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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 24. Выпуск 1.
УДК 512 DOI 10.22405/2226-8383-2023-24-1-203-212
Когомологии де Рама алгебры полиномиальных функций на симплициальном комплексе1
И. С. Басков
Басков Игорь Сергеевич — Математический институт им. В. А. Стеклова РАН (г. Санкт-Петербург) .
e-mail: baskovigor<§pdmi. ras. ru
Аннотация
Мы рассматриваем алгебру А°(Х) полиномиальных функций на симплициальном комплексе X, которая является компонентой степени 0 введенной Сулливаном dg-алгебры А'(Х) полиномиальных форм. Все рассматриваемые алгебры над произвольным полем к 0
Нашей целью является вычисление когомологий де Рама алгебры А°(Х), то есть ко-гомологий универсальной dg-алгебры Q'A0(X). Имеется канонический морфизм dg-алгебр Р : П^о (Х) ^ А'(Х). Мы доказываем, что морфизм Р является квазиизоморфизмом. Таким образом, когомологии де Рама алгебры А°(Х) канонически изоморфны когомологиям симлициального комплекса X с коэффициентами в поле к. Более того, для к = Q, dg-алгебра О.'А0(Х) служит моделью симплициального комплекса X в смысле рациональной теории гомотопий. Наш результат показывает, что для алгебры А°(Х) верно утверждение теоремы сравнения Гротендика (доказанной им для гладких алгебр).
Для доказательства мы рассматриваем резольвенты Чеха, ассоциированные с покрытием симплициального комплекса звездами вершин.
Ранее Кан — Миллер доказали, что морфизм Р сюръективен, а также описали его ядро. Другое описание ядра дали Сулливан и Феликс — Джессап — Паран.
Ключевые слова: когомологии де Рама алгебры, универсальная dg-алгебра, алгебра полиномиальных функций, dg-алгебра полиномиальных форм, рациональная теория гомотопий.
Библиография: 15 названий. Для цитирования:
И. С. Басков. Когомологии де Рама алгебры полиномиальных функций на симплициальном комплексе // Чебышевский сборник, 2023, т. 24, вып. 1, с. 203-212.
1Данная работа была поддержана Санкт-Петербургским международным математическим Институтом имени Леонарда Эйлера, грантовое соглашение NN 075-15-2019-1620 от 08.11.2019 и 075-15-2022-289 от 06.04.2022.
CHEBYSHEVSKII SBORNIK Vol. 24. No. 1.
UDC 512
DOI 10.22405/2226-8383-2023-24-1-203-212
The de Rham cohomology of the algebra of polynomial functions
on a simplicial complex
I. S. Baskov
Baskov Igor Sergeevich — Steklov Mathematical Institute of Russian Academy of Sciences (St. Petersburg).
e-mail: baskovigorQpdmi. ras. ru
We consider the algebra A0(X) of polynomial functions on a simplicial complex X. The algebra A0(X) is the Oth component of Sullivan's dg-algebra A'(X) of polynomial forms on X. All algebras are over an arbitrary field k of characteristic O.
Our main interest lies in computing the de Rham cohomology of the algebra A0(X ), that is, the cohomology of the universal dg-algebra Q*A0(X)- There is a canonical morphism of dg-algebras P : ^ A'(X). We prove that P is a quasi-isomorphism. Therefore, the de
Rham cohomology of the algebra A0 (X) is canonically isomorphic to the cohomology of the simplicial complex X with coefficients in k. Moreover, for k = Q the dg-algebra Q^x) is a model of the simplicial complex X in the sense of rational homotopy theory. Our result shows that for the algebra A0(X) the statement of Grothendieck's comparison theorem holds (proved by him for smooth algebras).
In order to prove the statement we consider Cech resolution associated to the cover of the simplicial complex by the stars of the vertices.
Earlier, Kan-Miller proved that the morphism P is surjective and gave a description of its kernel. Another description of the kernel was given by Sullivan and Félix-Jessup-Parent.
Keywords: algebraic de Rham cohomology, universal dg-algebra, algebra of polynomial functions, dg-algebra of polynomial forms, rational homotopy theory.
Bibliography: 15 titles. For citation:
I. S. Baskov, 2023"The de Rham cohomology of the algebra of polynomial functions on a simplicial complex" , Chebyshevskii sbornik, vol. 24, no. 1, pp. 203-212.
1. Introduction
All algebras and dg-algebras are commutative over a field k of characteristic 0. In [15, Section 7] Sullivan introduces the dg-algebra A^(X) of polynomial forms on a simplicial complex X. The algebra A°(X) of the degree zero elements of A^(X) is the algebra of polynomial funotions on X. The cohomology of the dg-algebra A^(X) is isomorphic to H•(X,k). One can ask what natural dg-algebras are weakly equivalent to A^(X). One such candidate is the universal dg-algebra Q*A0(X) on the algebra A°(X) of polynomial functions on X. There is a canonical morphism of dg-algebras
The main result is Theorem 1, where we prove that P is a quasi-isomorphism. In [12] the authors prove that the morphism P is surjective and give a description of its kernel. In [7] and [14, Appendix G(i)] another description of the kernel is given. In [8, Example 3.8], Gomez
Abstract
establishes that the morphism P is not a quasi-isomorphism, which contradicts our main result, Theorem 1. We were able to correct the erroneous computation of Gomez in Remark 4.
Grothendieck proved that for a smooth C-algebra A the cohomologv groups of the algebraic de Rham complex are isomorphic to the cohomologv groups of the space Spec A with complex analytic topology, sgg [10, Theorem 1]. The algebra A0(X) is not smooth in general and the result of Grothendieck does not hold for general algebras, see fl, Example 4.4].
The result of this paper can be used in order to give another proof of the similar result for the algebra of piecewise polynomial functions on a polyhedron, which is known due to [2, Theorem 51].
Acknowledgement
I would like to thank Dr. Semen Podkorvtov for his immense help, fruitful discussions and the idea to consider Cech resolution in order to prove the result. I am grateful to the St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences for their financial assistance.
2. Simplicial complexes
Definition 1. We call a set X of finite non-empty subsets of a finite set E a simplicial complex if for every v £ E we ha ve {«} £ X and for every s £ X and every non-empty subset s' C s we have s' £ X. We denote by V(X) the set E and call its elements vertices of X. The sets s £ X of cardinality m + 1 are called m-simplices. A simplicial complex Y is a subcomplex of X if for every s £ Y we have s £ X.
We denote by TP(X) the set of all sequences u = (u0,..., up) of vertices of X for p > — 1. We denote by diU the sequence (u0,... ,Ui,..., up). For a vertex v we denote by v * u the sequence (v, u0,..., Up). The symmetric group Xp+1 acts on TP(X).
Consider a sequence of vertices u £ TP(X). We denote by St« the star of u, that is the smallest subcomplex of X containing all the simplices containing the vertices Ui. If p = — 1, we have St« = X. If the sequence « spans a simplex in X, then St« is the star of this simplex. Hp > 0 and the sequence u does not span a simplex, we have St« = 0. For a subcomplex Y of X we denote by Styu the smallest subcomplex of y containing all the simplices containing the vertices «¿.If u £ TP(Y) then Sty u = 0.
3. Sullivan's dg-algebra of polynomial forms
For a simplicial complex X we define the dg-algebra A^(X) following Sullivan, see [15, Section 7], [5], [11], [9]. For an m-simplex a consider the dg-algebra
:= A (tv, dtv | deg(tv) = 0, deg(dtv) = 1, v £ a) {J2vea t"" — 1, vEa dtv)
with the differential tv ^ dtv for v £ a.
For a simplex & such that b C a one has a natural morphism of dg-algebras
1ь : А» ^ A*(b), tv ^
0,v/b,
tv, V £ b.
Then an element w = (wa)aEx of A^(X) is a collection of elements wa £ A^(a) such that for two simplices b C a one has ua = ^b-
We call the algebra A0 (X) the algebra of polynomial functions on X. This algebra has another description as a quotient of a Stanlev-Reisner algebra, see [3].
An inclusion of simplicial complexes Y C X gives rise to the restriction morphism of dg-algebras
|y : A\X) ^ A\Y).
Lemma 1. The above restriction morphism is surjective.
proof. This fact is quite nontrivial, see [15, Section 7]. □
We introduce the double graded vector space Vp'q, p,q e Z as follows. For p < — 2 set Vp'q = 0 and for p > — 1, we define Vp'q as the subspace of
^ Aq (Stu)
u£Tp(X)
consisting of families of forms e Aq(Stu), such that for any a e Sp+i we have
Uau = (sgn a)uu.
We define the linear map
S : Vp'q ^ Vp+1'q.
For p > — 1 and for
w = (uu)ueTp[x) e Vp,q we set the value of ¿^ns e TP+1(X) as
p+i
(Su)s = 1)'wSiS|sts.
i=0
The differential ^n A^(Sta) gives rise to a differential ^n Dp'• for each p.
Proposition 1. The map 5 is a differential on T>^,q for each q e Z. Moreover, the double graded vector space T)%,% together with 5 and d forms a double complex in the sense that d5 = 5 d.
Proposition 2. The. complex is exact.
Proof. The proof is similar to that of Proposition 4 below and relies on Lemma 1. In this case one can use the partition of unity tv, v e V(X), instead of pv,v e V(X). □
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207
4. The dg-algebra of de Rham forms
To a fc-algebra A one associates the commutative dg-algebra QA ([13, Theorem 3.2]) with QA = A ft has the following universal property: for any dg-algebra E* and any algebra homomorphism f : A ^ E° there exists a unique morphism of dg-algebras F : QA ^ such that F|a = /:
The elements of QA are called algebraic g-forms. The dg-algebra QA '1S covariant in the algebra A We will simply write Q^(X) for the dg-algebra QA0(x)-
Inclusion of simplicial complexes Y C X gives rise to the restriction morphism of dg-algebras
|y : Q^(X) ^ Q%Y).
Lemma 2. Suppose A and B are k-algebras and ¡p : A ^ B is a surjective homomorphism of algebras. Then the induced morphism Qv : QA ^ Q% is surjective and its kernel is the ideal of QA generated by Ker ip and d(Ker (p).
From this and Lemma 1 it follows that for an inclusion of simplicial complexes Y C X the restriction morphism |y : Q^(X) ^ Q^(Y) is surjective. Proof. See [2, Lemma 6]. □
Let us introduce the following elements tv of A°(X). For a vertex v £ V(X) and an m-simplex se^t (tv)a = 0 if v £ a and (tv)a = tv £ A (a) if v £ a.
Lemma 3. Take a simplicial complex X and a subcomplex Y C X. Suppose w £ Qq(Y) is such that w|sty(v) = 0 for a vertex v £ V(X). Then t2|y w = 0.
Proof. First, if v £ V(Y), then tv|y = 0 and the claim follows.
Assume v £ V(Y). By Lemma 1 and Lemma 2, the form w lies in the dg-ideal I of Q^(X) generated by the elements m £ A°(X) with the restriction to Sty(v) being zero, therefore tv|y m = 0. ft is enough to consider the cases w = m and w = dm. We have i2|y m = 0 and i2|y dm = tv|y d(tv|y m) — tv|y mdtv|y = 0 □
We introduce the double graded vector space Cp'q, p,q £ Z as follows. For p < —2 set Cp'q = 0 and for p > — 1, we define Cp'q as the subspace of
n Qq(St«)
u£Tp(X)
consisting of families of forms £ Qq(St«) such that for any a £ Sp+i we have
= (sgn a)wu.
We define the linear map
S : Cp'q ^ Cp+1'q.
For p > — 1 and for
U = (^u)uETp(X) £ Cp'q
we set the value of 5u on s £ Tp+1(X) as
p+i
(5w)s = ^2(—1)гwдiS|sts. i=0
The differential d on Q^(St-u) gives rise to a differential d on Cp'• for each p.
Proposition 3. The map 5 is a differential on C^'q for each q e Z. Moreover, the double graded vector space C •'•together w ith 5 and d forms a double complex in the sense that d5 = 5 d.
Lemma 4. There exist elements pv e A°(X), v e V(X), such that
£ Pv = 1
veV (X)
PROOF. In A0(X) we have the equality
£ ^ = 1.
vev (X)
□
We put pv = pv t2u.
For an inclusion of simplicial complexes Y C X we choose a linear map, the distinguished "extension",
[—] : Qq(Y) ^ Qq(X),
such that [w]|y = w. Such an extension exists bv Lemma 1 applied to A°(X) and Lemma 2. Lemma 5. For an inclusion of simplicial complexes Y C X and a form w e Qq(Y) we have
£ Pv Msty (^)]|V = U. vev (x)
Proof. As ^ pv = 1 in A0(X) we have
£ Pv Msty (^)]|V — W = Pv Iy (Msty («)]|V — u).
vev (X) vev(x)
We have
(Msty(^)] |y — v) |sty(v) = W|sty(v) — W|sty(v) = Hence, by Lemma 3 we have pv|y ([w|sty(^)]|y — w) = 0. □
Proposition 4. The. complex is exact.
The proof follows the proof of [4, Proposition 8.5]. Proof. First, notice that
Stv * u = Ststu(v). For u e TP(X) and w e Qq(St«) by Lemma 5 we have
£ Pv Msto*«]|st« = U. (1)
vev (X)
For w e Qq(Z), where Z is a subcomplex of X such that Stv * u C Z, bv Lemma 3, we have
pv ([w] — Msw]) 1st« = °. (2)
We construct a cochain homotopy
K : Cp'q ^Cp-1'q. For p > ^d w e and w e TP-1(X) put
(Kw)w := ^ pv [Wv*w]|stw. vev (x)
By Lemma 3 this map does not depend on the choice of the distinguished extension. Let us check that 5K + Kb = 1. For p > — 1 and
u = KWW eCp'q C n &(St«),
ueTp(x)
where e Qq(St«), we have
p p
(5Kw)u = 1)%(Kw)diV\stu = £(—1)' £ Pv [w^JIst«, i=o i=o vev(x)
and
(К5ш)и = ^ pv [(M^*«]|St« =
veV (X) P+l
vev(x) i=o
P+l ^
vev(x) vev(x) i=i
p
= Ши — £ ^ [ £( — |St^*«]|St«
vev(x) i=o
p
i=0 vev(x)
Hence,
p ^
((5К + К5)ш)и = + £(-1)г £ P^ - [w^^Sto*«]) |St« "= uu.
i=o vev (X)
□
5. The morphism P : Q'(X) ^ A*(X)
For a simplicial complex X, by the universal propertv of Q^(X), there is a canonical morphism dg-algebras
P : Q?(X) ^ A^(X),
which is the identity in degree 0.
We denote by fc[0] the complex with the № term k and the others zero. An element of k gives rise to a constant function in A°(X), hence, there are morphisms of complexes e : fc[0] ^ Q^ (X) and e : k[0] ^ A^(X) such that e = P o e.
Proposition 5. For a sequence u <E TP(X) p > 0, the commutative diagram
fc[0] —^ Q*(St u)
A*(Stu)
consists of quasi-isomorphisms.
PROOF. The map e is a quasi-isomorphism bv [2, Corollary 47]. The mар e is a quasi-isomorphism bv [6, Theorem 10.9]. Hence, the morphism P is a quasi-isomorphism. □
Theorem 1. The. natural map
P : tt^(X) ^ A^(X)
is a quasi-isomorphism.
PROOF. The morphism P от stars gives rise to the maps кр : Cp^ for each p > 0. We have the following commutative diagram of non-negative complexes
0-* fi"(X)
P
0-*
The vertical arrows np, p > 0 are quasi-isomorphisms by Proposition 5. The first row is exact by Proposition 4. The second row is exact by Proposition 2. Hence, the map P is also a quasi-isomorphism. □
Замечание 4. A,? was said in the. introduction, the. paper [8] suggests that Theorem 1 is false. Namely, in [8, Example 3.8], one considers the simplicial complex X corresponding to the boundary of a triangle on the vertices 1, 2, 3. The dg-algebra Q^(X) is generated by the elements t\, t2, t3 modulo the dg-ideal generated by tl + t2 + t3 — 1 and tlt2t3. Next, the author considers the form ¿2t2dt3 and claims that this form is not zero. However, this form is zero, which can be seen as follows: applying the differential d to the equality tlt2t3 = 0 we get
t2t3dt\ + tit3dt2 + tit2dt3 = 0.
Next, we multiply this equality by tlt2 and get
0 = tit2t3dti + tjt2t3dt2 + t\t2dt3 = t21t22dt3.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Arapura, Donu and Kang, Su-Jeong, Kâhler-de Rham cohomology and Chern classes // Communications in Algebra 2011. Vol. 39, № 4. P. 1153-1167.
2. Baskov, I., The de Rham cohomology of soft function algebras // preprint 2022, https:// arxiv.org/abs/2208.11431.
3. Billera, Louis J., The algebra of continuous piecewise polynomials // Advances in Mathematics 1989. Vol. 76, № 2. P. 170-183.
4. Bott, Raoul and Tu, Loring W., Differential forms in algebraic topology // Springer 1982.
5. Bousfield, Aldridge Knight and Gugenheim, Victor K.A.M., On PL De Rham theory and rational homotopy type // Memoirs American Mathematical Soc. 1976. Vol. 179.
6. Félix, Yves and Halperin, Stephen and Thomas, Jean-Claude, Rational homotopy theory // Springer 2012.
7. Félix, Yves and Jessup, Barry and Parent, Paul-Eugène, The combinatorial model for the Sullivan functor on simplicial sets // Journal of Pure and Applied Algebra 2009. Vol. 213, № 2. P. 231-240.
8. Gomez, Francisco, Simplicial types and polynomial algebras // Archivum Mathematicum 2002. Vol. 38, № 1. P. 27-36.
9. Griffiths, Phillip and Morgan, John, Rational Homotopy Theory and Differential Forms // Birkhâuser 1981.
10. Grothendieck, Alexander, On the de Rham cohomology of algebraic varieties // Publications Mathématiques de l'IHÉS, 1966. Vol. 29, № 1. Р. 95-103.
11. Hess, Kathrvn, Rational Homotopy Theory //Interactions between Homotopy Theory and Algebra: Summer School on Interactions Between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago, Illinois. - 2007. - P. 175-202.
12. Kan, Daniel M. and Miller, Edward Y., Sullivan's de Rham complex is definable in terms of its 0-forms // Proceedings of the American Mathematical Society 1976. Vol. 57, № 2. P. 337-339.
13. Kunz, Ernst, Kâhler differentials // 1986, Friedr. Vieweg k, Sohn.
14. Sullivan, Dennis, Differential forms and the topology of manifolds // Manifolds Tokyo 1973. P. 37-49.
15. Sullivan, Dennis, Infinitesimal computations in topology // Publications Mathématiques de l'IHÉS 1977. Vol. 47, Р. 269-331.
REFERENCES
1. Arapura, Donu and Kang, Su-Jeong, 2011, "Kâhler-de Rham cohomology and Chern classes" // Communications in Algebra. Vol. 39, № 4. pp. 1153-1167.
2. Baskov, I., 2022, "The de Rham cohomology of soft function algebras" // preprint, https:// arxiv.org/abs/2208.11431.
3. Billera, Louis J., 1989, "The algebra of continuous piecewise polynomials" // Advances in Mathematics. Vol. 76, № 2. pp. 170-183.
4. Bott, Raoul and Tu, Loring W., 1982, "Differential forms in algebraic topology" // Springer.
5. Bousfield, Aldridge Knight and Gugenheim, Victor K.A.M., 1976, "On PL De Rham theory and rational homotopy type" // Memoirs American Mathematical Soc.. Vol. 179.
6. Félix, Yves and Halperin, Stephen and Thomas, Jean-Claude, 2012, "Rational homotopy theory" // Springer.
7. Félix, Yves and Jessup, Barry and Parent, Paul-Eugène, 2009, "The combinatorial model for the Sullivan functor on simplicial sets" // Journal of Pure and Applied Algebra. Vol. 213, № 2. pp. 231-240.
8. Gomez, Francisco, 2002, "Simplicial types and polynomial algebras" // Archivum Mathemati-cum. Vol. 38, № 1. pp. 27-36.
9. Griffiths, Phillip and Morgan, John, 1981, "Rational Homotopy Theory and Differential Forms" // Birkhduser.
10. Grothendieck, Alexander, 1966, "On the de Rham cohomology of algebraic varieties" // Publications Mathématiques de l'IHES. Vol. 29, № 1. pp. 95-103.
11. Hess, Kathrvn, 2007, "Rational Homotopy Theory" // Interactions between Homotopy Theory and Algebra: Summer School on Interactions Between HomMopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago, Illinois. - pp. 175-202.
12. Kan, Daniel M. and Miller, Edward Y., 1976, "Sullivan's de Rham complex is definable in terms of its 0-forms" // Proceedings of the American Mathematical Society. Vol. 57, № 2. pp. 337-339.
13. Kunz, Ernst, 1986, "Kâhler differentials" // Friedr. Vieweg k. Sohn.
14. Sullivan, Dennis, 1973, "Differential forms and the topology of manifolds" // Manifolds Tokyo. pp. 37-49.
15. Sullivan, Dennis, 1977, "Infinitesimal computations in topology" // Publications Mathématiques de l'IHÉS. Vol. 47, pp. 269-331.
Получено: 20.01.2023 Принято в печать: 24.04.2023
