under taking shifts, direct sums, and direct summands. Consider a distinguished triangle

\[ X \to Y \to Z \to X[1] \]

of $\mathcal{D}$. By Lemma 13.4.16 it suffices to show that if $X$ and $Y$ are in $\mathcal{A}^\perp $, then $Z$ is in $\mathcal{A}^\perp $. This is immediate from Lemma 13.39.2. $\square$

Lemma 13.39.4. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A}$ be a full triangulated subcategory of $\mathcal{D}$. For an object $X$ of $\mathcal{D}$ consider the property $P(X)$: there exists a distinguished triangle $A \to X \to B \to A[1]$ in $\mathcal{D}$ with $A$ in $\mathcal{A}$ and $B$ in $\mathcal{A}^\perp $.

  1. If $X_1 \to X_2 \to X_3 \to X_1[1]$ is a distinguished triangle and $P$ holds for two out of three, then it holds for the third.

  2. If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \oplus X_2$.

Proof. Let $X_1 \to X_2 \to X_3 \to X_1[1]$ be a distinguished triangle and assume $P$ holds for $X_1$ and $X_2$. Choose distinguished triangles

\[ A_1 \to X_1 \to B_1 \to A_1[1] \quad \text{and}\quad A_2 \to X_2 \to B_2 \to A_2[1] \]

as in condition $P$. Since $\mathop{\mathrm{Hom}}\nolimits (A_1, A_2) = \mathop{\mathrm{Hom}}\nolimits (A_1, X_2)$ by Lemma 13.39.2 there is a unique morphism $A_1 \to A_2$ such that the diagram

\[ \xymatrix{ A_1 \ar[d] \ar[r] & X_1 \ar[d] \\ A_2 \ar[r] & X_2 } \]

commutes. Choose an extension of this to a diagram

\[ \xymatrix{ A_1 \ar[r] \ar[d] & X_1 \ar[r] \ar[d] & Q_1 \ar[r] \ar[d] & A_1[1] \ar[d] \\ A_2 \ar[r] \ar[d] & X_2 \ar[r] \ar[d] & Q_2 \ar[r] \ar[d] & A_2[1] \ar[d] \\ A_3 \ar[r] \ar[d] & X_3 \ar[r] \ar[d] & Q_3 \ar[r] \ar[d] & A_3[1] \ar[d] \\ A_1[1] \ar[r] & X_1[1] \ar[r] & Q_1[1] \ar[r] & A_1[2] } \]

as in Proposition 13.4.23. By TR3 we see that $Q_1 \cong B_1$ and $Q_2 \cong B_2$ and hence $Q_1, Q_2 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}^\perp )$. As $Q_1 \to Q_2 \to Q_3 \to Q_1[1]$ is a distinguished triangle we see that $Q_3 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}^\perp )$ by Lemma 13.39.3. Since $\mathcal{A}$ is a full triangulated subcategory, we see that $A_3$ is isomorphic to an object of $\mathcal{A}$. Thus $X_3$ satisfies $P$. The other cases of (1) follow from this case by translation. Part (2) is a special case of (1) via Lemma 13.4.11. $\square$

Lemma 13.39.5. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

  1. the inclusion functor $\mathcal{A} \to \mathcal{D}$ has a right adjoint, and

  2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

    \[ A \to X \to B \to A[1] \]

    in $\mathcal{D}$ with $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ and $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}^\perp )$.

If this holds, then $\mathcal{A}$ is saturated (Definition 13.6.1) and if $\mathcal{A}$ is strictly full in $\mathcal{D}$, then $\mathcal{A} = {}^\perp (\mathcal{A}^\perp )$.

Proof. Assume (1) and denote $v : \mathcal{D} \to \mathcal{A}$ the right adjoint. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Set $A = v(X)$. We may extend the adjunction mapping $A \to X$ to a distinguished triangle $A \to X \to B \to A[1]$. Since

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(A', A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(A', v(X)) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(A', X) \]

for $A' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$, we conclude that $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}^\perp )$ by Lemma 13.39.2.

Assume (2). We will contruct the adjoint $v$ explictly. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Choose $A \to X \to B \to A[1]$ as in (2). Set $v(X) = A$. Let $f : X \to Y$ be a morphism in $\mathcal{D}$. Choose $A' \to Y \to B' \to A'[1]$ as in (2). Since $\mathop{\mathrm{Hom}}\nolimits (A, A') = \mathop{\mathrm{Hom}}\nolimits (A, Y)$ by Lemma 13.39.2 there is a unique morphism $f' : A \to A'$ such that the diagram

\[ \xymatrix{ A \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ A' \ar[r] & Y } \]

commutes. Hence we can set $v(f) = f'$ to get a functor. To see that $v$ is adjoint to the inclusion morphism use Lemma 13.39.2 again.

Proof of the final statement. In other to prove that $\mathcal{A}$ is saturated we may replace $\mathcal{A}$ by the strictly full subcategory having the same isomorphism classes as $\mathcal{A}$; details omitted. Assume $\mathcal{A}$ is strictly full. If we show that $\mathcal{A} = {}^\perp (\mathcal{A}^\perp )$, then $\mathcal{A}$ will be saturated by Lemma 13.39.3. Since the incusion $\mathcal{A} \subset {}^\perp (\mathcal{A}^\perp )$ is clear it suffices to prove the other inclusion. Let $X$ be an object of ${}^\perp (\mathcal{A}^\perp )$. Choose a distinguished triangle $A \to X \to B \to A[1]$ as in (2). As $\mathop{\mathrm{Hom}}\nolimits (X, B) = 0$ by assumption we see that $A \cong X \oplus B[-1]$ by Lemma 13.4.11. Since $\mathop{\mathrm{Hom}}\nolimits (A, B[-1]) = 0$ as $B \in \mathcal{A}^\perp $ this implies $B[-1] = 0$ and $A \cong X$ as desired. $\square$

Lemma 13.39.6. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent

  1. the inclusion functor $\mathcal{A} \to \mathcal{D}$ has a left adjoint, and

  2. for every $X$ in $\mathcal{D}$ there exists a distinguished triangle

    \[ B \to X \to A \to B[1] \]

    in $\mathcal{D}$ with $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ and $B \in \mathop{\mathrm{Ob}}\nolimits ({}^\perp \mathcal{A})$.

If this holds, then $\mathcal{A}$ is saturated (Definition 13.6.1) and if $\mathcal{A}$ is strictly full in $\mathcal{D}$, then $\mathcal{A} = ({}^\perp \mathcal{A})^\perp $.

Proof. Omitted. Dual to Lemma 13.39.5. $\square$

Definition 13.39.7. Let $\mathcal{D}$ be a triangulated category. A right admissible subcategory of $\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma 13.39.5. A left admissible subcategory of $\mathcal{D}$ is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma 13.39.6. A two-sided admissible subcategory is one which is both right and left admissible.

[1] Definition 13.6.1.

Comments (3)

Comment #6021 by Sandeep on

Two typos: In the statements of Lemmas 13.39.5 and 13.39.6, the distinguished triangles should be and respectively.

Comment #6022 by Sandeep on

Sorry, I mean .


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