jesse burke (joint with mark e. walker)jbergner/burke.pdf · matrix factorizations over projective...
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Matrix factorizations over projective schemes
Jesse Burke(joint with Mark E. Walker)
Department of MathematicsUniversity of California, Los Angeles
January 11, 2013
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Matrix factorizations
Let Q be a commutative ring and f an element of Q.
A matrix factorization of f consists of
free Q-modules G1,G0
maps (i.e., matrices) g1 : G1 → G0, g0 : G0 → G1
such that g1 ◦ g0 = f · 1G0 and g0 ◦ g1 = f · 1G1 .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Matrix factorizations
Let Q be a commutative ring and f an element of Q.
A matrix factorization of f consists of
free Q-modules G1,G0
maps (i.e., matrices) g1 : G1 → G0, g0 : G0 → G1
such that g1 ◦ g0 = f · 1G0 and g0 ◦ g1 = f · 1G1 .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 4: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/4.jpg)
Example
Q = C[[x , y ]] and f = x3 + y 2.
A matrix factorization of f is given by setting G1 = G0 = Q2 and
g1 =
[y xx2 −y
]g0 =
[y xx2 −y
]
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Example
Q = C[[x , y ]] and f = x3 + y 2.
A matrix factorization of f is given by setting G1 = G0 = Q2 and
g1 =
[y xx2 −y
]g0 =
[y xx2 −y
]
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 6: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/6.jpg)
Matrix factorizations were introduced by Eisenbud in 1980. Heshowed that if Q is a regular local ring, and we set R = Q/(f ),then the free resolution of every finitely generated R-module isdetermined after at most d = dim Q steps by a matrixfactorization of f .
Example: if R = C[[x , y ]]/(x3 + y 2), then the free resolution ofR/(x , y) has the form:
. . .→ R2
y xx2 −y
−−−−−−−−−→ R2
y xx2 −y
−−−−−−−−−→ R2
[y x
]−−−−−−→ R1 → 0
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Matrix factorizations were introduced by Eisenbud in 1980. Heshowed that if Q is a regular local ring, and we set R = Q/(f ),then the free resolution of every finitely generated R-module isdetermined after at most d = dim Q steps by a matrixfactorization of f .
Example: if R = C[[x , y ]]/(x3 + y 2), then the free resolution ofR/(x , y) has the form:
. . .→ R2
y xx2 −y
−−−−−−−−−→ R2
y xx2 −y
−−−−−−−−−→ R2
[y x
]−−−−−−→ R1 → 0
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 8: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/8.jpg)
Homotopy category of matrix factorizations
We can form a dg-category of matrix factorizations analogously tothe dg-category of chain complexes of a ring:
the objects are matrix factorizations G,H of f ∈ Q and we denotethe Hom-complexes as
HomMF (G,H).
The homotopy category of matrix factorizations, denoted[MF (Q, f )], is the homotopy category of this dg-category.Morphisms are given by:
Hom[MF ](G,H) = H0 HomMF (G,H).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Homotopy category of matrix factorizations
We can form a dg-category of matrix factorizations analogously tothe dg-category of chain complexes of a ring:the objects are matrix factorizations G,H of f ∈ Q and we denotethe Hom-complexes as
HomMF (G,H).
The homotopy category of matrix factorizations, denoted[MF (Q, f )], is the homotopy category of this dg-category.Morphisms are given by:
Hom[MF ](G,H) = H0 HomMF (G,H).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Homotopy category of matrix factorizations
We can form a dg-category of matrix factorizations analogously tothe dg-category of chain complexes of a ring:the objects are matrix factorizations G,H of f ∈ Q and we denotethe Hom-complexes as
HomMF (G,H).
The homotopy category of matrix factorizations, denoted[MF (Q, f )], is the homotopy category of this dg-category.Morphisms are given by:
Hom[MF ](G,H) = H0 HomMF (G,H).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 11: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/11.jpg)
To say this more concretely, if
G = (G1g1−→ G0
g0−→ G1)
H = (H1h1−→ H0
h0−→ H1)
are matrix factorizations of f , then an element α ofHom[MF ](G,H) is an equivalence class of a pair of maps
α1 : G1 → H1 α0 : G0 → H0
such that the following squares commute:
G1
α1
��
g1 // G0
α0
��
g0 // G1
α1
��H1
h1 // H0h0 // H1,
under the usual homotopy equivalence relation.Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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As in the case of complexes, the homotopy category of matrixfactorizations is triangulated.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Motivated in part by matrix factorizations, Buchweitz introducedin 1986 the stable or singularity category of a ring R:
Dsg(R) = Db(R)/Perf(R),
where Db(R) is the bounded derived category of finitely generatedR-modules and Perf(R) is the full subcategory of complexesquasi-isomorphic to a bounded complex of finitely generatedprojective R-modules, i.e., perfect complexes.
Note that this category is trivial if and only if R has finite globaldimension, so it in some sense captures information on thesingularity of R.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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Motivated in part by matrix factorizations, Buchweitz introducedin 1986 the stable or singularity category of a ring R:
Dsg(R) = Db(R)/Perf(R),
where Db(R) is the bounded derived category of finitely generatedR-modules and Perf(R) is the full subcategory of complexesquasi-isomorphic to a bounded complex of finitely generatedprojective R-modules, i.e., perfect complexes.
Note that this category is trivial if and only if R has finite globaldimension, so it in some sense captures information on thesingularity of R.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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If G = (G1g1−→ G0
g0−→ G1) is a matrix factorization of f ∈ Q, andR = Q/(f ), then coker g1 is naturally an R-module.(i.e., f · G0 ⊆ Image(g1).)
Buchweitz noted that in case Q is a regular local ring, the functor
[MF (Q, f )]Ψ−→ Dsg(R)
which acts on objects by
G = (G1g1−→ G0
g0−→ G1) 7→ coker g1 ∈ Dsg(R)
is a triangulated equivalence.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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If G = (G1g1−→ G0
g0−→ G1) is a matrix factorization of f ∈ Q, andR = Q/(f ), then coker g1 is naturally an R-module.(i.e., f · G0 ⊆ Image(g1).)
Buchweitz noted that in case Q is a regular local ring, the functor
[MF (Q, f )]Ψ−→ Dsg(R)
which acts on objects by
G = (G1g1−→ G0
g0−→ G1) 7→ coker g1 ∈ Dsg(R)
is a triangulated equivalence.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 17: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/17.jpg)
An idea of why the functor
[MF (Q, f )]Ψ−→ Dsg(R)
G = (G1g1−→ G0
g0−→ G1) 7→ coker g1 ∈ Dsg(R)
is an equivalence:
The essential surjectivity of Ψ follows from Eisenbud’s theoremthat every R-module is determined up to syzygy by a matrixfactorization and by design the syzygy functor is invertible on thecategory Dsg(R).
The kernel of Ψ is zero because if coker g1 = 0 ∈ Dsg(R), thencoker g1 is a projective R-module (easy Lemma) and one can usethe lifting property of projectives to find a null-homotopy of G.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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An idea of why the functor
[MF (Q, f )]Ψ−→ Dsg(R)
G = (G1g1−→ G0
g0−→ G1) 7→ coker g1 ∈ Dsg(R)
is an equivalence:
The essential surjectivity of Ψ follows from Eisenbud’s theoremthat every R-module is determined up to syzygy by a matrixfactorization and by design the syzygy functor is invertible on thecategory Dsg(R).
The kernel of Ψ is zero because if coker g1 = 0 ∈ Dsg(R), thencoker g1 is a projective R-module (easy Lemma) and one can usethe lifting property of projectives to find a null-homotopy of G.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 19: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/19.jpg)
An idea of why the functor
[MF (Q, f )]Ψ−→ Dsg(R)
G = (G1g1−→ G0
g0−→ G1) 7→ coker g1 ∈ Dsg(R)
is an equivalence:
The essential surjectivity of Ψ follows from Eisenbud’s theoremthat every R-module is determined up to syzygy by a matrixfactorization and by design the syzygy functor is invertible on thecategory Dsg(R).
The kernel of Ψ is zero because if coker g1 = 0 ∈ Dsg(R), thencoker g1 is a projective R-module (easy Lemma) and one can usethe lifting property of projectives to find a null-homotopy of G.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 20: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/20.jpg)
So when R = Q/(f ) for Q a regular local ring, we have a veryconcrete description of the “stable” homological algebra: it is allpackaged in matrix factorizations.
This has led to results such as Polishchuk and Vaintrob’s theory ofChern characters and Hirzebruch-Riemann-Roch theorem formodules over such a hypersurface (as discussed in Mark’s talk).These are very striking results motivated by physics that havesolved some open problems in commutative algebra.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
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So when R = Q/(f ) for Q a regular local ring, we have a veryconcrete description of the “stable” homological algebra: it is allpackaged in matrix factorizations.
This has led to results such as Polishchuk and Vaintrob’s theory ofChern characters and Hirzebruch-Riemann-Roch theorem formodules over such a hypersurface (as discussed in Mark’s talk).These are very striking results motivated by physics that havesolved some open problems in commutative algebra.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 22: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/22.jpg)
We want to consider the following generalization.
Let X be a regular closed subscheme of projective space over somecommutative Noetherian ring Q:
i : X ↪→ PnQ = Proj(Q[T1, . . . ,Tn]).
Let W ∈ Γ(X , i∗O(1)) be a nonzero global section, and let
Y = Proj(Q[T1, . . . ,Tc ]/(W )) ⊆ X
be the zero subscheme of W .
Example: If X = P0Q∼= Spec Q, W ∈ Γ(X ,O(1)) = Q, and
R = Q/(W ), then Y = Spec R ⊆ Spec Q = X .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 23: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/23.jpg)
We want to consider the following generalization.Let X be a regular closed subscheme of projective space over somecommutative Noetherian ring Q:
i : X ↪→ PnQ = Proj(Q[T1, . . . ,Tn]).
Let W ∈ Γ(X , i∗O(1)) be a nonzero global section, and let
Y = Proj(Q[T1, . . . ,Tc ]/(W )) ⊆ X
be the zero subscheme of W .
Example: If X = P0Q∼= Spec Q, W ∈ Γ(X ,O(1)) = Q, and
R = Q/(W ), then Y = Spec R ⊆ Spec Q = X .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 24: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/24.jpg)
We want to consider the following generalization.Let X be a regular closed subscheme of projective space over somecommutative Noetherian ring Q:
i : X ↪→ PnQ = Proj(Q[T1, . . . ,Tn]).
Let W ∈ Γ(X , i∗O(1)) be a nonzero global section, and let
Y = Proj(Q[T1, . . . ,Tc ]/(W )) ⊆ X
be the zero subscheme of W .
Example: If X = P0Q∼= Spec Q, W ∈ Γ(X ,O(1)) = Q, and
R = Q/(W ), then Y = Spec R ⊆ Spec Q = X .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 25: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/25.jpg)
We want to consider the following generalization.Let X be a regular closed subscheme of projective space over somecommutative Noetherian ring Q:
i : X ↪→ PnQ = Proj(Q[T1, . . . ,Tn]).
Let W ∈ Γ(X , i∗O(1)) be a nonzero global section, and let
Y = Proj(Q[T1, . . . ,Tc ]/(W )) ⊆ X
be the zero subscheme of W .
Example: If X = P0Q∼= Spec Q, W ∈ Γ(X ,O(1)) = Q, and
R = Q/(W ), then Y = Spec R ⊆ Spec Q = X .
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 26: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/26.jpg)
A matrix factorization of W consists of
locally free coherent sheaves E1, E0 on X
maps e1 : E1 → E0, e0 : E0 → E1 ⊗O(1)
such that
e0 ◦ e1 = 1⊗W : E1 → E1 ⊗O(1) and
(e1 ⊗ 1O(1)) ◦ e0 = 1⊗W : E0 → E0 ⊗O(1),
where we think of W as a map O → O(1).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 27: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/27.jpg)
A matrix factorization of W consists of
locally free coherent sheaves E1, E0 on X
maps e1 : E1 → E0, e0 : E0 → E1 ⊗O(1)
such that
e0 ◦ e1 = 1⊗W : E1 → E1 ⊗O(1) and
(e1 ⊗ 1O(1)) ◦ e0 = 1⊗W : E0 → E0 ⊗O(1),
where we think of W as a map O → O(1).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 28: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/28.jpg)
A matrix factorization of W consists of
locally free coherent sheaves E1, E0 on X
maps e1 : E1 → E0, e0 : E0 → E1 ⊗O(1)
such that
e0 ◦ e1 = 1⊗W : E1 → E1 ⊗O(1) and
(e1 ⊗ 1O(1)) ◦ e0 = 1⊗W : E0 → E0 ⊗O(1),
where we think of W as a map O → O(1).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 29: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/29.jpg)
We are interested in these non-affine matrix factorizations becauseof their relation to affine complete intersection rings (for details onthis see arXiv:1205.2552). Other people (Orlov,Polischuk/Vaintrob, Positselski,. . . ) have studied non-affine matrixfactorizations motivated by homological mirror symmetry.
Our goal is to generalize the equivalence of Buchweitz by giving arelation between the “homotopy category” of non-affine matrixfactorizations and the singularity category of Y ⊆ X (known in thecase X = P0
Q).
The singularity category of a scheme Y is
Dsg(Y ) = Db(coh Y )/Perf Y
where a complex of coherent sheaves on Y is perfect if it locallyquasi-isomorphic to a bounded complex of free modules.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 30: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/30.jpg)
We are interested in these non-affine matrix factorizations becauseof their relation to affine complete intersection rings (for details onthis see arXiv:1205.2552). Other people (Orlov,Polischuk/Vaintrob, Positselski,. . . ) have studied non-affine matrixfactorizations motivated by homological mirror symmetry.
Our goal is to generalize the equivalence of Buchweitz by giving arelation between the “homotopy category” of non-affine matrixfactorizations and the singularity category of Y ⊆ X (known in thecase X = P0
Q).
The singularity category of a scheme Y is
Dsg(Y ) = Db(coh Y )/Perf Y
where a complex of coherent sheaves on Y is perfect if it locallyquasi-isomorphic to a bounded complex of free modules.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 31: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/31.jpg)
We are interested in these non-affine matrix factorizations becauseof their relation to affine complete intersection rings (for details onthis see arXiv:1205.2552). Other people (Orlov,Polischuk/Vaintrob, Positselski,. . . ) have studied non-affine matrixfactorizations motivated by homological mirror symmetry.
Our goal is to generalize the equivalence of Buchweitz by giving arelation between the “homotopy category” of non-affine matrixfactorizations and the singularity category of Y ⊆ X (known in thecase X = P0
Q).
The singularity category of a scheme Y is
Dsg(Y ) = Db(coh Y )/Perf Y
where a complex of coherent sheaves on Y is perfect if it locallyquasi-isomorphic to a bounded complex of free modules.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 32: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/32.jpg)
What is the “homotopy” category of non-affine matrixfactorizations?
First we naively mimic the affine case: to matrix factorizations E,Fof W , we associate a chain complex HomMF (E,F) of coherentsheaves on X .
We define [MF (X ,O(1),W )]naive to be the category withmorphisms
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 33: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/33.jpg)
What is the “homotopy” category of non-affine matrixfactorizations?
First we naively mimic the affine case: to matrix factorizations E,Fof W , we associate a chain complex HomMF (E,F) of coherentsheaves on X .
We define [MF (X ,O(1),W )]naive to be the category withmorphisms
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 34: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/34.jpg)
What is the “homotopy” category of non-affine matrixfactorizations?
First we naively mimic the affine case: to matrix factorizations E,Fof W , we associate a chain complex HomMF (E,F) of coherentsheaves on X .
We define [MF (X ,O(1),W )]naive to be the category withmorphisms
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 35: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/35.jpg)
To say this more concretely,
if
E = (E1e1−→ E0
e0−→ E1(1))
F = (F1f1−→ F0
f0−→ F1(1))
are matrix factorizations of W , then an element h ofHom[MF ]naive
(E,F) is an equivalence class of a pair of maps
h1 : E1 → F1 h0 : E0 → F0
that make the obvious diagrams commute, under the usualhomotopy equivalence.
[MF (X ,O(1),W )]naive is a triangulated category.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 36: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/36.jpg)
To say this more concretely, if
E = (E1e1−→ E0
e0−→ E1(1))
F = (F1f1−→ F0
f0−→ F1(1))
are matrix factorizations of W
, then an element h ofHom[MF ]naive
(E,F) is an equivalence class of a pair of maps
h1 : E1 → F1 h0 : E0 → F0
that make the obvious diagrams commute, under the usualhomotopy equivalence.
[MF (X ,O(1),W )]naive is a triangulated category.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 37: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/37.jpg)
To say this more concretely, if
E = (E1e1−→ E0
e0−→ E1(1))
F = (F1f1−→ F0
f0−→ F1(1))
are matrix factorizations of W , then an element h ofHom[MF ]naive
(E,F) is an equivalence class of a pair of maps
h1 : E1 → F1 h0 : E0 → F0
that make the obvious diagrams commute, under the usualhomotopy equivalence.
[MF (X ,O(1),W )]naive is a triangulated category.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 38: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/38.jpg)
To say this more concretely, if
E = (E1e1−→ E0
e0−→ E1(1))
F = (F1f1−→ F0
f0−→ F1(1))
are matrix factorizations of W , then an element h ofHom[MF ]naive
(E,F) is an equivalence class of a pair of maps
h1 : E1 → F1 h0 : E0 → F0
that make the obvious diagrams commute, under the usualhomotopy equivalence.
[MF (X ,O(1),W )]naive is a triangulated category.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 39: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/39.jpg)
As in the affine case, if E = (E1e1−→ E0
e0−→ E1(1)) is a matrixfactorization of W , then coker e1 has support contained in Y ⊆ X ,i.e., is naturally a coherent sheaf on Y .
There is a triangulated functor
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
and it’s not too hard to show that this functor is essentiallysurjective.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 40: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/40.jpg)
As in the affine case, if E = (E1e1−→ E0
e0−→ E1(1)) is a matrixfactorization of W , then coker e1 has support contained in Y ⊆ X ,i.e., is naturally a coherent sheaf on Y .
There is a triangulated functor
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
and it’s not too hard to show that this functor is essentiallysurjective.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 41: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/41.jpg)
As in the affine case, if E = (E1e1−→ E0
e0−→ E1(1)) is a matrixfactorization of W , then coker e1 has support contained in Y ⊆ X ,i.e., is naturally a coherent sheaf on Y .
There is a triangulated functor
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
and it’s not too hard to show that this functor is essentiallysurjective.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 42: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/42.jpg)
Have the essentially surjective functor:
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
Unfortunately it is not always the case that the kernel of Ψ is zero:if coker e1 = 0 ∈ Dsg(Y ), then coker e1 is locally free, and so E willlocally be null-homotopic, but there are examples which show thatE need not be null-homotopic.
Thus Ψ will not in general be an equivalence. To find anequivalence between a category of matrix factorizations andDsg(Y ) we need to keep track of more data than is contained in[MF (X ,O(1),W )]naive.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 43: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/43.jpg)
Have the essentially surjective functor:
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
Unfortunately it is not always the case that the kernel of Ψ is zero:if coker e1 = 0 ∈ Dsg(Y ), then coker e1 is locally free, and so E willlocally be null-homotopic, but there are examples which show thatE need not be null-homotopic.
Thus Ψ will not in general be an equivalence. To find anequivalence between a category of matrix factorizations andDsg(Y ) we need to keep track of more data than is contained in[MF (X ,O(1),W )]naive.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 44: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/44.jpg)
Have the essentially surjective functor:
Ψ : [MF (X ,O(1),W )]naive → Dsg(Y )
E = (E1e1−→ E0
e0−→ E1(1)) 7→ coker e1
Unfortunately it is not always the case that the kernel of Ψ is zero:if coker e1 = 0 ∈ Dsg(Y ), then coker e1 is locally free, and so E willlocally be null-homotopic, but there are examples which show thatE need not be null-homotopic.
Thus Ψ will not in general be an equivalence. To find anequivalence between a category of matrix factorizations andDsg(Y ) we need to keep track of more data than is contained in[MF (X ,O(1),W )]naive.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 45: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/45.jpg)
Recall that the definition of morphisms in [MF (X ,O(1),W )]naive is
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
To remedy the deficiency of the “naive” homotopy category,instead of taking global sections Γ(X ,−) we should take derivedglobal sections or hypercohomology
H0RΓ(X ,HomMF (E,F)).
Hypercohomology may be computed using a semi-injectiveresolution (fibrant replacement) of Γ(X ,HomMF (E,F)) or thetotal complex of the bicomplex which is the Cech complex on everydegree of HomMF (E,F).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 46: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/46.jpg)
Recall that the definition of morphisms in [MF (X ,O(1),W )]naive is
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
To remedy the deficiency of the “naive” homotopy category,instead of taking global sections Γ(X ,−) we should take derivedglobal sections or hypercohomology
H0RΓ(X ,HomMF (E,F)).
Hypercohomology may be computed using a semi-injectiveresolution (fibrant replacement) of Γ(X ,HomMF (E,F)) or thetotal complex of the bicomplex which is the Cech complex on everydegree of HomMF (E,F).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 47: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/47.jpg)
Recall that the definition of morphisms in [MF (X ,O(1),W )]naive is
Hom[MF ]naive(E,F) = H0Γ(X ,HomMF (E,F)).
To remedy the deficiency of the “naive” homotopy category,instead of taking global sections Γ(X ,−) we should take derivedglobal sections or hypercohomology
H0RΓ(X ,HomMF (E,F)).
Hypercohomology may be computed using a semi-injectiveresolution (fibrant replacement) of Γ(X ,HomMF (E,F)) or thetotal complex of the bicomplex which is the Cech complex on everydegree of HomMF (E,F).
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 48: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/48.jpg)
Define the homotopy category of matrix factorizations of W to bethe category [MF (X ,O(1),W )] with objects the same as those of[MF (X ,O(1),W )]naive. Morphisms are given by
Hom[MF ](E,F) := H0RΓ(X ,HomMF (E,F)).
There is a natural transformation Γ(X ,−)→ RΓ(X ,−) whichgives rise to a functor
[MF (X ,O(1),W ]naive → [MF (X ,O(1),W )]
that is the identity on objects.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 49: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/49.jpg)
Define the homotopy category of matrix factorizations of W to bethe category [MF (X ,O(1),W )] with objects the same as those of[MF (X ,O(1),W )]naive. Morphisms are given by
Hom[MF ](E,F) := H0RΓ(X ,HomMF (E,F)).
There is a natural transformation Γ(X ,−)→ RΓ(X ,−) whichgives rise to a functor
[MF (X ,O(1),W ]naive → [MF (X ,O(1),W )]
that is the identity on objects.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 50: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/50.jpg)
Theorem ( -, Walker)
There is a triangulated equivalence
Φ : [MF (X ,O(1),W )]→ Dsg(Y ).
that fits into the following commutative diagram:
[MF (X ,O(1),W )]naiveΨ //
��
Dsg(Y )
[MF (X ,O(1),W )]
Φ
66
In particular, for E = (E1e1−→ E0
e0−→ E1(1),
Φ(E) = coker e1.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 51: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/51.jpg)
Idea of proof: the functor Φ is essentially surjective, so we need toshow that it’s fully faithful.
Let E,F be objects of [MF (X ,O(1),W )]. For every m ∈ Z, wehave a commutative diagram
HmΓ(X ,HomMF (E,F))
��
// HomDsg(Y )(cokerE[−m], cokerF)
HmRΓ(X ,HomMF (E,F))
The vertical arrow is an isomorphism for m� 0 by Serre Vanishing(uses that X is projective). The horizontal arrow is an isomorphismfor m� 0 by classical arguments for matrix factorizations.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 52: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/52.jpg)
So for all m� 0 we have isomorphisms:
HmRΓ(X , Γ(X ,HomMF (E,F))→ HomDsg(Y )(cokerE[−m], cokerF)
Now we can use a totaling of the Koszul complex (also uses that Xis projective) and the fact that RΓ respects quasi-isomorphisms toget an isomorphism in one degree less:
Hm−1RΓ(X , Γ(X ,HomMF (E,F))→ HomDsg(Y )(cokerE[−m+1], cokerF)
and by induction we have an isomorphism for m = 0.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 53: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/53.jpg)
So for all m� 0 we have isomorphisms:
HmRΓ(X , Γ(X ,HomMF (E,F))→ HomDsg(Y )(cokerE[−m], cokerF)
Now we can use a totaling of the Koszul complex (also uses that Xis projective) and the fact that RΓ respects quasi-isomorphisms toget an isomorphism in one degree less:
Hm−1RΓ(X , Γ(X ,HomMF (E,F))→ HomDsg(Y )(cokerE[−m+1], cokerF)
and by induction we have an isomorphism for m = 0.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 54: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/54.jpg)
Polishchuk and Vaintrob proved a similar theorem (which wasposted before ours, works for smooth Deligne-Mumford stacks, anddoes not require X to be projective). They showed that theVerdier quotient of [MF ]naive by the locally nullhomotopic objectsis equivalent to Dsg(Y ).
Our version has two advantages:
The Hom-spaces in our category have a very explicitdescription in terms of a Cech complex which gives rise to auseful spectral sequence.
Our methods let us prove a relative version in which X is notassumed to be regular.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 55: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/55.jpg)
Polishchuk and Vaintrob proved a similar theorem (which wasposted before ours, works for smooth Deligne-Mumford stacks, anddoes not require X to be projective). They showed that theVerdier quotient of [MF ]naive by the locally nullhomotopic objectsis equivalent to Dsg(Y ).
Our version has two advantages:
The Hom-spaces in our category have a very explicitdescription in terms of a Cech complex which gives rise to auseful spectral sequence.
Our methods let us prove a relative version in which X is notassumed to be regular.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 56: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/56.jpg)
Polishchuk and Vaintrob proved a similar theorem (which wasposted before ours, works for smooth Deligne-Mumford stacks, anddoes not require X to be projective). They showed that theVerdier quotient of [MF ]naive by the locally nullhomotopic objectsis equivalent to Dsg(Y ).
Our version has two advantages:
The Hom-spaces in our category have a very explicitdescription in terms of a Cech complex which gives rise to auseful spectral sequence.
Our methods let us prove a relative version in which X is notassumed to be regular.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes
![Page 57: Jesse Burke (joint with Mark E. Walker)jbergner/burke.pdf · Matrix factorizations over projective schemes Jesse Burke (joint with Mark E. Walker) Department of Mathematics University](https://reader031.vdocument.in/reader031/viewer/2022030220/5b8109d37f8b9a466b8b7517/html5/thumbnails/57.jpg)
References:
Ragnar-Olaf Buchweitz, “Maximal Cohen-Macaulay modulesand Tate cohomology over Gorenstein rings, ” 1986.Unpublished manuscript.Jesse Burke, Mark E. Walker, “Matrix factorizations overprojective schemes,” HHA, 2012.David Eisenbud, “Homological algebra on a completeintersection with applications to group representations,”TAMS, 1980.Dimitri Orlov, “Matrix factorizations for non-affine LGmodels,” Math. Ann., 2012.Alexander Polishchuk and Arkady Vaintrob, “Matrixfactorizations and singularity categories for stacks,”arXiv:1011.4544.Leonid Positselski, “Coherent analogues of matrixfactorizations and relative singularity categories,”arXiv:1102.0261.
Jesse Burke (joint with Mark E. Walker) UCLA
Matrix factorizations over projective schemes