Thom-Sebastiani & Duality for Matrix Factorizations

Anatoly Preygel

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Thom-Sebastiani & Duality for Matrix Factorizations

Anatoly Preygel

Algebraic Geometry · Submitted 30 Jan 2011 · 2 references

Pith's verdict: A real, well-executed k[[β]]-linear Thom-Sebastiani and duality theorem; the weak point is the identity-functor identification in 4.2.3(iv), not the f^! formalism the reader flagged.

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arXiv ID1101.5834 (version 1)

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CitationAnatoly Preygel. "Thom-Sebastiani & Duality for Matrix Factorizations." arXiv:1101.5834 (2011). https://pith.science/paper/LGD3FSH4

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The thing to know: this delivers the Thom-Sebastiani and duality statements people expected for matrix factorizations, without isolated-singularity hypotheses, and it does so by working k[[β]]-linearly on DCoh of the zero fiber rather than in the 2-periodic category. That refinement is the actual contribution — it lets the tensor-product and functor-category theorems be proved in ordinary coherent geometry. The geometric model of the k[[β]]-structure via B = 0×_{A1}0 (Construction 3.1.5, Prop. 3.1.4) is the trick that makes the rest go, and it is a good one. The applications in §8–9 are genuine consequences, not restatements; the metabolic Knörrer periodicity and the Clifford description (9.1.7, 9.3.4) are the parts I'd actually use. Note the classifier says no stated limitations; that's wrong — the author flags gaps in 1.4(i), Remark 3.4.5, Remark 4.2.5, and the sign warning in 8.2.4. He is unusually candid, which counts for something. On the reader's f^! worry: I think it lands less hard than stated. 4.1.3 reduces to the k-linear exterior-product theorem plus a two-parameter Cor. 3.2.4, and 4.2.2's perfect-pairing check reduces to classical Grothendieck duality on smooth M0 with standard base-change. The genuinely load-bearing and least-secure step is 4.2.3(iv): identifying ω_{Δ,Z} = Δ_* RΓ_Z ω_M with the identity functor. That uses the D1/D2/Δ1/Δ2 base-change diagram, the interaction of RΓ_Z with the k[[β]]-structure, and the author himself calls the proof 'somewhat opaque.' Remarks 4.2.6–4.2.7 make it worse: the two natural dualities (ω_{M0} vs ω_{M0/M}) give different identifications of trace and identity, so the formula is convention-dependent and needs checking against the specific equivalence chosen. If that is off by a shift or β-twist, 4.2.3(iv)–(v) and the trace-dependent applications (8.1.1 smoothness, 8.3.4 Calabi-Yau) weaken; 4.1.3 and 4.2.2 survive. §5 is sketch-level by the author's own admission, and 3.4.4's comparison to literal matrix factorizations is asserted without a natural functor (Remark 3.4.5 admits this). None of that is fatal. This is for people working in matrix factorizations, singularity categories, or the dg-category side of mirror symmetry. It deserves a serious referee, mainly to pin down 4.2.3(iv).

Referee report

Significance

If correct, the paper settles the expected Thom-Sebastiani and duality statements for matrix factorizations in a generality that goes well beyond isolated singularities, and the applications (smoothness, Calabi-Yau, Knörrer periodicity, Hochschild invariants) are substantial and were open in this generality. The organizing principle — the k[[β]]-linear structure as a derived group action, and the S^1-action / Tate-construction perspective on MF — is novel and unifying, and the paper is careful to reduce k[[β]]-linear statements to k-linear ones where possible. The main theorems 4.1.3 and 4.2.2/4.2.3 have real proofs in §4, and the author is explicit about which parts of the paper are sketch-level. The paper is a significant contribution to the field.

Major objections (4)

[§4.2.3(iv)-(v), Remarks 4.2.5-4.2.7] The identification of ω_{Δ,Z} = Δ_* RΓ_Z ω_M with the identity functor is the least-secure step in the paper. The author calls the proof 'somewhat opaque' (Remark 4.2.5) and warns (Remarks 4.2.6-4.2.7) that the two natural dualities (ω_{M0} vs ω_{M0/M}) give different explicit identifications of the trace and identity functors. This identification is load-bearing for Theorem 8.1.1(ii) (smoothness, which reduces to compactness of ω_{Δ,Z}), Theorem 8.2.6 (Hochschild invariants), and Theorem 8.3.4 (Calabi-Yau, whose non-degeneracy check uses [vol_M]^{Tate} being an equivalence). The paper should either verify the identification under the specific convention chosen, or show that the applications are insensitive to the choice of duality (e.g., that a shift or β-twist in ω_{Δ,Z} does not affect the compactness and non-degeneracy conclusions). As written, the reader cannot tell whether the formula in 4.2.3(iv) is correct under the chosen convention or under the other one.
[§1.4(i), Appendix B (Prop. B.3.2, Theorem B.2.4)] The Grothendieck duality / upper-shriek formalism for QC! on derived schemes and nice derived DM stacks is invoked without a published reference; the author states in §1.4(i) that 'a reference for this does not yet exist in the literature' and that it 'will probably appear in a later volume of Jacobi Lurie's DAG series.' Appendix B is the foundation for the tensor-product theorem and the functor-category description, and it is not clear from the text which parts of Appendix B are proved from scratch and which depend on the unpublished f^! formalism. The paper should state precisely which inputs are proved in Appendix B and which are conditional on the forthcoming reference, and ideally give a self-contained proof of the key inputs (or a precise statement of the result used from the forthcoming source). The concrete proof in §4 mitigates this for 4.1.3 and 4.2.2, but not for the parts of Appendix B that feed into 4.2.3.
[§5 (Part 2), Theorems 5.2.7, 5.2.8, Lemmas 5.5.1, 5.5.2] Section 5 is explicitly sketch-level: the author writes 'we will only sketch these proofs (being cavalier about these compatibilities)' and Theorem 5.2.8 and Lemma 5.5.2 are labelled 'Sketch.' While the main theorems 4.1.3 and 4.2.2/4.2.3 are proved in §4 independently of §5, §5 is the basis for the 'comparison of three viewpoints' (§7) and for the S^1-action perspective that the author says 'formed the basis for the author's thinking on the subject' (§1.5). The paper should either complete these proofs or clearly delimit which results depend on them, so that a reader can tell which statements are established and which are conditional.
[§3.4.4, Remark 3.4.5] The comparison to literal matrix factorizations via Orlov's theorem is only stated for flat f; the f=0 case is handled separately and the author acknowledges 'It seems likely that one can remedy these, but the author has not tried to check the details.' This is a gap in the justification of the notation MF(M,f), though not in the main theorems. The paper should either supply the missing comparison or state clearly that the identification with literal matrix factorizations is only established under the flatness hypothesis, and that the notation MF(M,f) is a definition in general.

Minor objections (5)

[Abstract] The abstract uses k[t] with deg t = -2, while the body uses k[[β]] with deg β = -2. The relationship between the two notations should be clarified in the abstract or made consistent with the body.
[§1.2, Lemma 3.1.9, proof of Lemma 3.1.9, Lemma 5.4.5, Theorem 9.1.7, proof of Theorem 8.3.4, Remark 8.2.5] There are several typographical errors: 'out study' in §1.2; 'k⊗k[ [bt] ]−' in Lemma 3.1.9 (should be k⊗k[[β]]−); 'once we show identify G' in the proof of Lemma 3.1.9 (should be 'once we show to identify G'); 'the conclusion of the conclusion' in Lemma 5.4.5; 'a a relative form' in Theorem 9.1.7; 'Will will produce' in the proof of Theorem 8.3.4; 'of of' and 'the the' in Remark 8.2.5.
[Remark 3.1.7, Proposition 8.2.4] The author flags sign uncertainties: Remark 3.1.7 says 'There is a choice of sign appearing here that will probably change at random later,' and Proposition 8.2.4 says '(Warning: I don't know the sign rules well enough to be sure of the sign.)' Given the sign-sensitive HKR and Calabi-Yau computations, these signs should be pinned down, or the dependence of the main results on the sign convention should be stated explicitly.
[§5.4] Section 5.4 is titled 'Graded speculation' and is explicitly conditional on a suitable Grothendieck duality over BGm. It should be marked more clearly as speculative (e.g., moved to a remark or an appendix) so that it is not read as a proved result.
[Throughout] The notation 'PreMF' and 'Pre MF' is used inconsistently (e.g., 'Pre MF' in §1.4(ii) and 'PreMF' elsewhere). The paper would benefit from a consistent choice.

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The paper

The paper claims

By keeping a formal degree-(-2) variable before inverting it, the paper reduces tensor products, functor categories, and Hochschild invariants of matrix factorizations to ordinary coherent sheaves on the zero loci of the sum and difference.

The reading

The paper's target is the 2-periodic category of matrix factorizations $MF(M,f)$ attached to a map $f: M to mathbb{A}^1$ — the algebraic shadow of a hypersurface singularity, or a Landau–Ginzburg model. The usual formulation is awkward because 2-periodicity throws away information and the standard tools (tensor products, duality, functor categories, Hochschild invariants) are only well understood when the critical locus is an isolated point. The author's move is to keep the bookkeeping: instead of working 2-periodically from the start, he attaches to $(M,f)$ a module structure over $k[[beta]]$ with $deg beta = -2$, calling the result $mathrm{PreMF}(M,f)$; its underlying category is nothing more exotic than coherent complexes on the zero fibre $f^{-1}(0)$, and inverting $beta$ recovers $MF(M,f)$. The main results say this refinement interacts exactly as one would want with the two operations that matter: tensoring two such categories gives the category for the sum potential $f boxplus g$ on the product (Thom–Sebastiani), and colimit-preserving functors between their Ind-completions form a matrix-factorization category for the difference potential $-f boxplus g$ (duality). If that is right, questions about matrix factorizations become questions about coherent sheaves on derived schemes, which is a far more familiar world.

What carries the argument

The load-bearing device is a derived group scheme $B = 0 times_{mathbb{A}^1} 0$, the derived self-intersection of the origin in the affine line, equipped simultaneously with a 'loop composition' and a 'pointwise addition' product. The author proves (Proposition 3.1.4) that $QC^!(B)$ with its convolution product is symmetric-monoidally equivalent to $k[[beta]]$-modules, via a Koszul–Tate resolution that identifies the circle action with multiplication by $beta$. The fibre $M_0 = f^{-1}(0)$ carries a right action of $B$ built from the homotopies $f(m) to 0$, and applying $QC^!$ turns this into the $k[[beta]]$-linear structure on $DCoh(M_0)$; an Eckmann–Hilton argument reconciles the two products. The second ingredient is the $QC^! = mathrm{Ind},DCoh$ formalism with its shriek pullback and integral transforms, developed in the appendices, which supplies the tensor-product theorem and the functor-category description used to state and prove the main results.

Extended reading notes

Core claim

The central claim is a pair of theorems. First (Theorem 4.1.3), for Landau–Ginzburg pairs $(M,f)$ and $(N,g)$ and closed subsets $Z_M subset f^{-1}(0)$, $Z_N subset g^{-1}(0)$, external tensor product followed by pushforward along the inclusion $ell: M_0 times N_0 to (f boxplus g)^{-1}(0)$ is a $k[[beta]]$-linear equivalence $mathrm{PreMF}_{Z_M}(M,f) otimes_{k[[beta]]} mathrm{PreMF}_{Z_N}(N,g) xrightarrow{sim} mathrm{PreMF}_{Z_M times Z_N}(M times N, f boxplus g)$, and after inverting $beta$ the analogous statement for the 2-periodic categories, with a version that sums over matching critical values. Second (Theorems 4.2.2 and 4.2.3), Grothendieck duality lifts to a $k[[beta]]$-linear anti-equivalence $mathrm{PreMF}(M,f)^{op} simeq mathrm{PreMF}(M,-f)$, and consequently the dg category of colimit-preserving $k[[beta]]$-linear functors between Ind-completions is equivalent to $mathrm{PreMF}^infty_{Z_M times Z_N}(M times N, -f boxplus g)$, with the identity functor and evaluation written explicitly as the diagonal structure sheaf and its dualizing complex. The author then harvests consequences: the expected 2-periodic Hochschild homology and cohomology computations, with $k[[beta]]$-linear refinements; smoothness of $MF(M,f)$ over $k((beta))$ and properness when $mathrm{crit}(f) cap f^{-1}(0)$ is proper; $m$-Calabi–Yau structures on $MF(M,f)$ from a volume form on an $m$-fold $M$; and a relative Knörrer periodicity that lets metabolic quadratic bundles be discarded, together with a description of matrix factorizations for a non-degenerate quadratic bundle as modules over its Clifford algebra.

Load-bearing premise

The paper depends on a Grothendieck duality and upper-shriek formalism for Ind-coherent complexes on derived schemes and derived Deligne–Mumford stacks that the author says has no written reference yet, and several supporting results (Theorem 5.2.7, Lemma 5.5.2, Proposition 4.1.6) are presented only as sketches.

Editorial extensions

If this is right

Tensor products of matrix factorization categories are computed by coherent complexes on the zero locus of the sum potential, with no isolated-singularity hypothesis and with support conditions tracked explicitly.
Colimit-preserving functors between matrix factorization categories are again a matrix factorization category — for the difference potential — so integral transforms, the identity functor, and evaluation acquire concrete geometric descriptions.
$MF(M,f)$ is smooth over $k((beta))$, and proper when the critical locus meeting $f^{-1}(0)$ is proper, giving a general smooth-and-proper statement beyond the isolated case.
A volume form on $M$ produces an $m$-Calabi–Yau structure on $MF(M,f)$, so Calabi–Yau structures on Landau–Ginzburg models come from honest differential forms on the total space.
Metabolic quadratic bundles can be eliminated: tensoring with the structure sheaf of a Lagrangian subbundle gives an equivalence from $mathrm{Perf}(X)((beta))$ to $MF(Q,q)$, and non-degenerate quadratic bundles give an equivalence with Clifford-algebra modules.
The 2-periodic Hochschild homology and cohomology of matrix factorizations are identified with Tate constructions on the Hochschild invariants of $DCoh(M)$, with $k[[beta]]$-linear refinements computed by HKR and local cohomology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the whole strategy is 'solve the $k[[beta]]$-linear problem, then base-change to $k((beta))$', it suggests looking for $k[[beta]]$-linear lifts of other 2-periodic invariants (negative cyclic, topological Hochschild) rather than computing them 2-periodically from scratch.
The paper notes that $mathrm{PreMF}$ is almost never smooth over $k[[beta]]$; this makes smoothness of $MF$ an artifact of inverting $beta$, and one might expect the same 'smooth only after localization' pattern in other periodic settings.
Remark 4.1.4's description of the total category as a convolution over critical values hints at a concrete computational tool: a convolution algebra on the set of critical values that could be used to assemble global invariants from local ones.
The metabolic Knörrer statement plus the Clifford-algebra description together suggest that the Witt-group invariant of a quadratic bundle is exactly its Clifford algebra class in a 2-periodic Brauer group; that identification could be tested against known Brauer-group computations.

Original abstract

The derived category of a hypersurface has an action by "cohomology operations" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Kn\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The derived category of a hypersurface has an action by "cohomology operations" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Kn\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The derived category of a hypersurface has an action by "cohomology operations" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations).	background 0.99	0.12	0.06	asserted 1.00
s1	We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition).	result 0.70	0.96	0.79	asserted 1.00
s2	These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations.	interpretation 0.88	0.83	0.18	asserted 0.99
s3	Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Kn\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.	result 0.74	0.95	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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levels run

P abstract

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S sections
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References

Works cited (2)

[1]

[BZFN] David Ben-Zvi, John Francis, and David Nadler, Integral transforms and Drinfeld centers in derived algebr aic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909–966, available at arxiv:0805.0157. [D] Vladimir Drinfeld, DG quotients of DG categories , J. Algebra 272 (2004), no. 2, 643–691, available at arxiv:math/0210114. [K] Alexander Kuznetsov, D...

View on Pith Pith review arXiv 2010
[2009]

[O1] Martin C. Olsson, On proper coverings of Artin stacks , Adv. Math. 198 (2005), no. 1, 93–106. [O2] Dmitri Orlov, Triangulated categories of singularities and D-branes in L andau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240–26 2, available at arxiv:math/0302304. [O3] , Derived categories of cohe...

View on Pith Pith review arXiv 2005

Forward citations

Cited by 5 Pith papers

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Models Rozansky-Witten theory of T*X via sheaves of categories from Perf(X×A¹), constructing hybrid Lagrangian objects whose Homs are matrix factorizations.
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How later papers used it

49 later papers cite it

Evidence: substantial. 49 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 36 of the 121 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 3 used its method or tool.

Uses its method or tool 3 of 4 papers with this label

Found the formalism of matrix factorizations via formal group actions on categories developed in this paper to be the most helpful for proving sufficiently natural and sheafified versions of such statements.

A microlocal Feigin-Tsygan-Preygel theorem (2023) · 1.00 · paraphrased: the licence does not allow quoting
Followed the steps of this paper to show that M lies in (X_0) if and only if there exists N≥0 such that u^N∼0.

Motivic Realizations of Singularity Categories and Vanishing Cycles (2016) · 0.99 · paraphrased: the licence does not allow quoting
Used this paper's approach to MF(X,f) to show how this works.

Fukaya categories of higher-genus surfaces and pants decompositions (2021) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

49 arXiv papers cite this in 123 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 80 sentences without a confident label

Builds on it 3 papers

In particular this extends [ref] to a base ring which we only require to be regular, instead of a field of characteristic zero.

Motivic Realizations of Singularity Categories and Vanishing Cycles · Section · 1.00
The following theorem is a variant over a more general base of the theorem stated and proven in appendix of [ref] .

A microlocal Feigin-Tsygan-Preygel theorem · Ind-coherent sheaves · 0.96
We also point to [ref] (and [ref] ) for results extending those in [ref] .

Fourier-Mukai functors: a survey · Introduction · 0.85

Uses its method or tool 5 papers

Then, for any $∈X_0X$, we have [equation] Notice that $q^*^∨_S≃^∨_X⊗__X_X_0$, thus we can continue [equation] We will construct, using $∙∙^⊗$, a lax monoidal $$-functor $∙,∙^⊗:_(S,_S)^,⊞→$, following [ref] and [ref] .

Motivic and $\ell$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle · The dg category of singularities of a twisted LG model · 0.88
In the task of proving sufficiently natural and sheafified versions of such statements, we found the formalism of matrix factorizations via formal group actions on categories developed in Preygel's thesis [ref] to be the most helpful.

A microlocal Feigin-Tsygan-Preygel theorem · Methods · 1.00
For this we follow the steps of [ref] .

Motivic Realizations of Singularity Categories and Vanishing Cycles · Section · 0.99

Compares with it 2 papers

This definition is essentially the same as given in [ref] , but not the same as the definition given in [ref] .

The moduli stack and motivic Hall algebra for the bounded derived category · Pseudo-coherence · 1.00
Preygel independently proved similar results for matrix factorizations on derived schemes, using a different set of tools from derived algebraic geometry [ref] .

Fractional Calabi-Yau Categories from Landau-Ginzburg Models · General Results · 0.86

Cites it as support 2 papers

We have an equivalence of $S$-dg-categories [equation] This is [ref] together with the fact that as $Y$ is smooth over $S$ and $S$ is assumed to be regular, $Y$ is regular and therefore $(Y)=(Y)$Here in fact we need to assume S is excellent.

Motivic Realizations of Singularity Categories and Vanishing Cycles · Section · 0.96
This functor is an equivalence (see e.g. [ref] ).

Bogomolov-Tian-Todorov theorem for Calabi-Yau categories · Section · 0.97

Cites it as background 18 papers

Segal and Căldăraru-Tu give such an HKR map, involving Hochschild homology of the second kind and without a support condition, in [ref] and [ref] , respectively; Efimov generalizes this result to the non-affine setting in [ref] ; and Preygel gives a map just as in ([ref]) (but also in the not-necessarily-affine setting), and proves it is a quasi-isomorphism, in [ref] .

A proof of a conjecture of Shklyarov · The HKR map · 0.88
We will briefly discuss this phenomenon, and various modern formulations of it that have appeared in recent years [ref] , and conclude by showing our first equivalence $Ψ_1$ in [ref].

The Pfaffian-Grassmannian equivalence revisited · Knörrer periodicity and the Grassmannian side · 0.88
The notion of very good stack was introduced in [ref] to refer to those stacks over which an integral kernel theorem for $$ holds, and over which the relative box product is an isomorphism.

A microlocal Feigin-Tsygan-Preygel theorem · Very good stacks · 0.96

Discussion

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[1] [1]

[BZFN] David Ben-Zvi, John Francis, and David Nadler, Integral transforms and Drinfeld centers in derived algebr aic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909–966, available at arxiv:0805.0157. [D] Vladimir Drinfeld, DG quotients of DG categories , J. Algebra 272 (2004), no. 2, 643–691, available at arxiv:math/0210114. [K] Alexander Kuznetsov, D...

View on Pith Pith review arXiv 2010

[2] [2009]

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