Derived noncommutative algebraic geometry

The derived category of $\backslash mathbb\{P\}^1$ is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure. Recall that the Euler sequence of $\backslash mathbb\{P\}^1$ is the short exact sequence

: $0\; \backslash to\; \backslash mathcal\{O\}(-2)\; \backslash to\; \backslash mathcal\{O\}(-1)^\{\backslash oplus\; 2\}\; \backslash to\; \backslash mathcal\{O\}\; \backslash to\; 0$

if we consider the two terms on the right as a complex, then we get the distinguished triangle

: $\backslash mathcal\{O\}(-1)^\{\backslash oplus\; 2\}\; \backslash overset\{\backslash phi\}\{\backslash rightarrow\}\; \backslash mathcal\{O\}\; \backslash to\; \backslash operatorname\{Cone\}(\backslash phi)\; \backslash overset\{+1\}\{\backslash rightarrow\}.$

Since $\backslash operatorname\{Cone\}(\backslash phi)\; \backslash cong\; \backslash mathcal\{O\}(-2)[+1]$ we have constructed this sheaf $\backslash mathcal\{O\}(-2)$ using only categorical tools. We could repeat this again by tensoring the Euler sequence by the flat sheaf $\backslash mathcal\{O\}(-1)$, and apply the cone construction again. If we take the duals of the sheaves, then we can construct all of the line bundles in $\backslash operatorname\{Coh\}(\backslash mathbb\{P\}^1)$ using only its triangulated structure. It turns out the correct way of studying derived categories from its objects and triangulated structure is with exceptional collections.

{{main|semiorthogonal decomposition}}

The technical tools for encoding this construction are semiorthogonal decompositions and exceptional collections.{{Cite book|last=Liu|first=Yijia|title=Superschool on Derived Categories|pages=35, 37, 38, 41|chapter=Semi-orthogonal Decompositions of Derived Categories}} A semiorthogonal decomposition of a triangulated category $\backslash mathcal\{T\}$ is a collection of full triangulated subcategories $\backslash mathcal\{T\}\_1,\backslash ldots,\; \backslash mathcal\{T\}\_n$ such that the following two properties hold

(1) For objects $T\_i\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{T\}\_i)$ we have $\backslash operatorname\{Hom\}(T\_i,\; T\_j)\; =\; 0$ for $i\; >\; j$

(2) The subcategories $\backslash mathcal\{T\}\_i$ generate $\backslash mathcal\{T\}$, meaning every object $T\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{T\})$ can be decomposed in to a sequence of $T\_i\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{T\})$,

:$0\; =\; T\_n\; \backslash to\; T\_\{n-1\}\; \backslash to\; \backslash cdots\; \backslash to\; T\_1\; \backslash to\; T\_0\; =\; T$

such that $\backslash operatorname\{Cone\}(T\_i\; \backslash to\; T\_\{i-1\})\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{T\}\_i)$. Notice this is analogous to a filtration of an object in an abelian category such that the cokernels live in a specific subcategory.

We can specialize this a little further by considering exceptional collections of objects, which generate their own subcategories. An object $E$ in a triangulated category is called exceptional if the following property holds

:$\backslash operatorname\{Hom\}(E,E[+\backslash ell])\; =\; \backslash begin\{cases\}$

k &\text{if } \ell = 0 \\

0 &\text{if } \ell \neq 0

\end{cases}

where $k$ is the underlying field of the vector space of morphisms. A collection of exceptional objects $E\_1,\; \backslash ldots,\; E\_r$ is an exceptional collection of length $r$ if for any $i\; >\; j$ and any $\backslash ell$, we have

:$\backslash operatorname\{Hom\}(E\_i,\; E\_j[+\backslash ell])\; =\; 0$

and is a strong exceptional collection if in addition, for any $\backslash ell\; \backslash neq\; 0$ and any $i,\; j$, we have

:$\backslash operatorname\{Hom\}(E\_i,\; E\_j[+\backslash ell])\; =\; 0$

We can then decompose our triangulated category into the semiorthogonal decomposition

:$\backslash mathcal\{T\}\; =\; \backslash langle\; \backslash mathcal\{T\}\text{'},\; E\_1,\; \backslash ldots,\; E\_r\; \backslash rangle$

where $\backslash mathcal\{T\}\text{'}\; =\; \backslash langle\; E\_1,\; \backslash ldots,\; E\_r\; \backslash rangle^\backslash perp$, the subcategory of objects in $E\; \backslash in\; \backslash operatorname\{Ob\}(\backslash mathcal\{T\})$ such that $\backslash operatorname\{Hom\}(E,\; E\_i[+\backslash ell])\; =\; 0$. If in addition $\backslash mathcal\{T\}\text{'}\; =\; 0$ then the strong exceptional collection is called full.

Beilinson provided the first example of a full strong exceptional collection. In the derived category $D^b(\backslash mathbb\{P\}^n)$ the line bundles $\backslash mathcal\{O\}(-n),\; \backslash mathcal\{O\}(-n+1),\; \backslash ldots,\; \backslash mathcal\{O\}(-1),\; \backslash mathcal\{O\}$ form a full strong exceptional collection. He proves the theorem in two parts. First showing these objects are an exceptional collection and second by showing the diagonal $\backslash mathcal\{O\}\_\backslash Delta$ of $\backslash mathbb\{P\}^n\; \backslash times\; \backslash mathbb\{P\}^n$ has a resolution whose compositions are tensors of the pullback of the exceptional objects.

Technical Lemma

An exceptional collection of sheaves $E\_1,\; E\_2,\; \backslash ldots,\; E\_r$ on $X$ is full if there exists a resolution

:$0\; \backslash to\; p\_1^*E\_1\; \backslash otimes\; p\_2^*F\_1\; \backslash to\; \backslash cdots\; \backslash to\; p\_1^*E\_n\; \backslash otimes\; p\_2^*F\_n\; \backslash to\; \backslash mathcal\{O\}\_\backslash Delta\; \backslash to\; 0$

in $D^b(X\backslash times\; X)$ where $F\_i$ are arbitrary coherent sheaves on $X$.

Another way to reformulate this lemma for $X\; =\; \backslash mathbb\{P\}^n$ is by looking at the Koszul complex associated to

$\backslash bigoplus\_\{i=0\}^n\; \backslash mathcal\{O\}(-D\_i)\; \backslash xrightarrow\{\backslash phi\}\; \backslash mathcal\{O\}$

$0\; \backslash to\; \backslash mathcal\{O\}\backslash left(-\backslash sum\_\{i=1\}^n\; D\_i\; \backslash right)\; \backslash to\; \backslash cdots\; \backslash to$

\bigoplus_{i \neq j}\mathcal{O}(-D_i - D_j) \to

\bigoplus_{i=1}^n\mathcal{O}(-D_i) \to \mathcal{O} \to 0

$\backslash mathcal\{O\}\backslash left(-\backslash sum\_\{i=0\}^n\; D\_i\; \backslash right)\; \backslash cong\; \backslash mathcal\{O\}(-n-1)$

which gives a derived equivalence of the rest of the terms of the above complex with $\backslash mathcal\{O\}(-n-1)$. For $n=2$ the Koszul complex above is the exact complex

$0\; \backslash to\; \backslash mathcal\{O\}(-3)\; \backslash to\; \backslash mathcal\{O\}(-2)\backslash oplus\backslash mathcal\{O\}(-2)\; \backslash to\; \backslash mathcal\{O\}(-1)\backslash oplus\backslash mathcal\{O\}(-1)\; \backslash to\; \backslash mathcal\{O\}\; \backslash to\; 0$

$0\; \backslash to\; \backslash mathcal\{O\}(-2)\backslash oplus\backslash mathcal\{O\}(-2)\; \backslash to\; \backslash mathcal\{O\}(-1)\backslash oplus\backslash mathcal\{O\}(-1)\; \backslash to\; \backslash mathcal\{O\}\; \backslash to\; 0$

If $X$ is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categories $F:\; D^b(X)\; \backslash to\; D^b(Y)$, then there is an isomorphism of the underlying varieties.{{Cite book|last=Dolgachev|first=Igor|url=http://www.math.lsa.umich.edu/~idolga/derived9.pdf|title=Derived categories|pages=105–112}}

The proof starts out by analyzing two induced Serre functors on $D^b(Y)$ and finding an isomorphism between them. It particular, it shows there is an object $\backslash omega\_Y\; =\; F(\backslash omega\_X)$ which acts like the dualizing sheaf on $Y$. The isomorphism between these two functors gives an isomorphism of the set of underlying points of the derived categories. Then, what needs to be check is an ismorphism $F(\backslash omega\_X^\{\backslash otimes\; k\})\; \backslash cong\; \backslash omega\_Y^\{\backslash otimes\; k\}$, for any $k\; \backslash in\; \backslash mathbb\{N\}$, giving an isomorphism of canonical rings

: $A(X)\; =\; \backslash bigoplus\_\{k=0\}^\backslash infty\; H^0(X,\backslash omega\_X^\{\backslash otimes\; k\})\; \backslash cong\; \backslash bigoplus\_\{k=0\}^\backslash infty\; H^0(Y,\backslash omega\_Y^\{\backslash otimes\; k\})$

If $\backslash omega\_Y$ can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism $X\; \backslash to\; Y$. All of the details are contained in Dolgachev's notes.

This theorem fails in the case $X$ is Calabi-Yau, since $\backslash omega\_X\; \backslash cong\; \backslash mathcal\{O\}\_X$, or is the product of a variety which is Calabi-Yau. Abelian varieties are a class of examples where a reconstruction theorem could never hold. If $X$ is an abelian variety and $\backslash hat\{X\}$ is its dual, the Fourier–Mukai transform with kernel $\backslash mathcal\{P\}$, the Poincare bundle,The poincare bundle $\backslash mathcal\{P\}$ on $X\backslash times\; \backslash hat\{X\}$ is a line bundle which is trivial on $\backslash \{0\backslash \}\backslash times\backslash hat\{X\}$ and $X\backslash times\backslash \{0\backslash \}$ and has the property $\backslash mathcal\{P\}|\_\{X\backslash times\backslash \{x\backslash \}\}$ is the line bundle represented by the point $x\; \backslash in\; \backslash hat\{X\}$. gives an equivalence

:$FM\_\{\backslash mathcal\{P\}\}:D^b(X)\; \backslash to\; D^b(\backslash hat\{X\})$

of derived categories. Since an abelian variety is generally not isomorphic to its dual, there are derived equivalent derived categories without isomorphic underlying varieties.{{Cite journal|last=Mukai|first=Shigeru|date=1981|title=Duality between D(X) and D(X^) with its application to Picard sheaves|url=https://projecteuclid.org/euclid.nmj/1118786312|journal=Nagoya Math. J.|volume=81|pages=153–175|via=Project Euclid|doi=10.1017/S002776300001922X|doi-access=free}} There is an alternative theory of tensor triangulated geometry where we consider not only a triangulated category, but also a monoidal structure, i.e. a tensor product. This geometry has a full reconstruction theorem using the spectrum of categories.{{Cite journal|url=https://www.math.ucla.edu/~balmer/Pubfile/TTG.pdf|title=Tensor triangulated geometry|last=Balmer|first=Paul|date=2010|journal=Proceedings of the International Congress of Mathematicians}}

K3 surfaces are another class of examples where reconstruction fails due to their Calabi-Yau property. There is a criterion for determining whether or not two K3 surfaces are derived equivalent: the derived category of the K3 surface $D^b(X)$ is derived equivalent to another K3 $D^b(Y)$ if and only if there is a Hodge isometry $H^2(X,\; \backslash mathbb\{Z\})\; \backslash to\; H^2(Y,\; \backslash mathbb\{Z\})$, that is, an isomorphism of Hodge structure. Moreover, this theorem is reflected in the motivic world as well, where the Chow motives are isomorphic if and only if there is an isometry of Hodge structures.{{cite arXiv|eprint=1705.04063|title=Motives of isogenous K3 surfaces|last=Huybrechts|first=Daniel|date=2018|class=math.AG}}

One nice application of the proof of this theorem is the identification of autoequivalences of the derived category of a smooth projective variety with ample (anti-)canonical sheaf. This is given by

:$\backslash operatorname\{Auteq\}(D^b(X))\; \backslash cong\; (\backslash operatorname\{Pic\}(X)\backslash rtimes\; \backslash operatorname\{Aut\}(X))\backslash times\backslash mathbb\{Z\}$

Where an autoequivalence $F$ is given by an automorphism $f:X\backslash to\; X$, then tensored by a line bundle $\backslash mathcal\{L\}\; \backslash in\; \backslash operatorname\{Pic\}(X)$ and finally composed with a shift. Note that $\backslash operatorname\{Aut\}(X)$ acts on $\backslash operatorname\{Pic\}(X)$ via the polarization map, $g\; \backslash mapsto\; g^*(L)\backslash otimes\; L^\{-1\}$.{{Cite web|url=https://www-fourier.ujf-grenoble.fr/~mbrion/autos.pdf|title=Notes on Automorphism Groups of Projective Varieties|last=Brion|first=Michel|page=8|url-status=live|archive-url=https://web.archive.org/web/20200213231444/https://www-fourier.ujf-grenoble.fr/~mbrion/autos.pdf|archive-date=13 February 2020}}

The bounded derived category $D^b(X)$ was used extensively in SGA6 to construct an intersection theory with $K(X)$ and $Gr\_\backslash gamma\; K(X)\backslash otimes\backslash mathbb\{Q\}$. Since these objects are intimately relative with the Chow ring of $X$, its chow motive, Orlov asked the following question: given a fully-faithful functor

: $F:D^b(X)\; \backslash to\; D^b(Y)$

is there an induced map on the chow motives

: $f:M(X)\; \backslash to\; M(Y)$

such that $M(X)$ is a summand of $M(Y)$?{{cite journal|arxiv=math/0512620|title=Derived categories of coherent sheaves and motives|last=Orlov|first=Dmitri|journal=Russian Mathematical Surveys|volume=60|issue=6|pages=1242–1244|date=2011|doi=10.1070/RM2005v060n06ABEH004292|s2cid=11484447}} In the case of K3 surfaces, a similar result has been confirmed since derived equivalent K3 surfaces have an isometry of Hodge structures, which gives an isomorphism of motives.

On a smooth variety there is an equivalence between the derived category $D^b(X)$ and the thickMeaning it is closed under extensions. Given any two objects $\backslash mathcal\{E\}\text{'},\; \backslash mathcal\{E\}$ in the subcategory, any object $\backslash mathcal\{E\}$ fitting into an exact sequence $0\; \backslash to\; \backslash mathcal\{E\}\text{'}\; \backslash to\; \backslash mathcal\{E\}\; \backslash to\; \backslash mathcal\{E\}$ \to 0 is also in the subcategory. In the triangulated case, this translates to the same conditions, but instead of an exact sequence, it is a distinguished triangle $\backslash mathcal\{E\}\text{'}\; \backslash to\; \backslash mathcal\{E\}\; \backslash to\; \backslash mathcal\{E\}\text{'}\text{'}\; \backslash to\; \backslash mathcal\{E\}[+1]${{Cite web|url=https://www.gwern.net/docs/math/1990-thomason.pdf|title=Higher Algebraic K-Theory of Schemes and of Derived Categories|last1=Thomason|first1=R.W.|last2=Trobaugh|first2=Thomas|url-status=live|archive-url=https://web.archive.org/web/20190130235347/https://www.gwern.net/docs/math/1990-thomason.pdf|archive-date=30 January 2019}} full triangulated $D\_\{\backslash operatorname\{perf\}\}(X)$ of perfect complexes. For separated, Noetherian schemes of finite Krull dimension (called the ELF condition)Which he uses because of its nice properties: in particular every bounded complex of coherent sheaves $C^\backslash bullet$ has a resolution from a bounded above complex $P^\backslash bullet\; \backslash xrightarrow\{\backslash simeq\}\; C^\backslash bullet$ such that $P^\backslash bullet$ is a complex of locally free sheaves of finite type. this is not the case, and Orlov defines the derived category of singularities as their difference using a quotient of categories. For an ELF scheme $X$ its derived category of singularities is defined as

: $D\_\{sg\}(X)\; :=\; D^b(X)/D\_\backslash text\{perf\}(X)${{Cite arXiv|title=Triangulated Categories of Singularities and D-Branes in Landau–Ginzburg Models|last=Orlov|first=Dmitri|eprint=math/0302304|year=2003}}

for a suitable definition of localization of triangulated categories.

Although localization of categories is defined for a class of morphisms $\backslash Sigma$ in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategory $\backslash mathcal\{N\}\; \backslash subset\; \backslash mathcal\{T\}$ the class of morphisms $\backslash Sigma(\backslash mathcal\{N\})$, $s$ in $\backslash mathcal\{T\}$ where $s$ fits into a distinguished triangle

$X\; \backslash xrightarrow\{s\}\; Y\; \backslash to\; N\; \backslash to\; X[+1]$

: $X\; \backslash xrightarrow\{s\}\; Y\; \backslash xrightarrow\{s\text{'}\}Z$

with distinguished triangles

: $X\; \backslash xrightarrow\{s\}Y\; \backslash to\; N\; \backslash to\; X[+1]$

: $Y\; \backslash xrightarrow\{s\text{'}\}\; Z\; \backslash to\; N\text{'}\; \backslash to\; Y[+1]$

where $N,N\text{'}\; \backslash in\; \backslash mathcal\{N\}$, then there are distinguished triangles

: $X\; \backslash to\; Z\; \backslash to\; M\; \backslash to\; X[+1]$

: $N\; \backslash to\; M\; \backslash to\; N\text{'}\; \backslash to\; N[+1]$ where $M\; \backslash in\; \backslash mathcal\{N\}$ since $\backslash mathcal\{N\}$ is closed under extensions. This new category has the following properties

• It is canonically triangulated where a triangle in $\backslash mathcal\{T\}/\backslash mathcal\{N\}$ is distinguished if it is isomorphic to the image of a triangle in $\backslash mathcal\{T\}$
• The category $\backslash mathcal\{T\}/\backslash mathcal\{N\}$ has the following universal property: any exact functor $F:\backslash mathcal\{T\}\; \backslash to\; \backslash mathcal\{T\}\text{'}$ where $F(N)\; \backslash cong\; 0$ where $N\; \backslash in\; \backslash mathcal\{N\}$, then it factors uniquely through the quotient functor $Q:\; \backslash mathcal\{T\}\; \backslash to\; \backslash mathcal\{T\}/\backslash mathcal\{N\}$, so there exists a morphism $\backslash tilde\{F\}:\; \backslash mathcal\{T\}/\backslash mathcal\{N\}\; \backslash to\; \backslash mathcal\{T\}\text{'}$ such that $\backslash tilde\{F\}\backslash circ\; Q\; \backslash simeq\; F$.

• If $X$ is a regular scheme, then every bounded complex of coherent sheaves is perfect. Hence the singularity category is trivial
• Any coherent sheaf $\backslash mathcal\{F\}$ which has support away from $\backslash operatorname\{Sing\}(X)$ is perfect. Hence nontrivial coherent sheaves in $D\_\{sg\}(X)$ have support on $\backslash operatorname\{Sing\}(X)$.
• In particular, objects in $D\_\{sg\}(X)$ are isomorphic to $\backslash mathcal\{F\}[+k]$ for some coherent sheaf $\backslash mathcal\{F\}$.

Kontsevich proposed a model for Landau–Ginzburg models which was worked out to the following definition:{{Cite journal|last1=Kapustin|first1=Anton|last2=Li|first2=Yi|date=2003-12-03|title=D-Branes in Landau–Ginzburg Models and Algebraic Geometry|arxiv=hep-th/0210296|journal=Journal of High Energy Physics|volume=2003|issue=12|pages=005|doi=10.1088/1126-6708/2003/12/005|bibcode=2003JHEP...12..005K|s2cid=11337046|issn=1029-8479}} a Landau–Ginzburg model is a smooth variety $X$ together with a morphism $W:X\; \backslash to\; \backslash mathbb\{A\}^1$ which is flat. There are three associated categories which can be used to analyze the D-branes in a Landau–Ginzburg model using matrix factorizations from commutative algebra.

With this definition, there are three categories which can be associated to any point $w\_0\; \backslash in\; \backslash mathbb\{A\}^1$, a $\backslash mathbb\{Z\}/2$-graded category $DG\_\{w\_0\}(W)$, an exact category $\backslash operatorname\{Pair\}\_\{w\_0\}(W)$, and a triangulated category $DB\_\{w\_0\}(W)$, each of which has objects

:$\backslash overline\{P\}\; =\; (p\_1:\; P\_1\; \backslash to\; P\_0,\; p\_0:\; P\_0\; \backslash to\; P\_1)$ where $p\_0\backslash circ\; p\_1,p\_1\backslash circ\; p\_0$ are multiplication by $W\; -\; w\_0$.

There is also a shift functor $[+1]$ send $\backslash overline\{P\}$ to

$\backslash overline\{P\}[+1]\; =\; (-p\_0:\; P\_0\; \backslash to\; P\_1,\; -p\_1:\; P\_1\; \backslash to\; P\_0)$.

:$\backslash operatorname\{Hom\}(\backslash overline\{P\},\backslash overline\{Q\})\; =\; \backslash bigoplus\_\{i,j\}\backslash operatorname\{Hom\}(P\_i,\; Q\_j)$

where the grading is given by $(i-j)\; \backslash bmod\; 2$ and differential acting on degree $d$ homogeneous elements by

:$Df\; =\; q\; \backslash circ\; f\; -\; (-1)^df\; \backslash circ\; p$

In $\backslash operatorname\{Pair\}\_\{w\_0\}(W)$ the morphisms are the degree $0$ morphisms in $DG\_\{w\_0\}(W)$. Finally, $DB\_\{w\_0\}(W)$ has the morphisms in $\backslash operatorname\{Pair\}\_\{w\_0\}(W)$ modulo the null-homotopies. Furthermore, $DB\_\{w\_0\}(W)$ can be endowed with a triangulated structure through a graded cone-construction in $\backslash operatorname\{Pair\}\_\{w\_0\}(W)$. Given $\backslash overline\{f\}:\backslash overline\{P\}\backslash to\backslash overline\{Q\}$ there is a mapping code $C(f)$ with maps

:$c\_1:\; Q\_1\backslash oplus\; P\_0\; \backslash to\; Q\_0\backslash oplus\; P\_1$ where

and

:$c\_0:\; Q\_0\backslash oplus\; P\_1\; \backslash to\; Q\_1\backslash oplus\; P\_0$ where

Then, a diagram $\backslash overline\{P\}\; \backslash to\; \backslash overline\{Q\}\; \backslash to\; \backslash overline\{R\}\; \backslash to\; \backslash overline\{P\}[+1]$ in $DB\_\{w\_0\}(W)$ is a distinguished triangle if it is isomorphic to a cone from $\backslash operatorname\{Pair\}\_\{w\_0\}(W)$.

Using the construction of $DB\_\{w\_0\}(W)$ we can define the category of D-branes of type B on $X$ with superpotential $W$ as the product category

: $DB(W)\; =\; \backslash prod\_\{w\; \backslash in\; \backslash mathbb\{A\}\_1\}DB\_\{w\_0\}(W).$

This is related to the singularity category as follows: Given a superpotential $W$ with isolated singularities only at $0$, denote $X\_0\; =\; W^\{-1\}(0)$. Then, there is an exact equivalence of categories

: $DB\_\{w\_0\}(W)\; \backslash cong\; D\_\{sg\}(X\_0)$

given by a functor induced from cokernel functor $\backslash operatorname\{Cok\}$ sending a pair $\backslash overline\{P\}\; \backslash mapsto\; \backslash operatorname\{Coker\}(p\_1)$. In particular, since $X$ is regular, Bertini's theorem shows $DB(W)$ is only a finite product of categories.

There is a Fourier-Mukai transform $\backslash Phi\_Z$ on the derived categories of two related varieties giving an equivalence of their singularity categories. This equivalence is called Knörrer periodicity. This can be constructed as follows: given a flat morphism $f:X\backslash to\backslash mathbb\{A\}^1$ from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme $Y\; =\; X\backslash times\; \backslash mathbb\{A\}^2$ and morphism $g:Y\; \backslash to\; \backslash mathbb\{A\}^1$ such that $g\; =\; f\; +\; xy$ where $xy$ are the coordinates of the $\backslash mathbb\{A\}^2$-factor. Consider the fibers $X\_0\; =\; f^\{-1\}(0)$, $Y\_0\; =\; g^\{-1\}(0)$, and the induced morphism $x:\; Y\_0\; \backslash to\; \backslash mathbb\{A\}^1$. And the fiber $Z\; =\; x^\{-1\}(0)$. Then, there is an injection $i:Z\; \backslash to\; Y\_0$ and a projection $q:\; Z\; \backslash to\; X\_0$ forming an $\backslash mathbb\{A\}^1$-bundle. The Fourier-Mukai transform

:$\backslash Phi\_Z(\backslash cdot)\; =\; \backslash mathbf\{R\}i\_*q^*(\backslash cdot)$

induces an equivalence of categories

:$D\_\{sg\}(X\_0)\; \backslash to\; D\_\{sg\}(Y\_0)$

called Knörrer periodicity. There is another form of this periodicity where $xy$ is replaced by the polynomial $x^2\; +\; y^2$.{{cite arXiv|last1=Brown|first1=Michael K.|last2=Dyckerhoff|first2=Tobias|date=2019-09-15|title=Topological K-theory of Equivariant Singularity Categories|eprint=1611.01931|pages=11|class=math.AG}}{{Cite web|url=https://gdz.sub.uni-goettingen.de/id/PPN356556735_0088?tify={%22pages%22:[159],%22panX%22:0.538,%22panY%22:0.719,%22view%22:%22info%22,%22zoom%22:0.583}|title=Cohen-Macaulay modules on hypersurface singularities I|last=Knörrer|first=Horst}} These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories.

If we take the Landau–Ginzburg model $(\backslash mathbb\{C\}^\{2k+1\},\; W)$ where $W\; =\; z\_0^n\; +\; z\_1^2\; +\; \backslash cdots\; +\; z\_\{2k\}^2$, then the only fiber singular fiber of $W$ is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category $D\_\backslash text\{sing\}(\backslash operatorname\{Spec\}(\backslash mathbb\{C\}[z]/(z^n)))$. Over the algebra $A\; =\; \backslash mathbb\{C\}[z]/(z^n)$ there are indecomposable objects

: $V\_i\; =\; \backslash operatorname\{Coker\}(A\; \backslash xrightarrow\{z^i\}\; A)\; =\; A\; /\; z^i$

whose morphisms can be completely understood. For any pair $i,j$ there are morphisms $\backslash alpha\_j^i:\; V\_i\; \backslash to\; V\_j$ where

• for $i\; \backslash geq\; j$ these are the natural projections
• for these are multiplication by $z^\{j-i\}$

where every other morphism is a composition and linear combination of these morphisms. There are many other cases which can be explicitly computed, using the table of singularities found in Knörrer's original paper.

• Derived category
• Triangulated category
• Perfect complex
• Semiorthogonal decomposition
• Fourier–Mukai transform
• Bridgeland stability condition
• Homological mirror symmetry
• Derived Categories notes - http://www.math.lsa.umich.edu/~idolga/derived9.pdf

{{reflist}}

• [https://www.sciencedirect.com/science/article/pii/S0021869308001427 A noncommutative version of Beilinson's theorem]
• Derived Categories of Toric Varieties
• Derived Categories of Toric Varieties II

Category:Algebraic geometry

Category:Noncommutative geometry